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The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organizations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organize and summarize the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalize your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarize your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, other interesting articles.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

• Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
• Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
• Null hypothesis: Parental income and GPA have no relationship with each other in college students.
• Alternative hypothesis: Parental income and GPA are positively correlated in college students.

Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

• In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
• In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
• In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

• In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
• In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
• In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
• Experimental
• Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

Measuring variables

When planning a research design, you should operationalize your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

• Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
• Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

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In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

Sampling for statistical analysis

There are two main approaches to selecting a sample.

• Probability sampling: every member of the population has a chance of being selected for the study through random selection.
• Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalizable findings, you should use a probability sampling method. Random selection reduces several types of research bias , like sampling bias , and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to at risk for biases like self-selection bias , they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

• your sample is representative of the population you’re generalizing your findings to.
• your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalize your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialized, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalized in your discussion section .

Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

• Will you have resources to advertise your study widely, including outside of your university setting?
• Will you have the means to recruit a diverse sample that represents a broad population?
• Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

• Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
• Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
• Expected effect size : a standardized indication of how large the expected result of your study will be, usually based on other similar studies.
• Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarize them.

Inspect your data

There are various ways to inspect your data, including the following:

• Organizing data from each variable in frequency distribution tables .
• Displaying data from a key variable in a bar chart to view the distribution of responses.
• Visualizing the relationship between two variables using a scatter plot .

By visualizing your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

• Mode : the most popular response or value in the data set.
• Median : the value in the exact middle of the data set when ordered from low to high.
• Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

• Range : the highest value minus the lowest value of the data set.
• Interquartile range : the range of the middle half of the data set.
• Standard deviation : the average distance between each value in your data set and the mean.
• Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

• Estimation: calculating population parameters based on sample statistics.
• Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

• A point estimate : a value that represents your best guess of the exact parameter.
• An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

• A test statistic tells you how much your data differs from the null hypothesis of the test.
• A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

• Comparison tests assess group differences in outcomes.
• Regression tests assess cause-and-effect relationships between variables.
• Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

• A simple linear regression includes one predictor variable and one outcome variable.
• A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

• A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
• A z test is for exactly 1 or 2 groups when the sample is large.
• An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

• If you have only one sample that you want to compare to a population mean, use a one-sample test .
• If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
• If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
• If you expect a difference between groups in a specific direction, use a one-tailed test .
• If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

• a t value (test statistic) of 3.00
• a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

• a t value of 3.08
• a p value of 0.001

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The final step of statistical analysis is interpreting your results.

Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimize the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasizes null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval

Methodology

• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Likert scale

Research bias

• Implicit bias
• Framing effect
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hostile attribution bias
• Affect heuristic

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Introduction to Statistics

(15 reviews)

David Lane, Rice University

Copyright Year: 2003

Publisher: David Lane

Language: English

Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Terri Torres, professor, Oregon Institute of Technology on 8/17/23

This author covers all the topics that would be covered in an introductory statistics course plus some. I could imagine using it for two courses at my university, which is on the quarter system. I would rather have the problem of too many topics... read more

Comprehensiveness rating: 5 see less

This author covers all the topics that would be covered in an introductory statistics course plus some. I could imagine using it for two courses at my university, which is on the quarter system. I would rather have the problem of too many topics rather than too few.

Content Accuracy rating: 5

Yes, Lane is both thorough and accurate.

Relevance/Longevity rating: 5

What is covered is what is usually covered in an introductory statistics book. The only topic I may, given sufficient time, cover is bootstrapping.

Clarity rating: 5

The book is clear and well-written. For the trickier topics, simulations are included to help with understanding.

Consistency rating: 5

All is organized in a way that is consistent with the previous topic.

Modularity rating: 5

The text is organized in a way that easily enables navigation.

Organization/Structure/Flow rating: 5

The text is organized like most statistics texts.

Interface rating: 5

Easy navigation.

Grammatical Errors rating: 5

I didn't see any grammatical errors.

Cultural Relevance rating: 5

Nothing is included that is culturally insensitive.

The videos that accompany this text are short and easy to watch and understand. Videos should be short enough to teach, but not so long that they are tiresome. This text includes almost everything: videos, simulations, case studies---all nicely organized in one spot. In addition, Lane has promised to send an instructor's manual and slide deck.

Reviewed by Professor Sandberg, Professor, Framingham State University on 6/29/21

This text covers all the usual topics in an Introduction to Statistics for college students. In addition, it has some additional topics that are useful. read more

This text covers all the usual topics in an Introduction to Statistics for college students. In addition, it has some additional topics that are useful.

I did not find any errors.

Some of the examples are dated. And the frequent use of male/female examples need updating in terms of current gender splits.

I found it was easy to read and understand and I expect that students would also find the writing clear and the explanations accessible.

Even with different authors of chapter, the writing is consistent.

The text is well organized into sections making it easy to assign individual topics and sections.

The topics are presented in the usual order. Regression comes later in the text but there is a difference of opinions about whether to present it early with descriptive statistics for bivariate data or later with inferential statistics.

I had no problem navigating the text online.

The writing is grammatical correct.

I saw no issues that would be offensive.

I did like this text. It seems like it would be a good choice for most introductory statistics courses. I liked that the Monty Hall problem was included in the probability section. The author offers to provide an instructor's manual, PowerPoint slides and additional questions. These additional resources are very helpful and not always available with online OER texts.

Reviewed by Emilio Vazquez, Associate Professor, Trine University on 4/23/21

This appears to be an excellent textbook for an Introductory Course in Statistics. It covers subjects in enough depth to fulfill the needs of a beginner in Statistics work yet is not so complex as to be overwhelming. read more

This appears to be an excellent textbook for an Introductory Course in Statistics. It covers subjects in enough depth to fulfill the needs of a beginner in Statistics work yet is not so complex as to be overwhelming.

I found no errors in their discussions. Did not work out all of the questions and answers but my sampling did not reveal any errors.

Some of the examples may need updating depending on the times but the examples are still relevant at this time.

This is a Statistics text so a little dry. I found that the derivation of some of the formulas was not explained. However the background is there to allow the instructor to derive these in class if desired.

The text is consistent throughout using the same verbiage in various sections.

The text dose lend itself to reasonable reading assignments. For example the chapter (Chapter 3) on Summarizing Distributions covers Central Tendency and its associated components in an easy 20 pages with Measures of Variability making up most of the rest of the chapter and covering approximately another 20 pages. Exercises are available at the end of each chapter making it easy for the instructor to assign reading and exercises to be discussed in class.

The textbook flows easily from Descriptive to Inferential Statistics with chapters on Sampling and Estimation preceding chapters on hypothesis testing

I had no problems with navigation

All textbooks have a few errors but certainly nothing glaring or making text difficult

I saw no issues and I am part of a cultural minority in the US

Overall I found this to be a excellent in-depth overview of Statistical Theory, Concepts and Analysis. The length of the textbook appears to be more than adequate for a one-semester course in Introduction to Statistics. As I no longer teach a full statistics course but simply a few lectures as part of our Research Curriculum, I am recommending this book to my students as a good reference. Especially as it is available on-line and in Open Access.

Reviewed by Audrey Hickert, Assistant Professor, Southern Illinois University Carbondale on 3/29/21

All of the major topics of an introductory level statistics course for social science are covered. Background areas include levels of measurement and research design basics. Descriptive statistics include all major measures of central tendency and... read more

All of the major topics of an introductory level statistics course for social science are covered. Background areas include levels of measurement and research design basics. Descriptive statistics include all major measures of central tendency and dispersion/variation. Building blocks for inferential statistics include sampling distributions, the standard normal curve (z scores), and hypothesis testing sections. Inferential statistics include how to calculate confidence intervals, as well as conduct tests of one-sample tests of the population mean (Z- and t-tests), two-sample tests of the difference in population means (Z- and t-tests), chi square test of independence, correlation, and regression. Doesn’t include full probability distribution tables (e.g., t or Z), but those can be easily found online in many places.

I did not find any errors or issues of inaccuracy. When a particular method or practice is debated in the field, the authors acknowledge it (and provide citations in some circumstances).

Relevance/Longevity rating: 4

Basic statistics are standard, so the core information will remain relevant in perpetuity. Some of the examples are dated (e.g., salaries from 1999), but not problematic.

Clarity rating: 4

All of the key terms, formulas, and logic for statistical tests are clearly explained. The book sometimes uses different notation than other entry-level books. For example, the variance formula uses "M" for mean, rather than x-bar.

The explanations are consistent and build from and relate to corresponding sections that are listed in each unit.

Modularity is a strength of this text in both the PDF and interactive online format. Students can easily navigate to the necessary sections and each starts with a “Prerequisites” list of other sections in the book for those who need the additional background material. Instructors could easily compile concise sub-sections of the book for readings.

The presentation of topics differs somewhat from the standard introductory social science statistics textbooks I have used before. However, the modularity allows the instructor and student to work through the discrete sections in the desired order.

Interface rating: 4

For the most part the display of all images/charts is good and navigation is straightforward. One concern is that the organization of the Table of Contents does not exactly match the organizational outline at the start of each chapter in the PDF version. For example, sometimes there are more detailed sub-headings at the start of chapter and occasionally slightly different section headings/titles. There are also inconsistencies in section listings at start of chapters vs. start of sub-sections.

The text is easy to read and free from any obvious grammatical errors.

Although some of the examples are outdated, I did not review any that were offensive. One example of an outdated reference is using descriptive data on “Men per 100 Women” in U.S. cities as “useful if we are looking for an opposite-sex partner”.

This is a good introduction level statistics text book if you have a course with students who may be intimated by longer texts with more detailed information. Just the core basics are provided here and it is easy to select the sections you need. It is a good text if you plan to supplement with an array of your own materials (lectures, practice, etc.) that are specifically tailored to your discipline (e.g., criminal justice and criminology). Be advised that some formulas use different notation than other standard texts, so you will need to point that out to students if they differ from your lectures or assessment materials.

Reviewed by Shahar Boneh, Professor, Metropolitan State University of Denver on 3/26/21, updated 4/22/21

The textbook is indeed quite comprehensive. It can accommodate any style of introductory statistics course. read more

The textbook is indeed quite comprehensive. It can accommodate any style of introductory statistics course.

The text seems to be statistically accurate.

It is a little too extensive, which requires instructors to cover it selectively, and has a potential to confuse the students.

It is written clearly.

Consistency rating: 4

The terminology is fairly consistent. There is room for some improvement.

By the nature of the subject, the topics have to be presented in a sequential and coherent order. However, the book breaks things down quite effectively.

Organization/Structure/Flow rating: 3

Some of the topics are interleaved and not presented in the order I would like to cover them.

Good interface.

The grammar is ok.

The book seems to be culturally neutral, and not offensive in any way.

I really liked the simulations that go with the book. Parts of the book are a little too advanced for students who are learning statistics for the first time.

Reviewed by Julie Gray, Adjunct Assistant Professor, University of Texas at Arlington on 2/26/21

The textbook is for beginner-level students. The concept development is appropriate--there is always room to grow to high higher level, but for an introduction, the basics are what is needed. This is a well-thought-through OER textbook project by... read more

The textbook is for beginner-level students. The concept development is appropriate--there is always room to grow to high higher level, but for an introduction, the basics are what is needed. This is a well-thought-through OER textbook project by Dr. Lane and colleagues. It is obvious that several iterations have only made it better.

I found all the material accurate.

Essentially, statistical concepts at the introductory level are accepted as universal. This suggests that the relevance of this textbook will continue for a long time.

The book is well written for introducing beginners to statistical concepts. The figures, tables, and animated examples reinforce the clarity of the written text.

Yes, the information is consistent; when it is introduced in early chapters it ties in well in later chapters that build on and add more understanding for the topic.

Modularity rating: 4

The book is well-written with attention to modularity where possible. Due to the nature of statistics, that is not always possible. The content is presented in the order that I usually teach these concepts.

The organization of the book is good, I particularly like the sample lecture slide presentations and the problem set with solutions for use in quizzes and exams. These are available by writing to the author. It is wonderful to have access to these helpful resources for instructors to use in preparation.

I did not find any interface issues.

The book is well written. In my reading I did not notice grammatical errors.

For this subject and in the examples given, I did not notice any cultural issues.

For the field of social work where qualitative data is as common as quantitative, the importance of giving students the rationale or the motivation to learn the quantitative side is understated. To use this text as an introductory statistics OER textbook in a social work curriculum, the instructor will want to bring in field-relevant examples to engage and motivate students. The field needs data-driven decision making and evidence-based practices to become more ubiquitous than not. Preparing future social workers by teaching introductory statistics is essential to meet that goal.

Reviewed by Mamata Marme, Assistant Professor, Augustana College on 6/25/19

This textbook offers a fairly comprehensive summary of what should be discussed in an introductory course in Statistics. The statistical literacy exercises are particularly interesting. It would be helpful to have the statistical tables... read more

Comprehensiveness rating: 4 see less

This textbook offers a fairly comprehensive summary of what should be discussed in an introductory course in Statistics. The statistical literacy exercises are particularly interesting. It would be helpful to have the statistical tables attached in the same package, even though they are available online.

The terminology and notation used in the textbook is pretty standard. The content is accurate.

The statistical literacy example are up to date but will need to be updated fairly regularly to keep the textbook fresh. The applications within the chapter are accessible and can be used fairly easily over a couple of editions.

The textbook does not necessarily explain the derivation of some of the formulae and this will need to be augmented by the instructor in class discussion. What is beneficial is that there are multiple ways that a topic is discussed using graphs, calculations and explanations of the results. Statistics textbooks have to cover a wide variety of topics with a fair amount of depth. To do this concisely is difficult. There is a fine line between being concise and clear, which this textbook does well, and being somewhat dry. It may be up to the instructor to bring case studies into the readings we are going through the topics rather than wait until the end of the chapter.

The textbook uses standard notation and terminology. The heading section of each chapter is closely tied to topics that are covered. The end of chapter problems and the statistical literacy applications are closely tied to the material covered.

The authors have done a good job treating each chapter as if they stand alone. The lack of connection to a past reference may create a sense of disconnect between the topics discussed

The text's "modularity" does make the flow of the material a little disconnected. If would be better if there was accountability of what a student should already have learnt in a different section. The earlier material is easy to find but not consistently referred to in the text.

I had no problem with the interface. The online version is more visually interesting than the pdf version.

I did not see any grammatical errors.

Cultural Relevance rating: 4

I am not sure how to evaluate this. The examples are mostly based on the American experience and the data alluded to mostly domestic. However, I am not sure if that creates a problem in understanding the methodology.

Overall, this textbook will cover most of the topics in a survey of statistics course.

Reviewed by Alexandra Verkhovtseva, Professor, Anoka-Ramsey Community College on 6/3/19

This is a comprehensive enough text, considering that it is not easy to create a comprehensive statistics textbook. It is suitable for an introductory statistics course for non-math majors. It contains twenty-one chapters, covering the wide range... read more

This is a comprehensive enough text, considering that it is not easy to create a comprehensive statistics textbook. It is suitable for an introductory statistics course for non-math majors. It contains twenty-one chapters, covering the wide range of intro stats topics (and some more), plus the case studies and the glossary.

The content is pretty accurate, I did not find any biases or errors.

The book contains fairly recent data presented in the form of exercises, examples and applications. The topics are up-to-date, and appropriate technology is used for examples, applications, and case studies.

The language is simple and clear, which is a good thing, since students are usually scared of this class, and instructors are looking for something to put them at ease. I would, however, try to make it a little more interesting, exciting, or may be even funny.

Consistency is good, the book has a great structure. I like how each chapter has prerequisites and learner outcomes, this gives students a good idea of what to expect. Material in this book is covered in good detail.

The text can be easily divided into sub-sections, some of which can be omitted if needed. The chapter on regression is covered towards the end (chapter 14), but part of it can be covered sooner in the course.

The book contains well organized chapters that makes reading through easy and understandable. The order of chapters and sections is clear and logical.

The online version has many functions and is easy to navigate. This book also comes with a PDF version. There is no distortion of images or charts. The text is clean and clear, the examples provided contain appropriate format of data presentation.

No grammatical errors found.

The text uses simple and clear language, which is helpful for non-native speakers. I would include more culturally-relevant examples and case studies. Overall, good text.

In all, this book is a good learning experience. It contains tools and techniques that free and easy to use and also easy to modify for both, students and instructors. I very much appreciate this opportunity to use this textbook at no cost for our students.

Reviewed by Dabrina Dutcher, Assistant Professor, Bucknell University on 3/4/19

This is a reasonably thorough first-semester statistics book for most classes. It would have worked well for the general statistics courses I have taught in the past but is not as suitable for specialized introductory statistics courses for... read more

This is a reasonably thorough first-semester statistics book for most classes. It would have worked well for the general statistics courses I have taught in the past but is not as suitable for specialized introductory statistics courses for engineers or business applications. That is OK, they have separate texts for that! The only sections that feel somewhat light in terms of content are the confidence intervals and ANOVA sections. Given that these topics are often sort of crammed in at the end of many introductory classes, that might not be problematic for many instructors. It should also be pointed out that while there are a couple of chapters on probability, this book spends presents most formulas as "black boxes" rather than worry about the derivation or origin of the formulas. The probability sections do not include any significant combinatorics work, which is sometimes included at this level.

I did not find any errors in the formulas presented but I did not work many end-of-chapter problems to gauge the accuracy of their answers.

There isn't much changing in the introductory stats world, so I have no concerns about the book becoming outdated rapidly. The examples and problems still feel relevant and reasonably modern. My only concern is that the statistical tool most often referenced in the book are TI-83/84 type calculators. As students increasingly buy TI-89s or Inspires, these sections of the book may lose relevance faster than other parts.

Solid. The book gives a list of key terms and their definitions at the end of each chapter which is a nice feature. It also has a formula review at the end of each chapter. I can imagine that these are heavily used by students when studying! Formulas are easy to find and read and are well defined. There are a few areas that I might have found frustrating as a student. For example, the explanation for the difference in formulas for a population vs sample standard deviation is quite weak. Again, this is a book that focuses on sort of a "black-box" approach but you may have to supplement such sections for some students.

I did not detect any problems with inconsistent symbol use or switches in terminology.

Modularity rating: 3

This low rating should not be taken as an indicator of an issue with this book but would be true of virtually any statistics book. Different books still use different variable symbols even for basic calculated statistics. So trying to use a chapter of this book without some sort of symbol/variable cheat-sheet would likely be frustrating to the students.

However, I think it would be possible to skip some chapters or use the chapters in a different order without any loss of functionality.

This book uses a very standard order for the material. The chapter on regressions comes later than it does in some texts but it doesn't really matter since that chapter never seems to fit smoothly anywhere.

There are numerous end of chapter problems, some with answers, available in this book. I'm vacillating on whether these problems would be more useful if they were distributed after each relevant section or are better clumped at the end of the whole chapter. That might be a matter of individual preference.

I did not detect any problems.

I found no errors. However, there were several sections where the punctuation seemed non-ideal. This did not affect the over-all useability of the book though

I'm not sure how well this book would work internationally as many of the examples contain domestic (American) references. However, I did not see anything offensive or biased in the book.

Reviewed by Ilgin Sager, Assistant Professor, University of Missouri - St. Louis on 1/14/19

As the title implies, this is a brief introduction textbook. It covers the fundamental of the introductory statistics, however not a comprehensive text on the subject. A teacher can use this book as the sole text of an introductory statistics.... read more

As the title implies, this is a brief introduction textbook. It covers the fundamental of the introductory statistics, however not a comprehensive text on the subject. A teacher can use this book as the sole text of an introductory statistics. The prose format of definitions and theorems make theoretical concepts accessible to non-math major students. The textbook covers all chapters required in this level course.

It is accurate; the subject matter in the examples to be up to date, is timeless and wouldn't need to be revised in future editions; there is no error except a few typographical errors. There are no logic errors or incorrect explanations.

This text will remain up to date for a long time since it has timeless examples and exercises, it wouldn't be outdated. The information is presented clearly with a simple way and the exercises are beneficial to follow the information.

The material is presented in a clear, concise manner. The text is easy readable for the first time statistics student.

The structure of the text is very consistent. Topics are presented with examples, followed by exercises. Problem sets are appropriate for the level of learner.

When the earlier matters need to be referenced, it is easy to find; no trouble reading the book and finding results, it has a consistent scheme. This book is set very well in sections.

The text presents the information in a logical order.

The learner can easily follow up the material; there is no interface problem.

There is no logic errors and incorrect explanations, a few typographical errors is just to be ignored.

Not applicable for this textbook.

Reviewed by Suhwon Lee, Associate Teaching Professor, University of Missouri on 6/19/18

This book is pretty comprehensive for being a brief introductory book. This book covers all necessary content areas for an introduction to Statistics course for non-math majors. The text book provides an effective index, plenty of exercises,... read more

This book is pretty comprehensive for being a brief introductory book. This book covers all necessary content areas for an introduction to Statistics course for non-math majors. The text book provides an effective index, plenty of exercises, review questions, and practice tests. It provides references and case studies. The glossary and index section is very helpful for students and can be used as a great resource.

Content appears to be accurate throughout. Being an introductory book, the book is unbiased and straight to the point. The terminology is standard.

The content in textbook is up to date. It will be very easy to update it or make changes at any point in time because of the well-structured contents in the textbook.

The author does a great job of explaining nearly every new term or concept. The book is easy to follow, clear and concise. The graphics are good to follow. The language in the book is easily understandable. I found most instructions in the book to be very detailed and clear for students to follow.

Overall consistency is good. It is consistent in terms of terminology and framework. The writing is straightforward and standardized throughout the text and it makes reading easier.

The authors do a great job of partitioning the text and labeling sections with appropriate headings. The table of contents is well organized and easily divisible into reading sections and it can be assigned at different points within the course.

Organization/Structure/Flow rating: 4

Overall, the topics are arranged in an order that follows natural progression in a statistics course with some exception. They are addressed logically and given adequate coverage.

The text is free of any issues. There are no navigation problems nor any display issues.

The text contains no grammatical errors.

The text is not culturally insensitive or offensive in any way most of time. Some examples might need to consider citing the sources or use differently to reflect current inclusive teaching strategies.

Overall, it's well-written and good recourse to be an introduction to statistical methods. Some materials may not need to be covered in an one-semester course. Various examples and quizzes can be a great recourse for instructor.

Reviewed by Jenna Kowalski, Mathematics Instructor, Anoka-Ramsey Community College on 3/27/18

The text includes the introductory statistics topics covered in a college-level semester course. An effective index and glossary are included, with functional hyperlinks. read more

The text includes the introductory statistics topics covered in a college-level semester course. An effective index and glossary are included, with functional hyperlinks.

Content Accuracy rating: 3

The content of this text is accurate and error-free, based on a random sampling of various pages throughout the text. Several examples included information without formal citation, leading the reader to potential bias and discrimination. These examples should be corrected to reflect current values of inclusive teaching.

The text contains relevant information that is current and will not become outdated in the near future. The statistical formulas and calculations have been used for centuries. The examples are direct applications of the formulas and accurately assess the conceptual knowledge of the reader.

The text is very clear and direct with the language used. The jargon does require a basic mathematical and/or statistical foundation to interpret, but this foundational requirement should be met with course prerequisites and placement testing. Graphs, tables, and visual displays are clearly labeled.

The terminology and framework of the text is consistent. The hyperlinks are working effectively, and the glossary is valuable. Each chapter contains modules that begin with prerequisite information and upcoming learning objectives for mastery.

The modules are clearly defined and can be used in conjunction with other modules, or individually to exemplify a choice topic. With the prerequisite information stated, the reader understands what prior mathematical understanding is required to successfully use the module.

The topics are presented well, but I recommend placing Sampling Distributions, Advanced Graphs, and Research Design ahead of Probability in the text. I think this rearranged version of the index would better align with current Introductory Statistics texts. The structure is very organized with the prerequisite information stated and upcoming learner outcomes highlighted. Each module is well-defined.

Adding an option of returning to the previous page would be of great value to the reader. While progressing through the text systematically, this is not an issue, but when the reader chooses to skip modules and read select pages then returning to the previous state of information is not easily accessible.

No grammatical errors were found while reviewing select pages of this text at random.

Cultural Relevance rating: 3

Several examples contained data that were not formally cited. These examples need to be corrected to reflect current inclusive teaching strategies. For example, one question stated that “while men are XX times more likely to commit murder than women, …” This data should be cited, otherwise the information can be interpreted as biased and offensive.

An included solutions manual for the exercises would be valuable to educators who choose to use this text.

Reviewed by Zaki Kuruppalil, Associate Professor, Ohio University on 2/1/18

This is a comprehensive book on statistical methods, its settings and most importantly the interpretation of the results. With the advent of computers and software’s, complex statistical analysis can be done very easily. But the challenge is the... read more

This is a comprehensive book on statistical methods, its settings and most importantly the interpretation of the results. With the advent of computers and software’s, complex statistical analysis can be done very easily. But the challenge is the knowledge of how to set the case, setting parameters (for example confidence intervals) and knowing its implication on the interpretation of the results. If not done properly this could lead to deceptive inferences, inadvertently or purposely. This book does a great job in explaining the above using many examples and real world case studies. If you are looking for a book to learn and apply statistical methods, this is a great one. I think the author could consider revising the title of the book to reflect the above, as it is more than just an introduction to statistics, may be include the word such as practical guide.

The contents of the book seems accurate. Some plots and calculations were randomly selected and checked for accuracy.

The book topics are up to date and in my opinion, will not be obsolete in the near future. I think the smartest thing the author has done is, not tied the book with any particular software such as minitab or spss . No matter what the software is, standard deviation is calculated the same way as it is always. The only noticeable exception in this case was using the Java Applet for calculating Z values in page 261 and in page 416 an excerpt of SPSS analysis is provided for ANOVA calculations.

The contents and examples cited are clear and explained in simple language. Data analysis and presentation of the results including mathematical calculations, graphical explanation using charts, tables, figures etc are presented with clarity.

Terminology is consistant. Framework for each chapter seems consistent with each chapter beginning with a set of defined topics, and each of the topic divided into modules with each module having a set of learning objectives and prerequisite chapters.

The text book is divided into chapters with each chapter further divided into modules. Each of the modules have detailed learning objectives and prerequisite required. So you can extract a portion of the book and use it as a standalone to teach certain topics or as a learning guide to apply a relevant topic.

Presentation of the topics are well thought and are presented in a logical fashion as if it would be introduced to someone who is learning the contents. However, there are some issues with table of contents and page numbers, for example chapter 17 starts in page 597 not 598. Also some tables and figures does not have a number, for instance the graph shown in page 114 does not have a number. Also it would have been better if the chapter number was included in table and figure identification, for example Figure 4-5 . Also in some cases, for instance page 109, the figures and titles are in two different pages.

No major issues. Only suggestion would be, since each chapter has several modules, any means such as a header to trace back where you are currently, would certainly help.

Grammatical Errors rating: 4

Easy to read and phrased correctly in most cases. Minor grammatical errors such as missing prepositions etc. In some cases the author seems to have the habbit of using a period after the decimal. For instance page 464, 467 etc. For X = 1, Y' = (0.425)(1) + 0.785 = 1.21. For X = 2, Y' = (0.425)(2) + 0.785 = 1.64.

However it contains some statements (even though given as examples) that could be perceived as subjective, which the author could consider citing the sources. For example from page 11: Statistics include numerical facts and figures. For instance: • The largest earthquake measured 9.2 on the Richter scale. • Men are at least 10 times more likely than women to commit murder. • One in every 8 South Africans is HIV positive. • By the year 2020, there will be 15 people aged 65 and over for every new baby born.

Solutions for the exercises would be a great teaching resource to have

Reviewed by Randy Vander Wal, Professor, The Pennsylvania State University on 2/1/18

As a text for an introductory course, standard topics are covered. It was nice to see some topics such as power, sampling, research design and distribution free methods covered, as these are often omitted in abbreviated texts. Each module... read more

As a text for an introductory course, standard topics are covered. It was nice to see some topics such as power, sampling, research design and distribution free methods covered, as these are often omitted in abbreviated texts. Each module introduces the topic, has appropriate graphics, illustration or worked example(s) as appropriate and concluding with many exercises. An instructor’s manual is available by contacting the author. A comprehensive glossary provides definitions for all the major terms and concepts. The case studies give examples of practical applications of statistical analyses. Many of the case studies contain the actual raw data. To note is that the on-line e-book provides several calculators for the essential distributions and tests. These are provided in lieu of printed tables which are not included in the pdf. (Such tables are readily available on the web.)

The content is accurate and error free. Notation is standard and terminology is used accurately, as are the videos and verbal explanations therein. Online links work properly as do all the calculators. The text appears neutral and unbiased in subject and content.

The text achieves contemporary relevance by ending each section with a Statistical Literacy example, drawn from contemporary headlines and issues. Of course, the core topics are time proven. There is no obvious material that may become “dated”.

The text is very readable. While the pdf text may appear “sparse” by absence varied colored and inset boxes, pictures etc., the essential illustrations and descriptions are provided. Meanwhile for this same content the on-line version appears streamlined, uncluttered, enhancing the value of the active links. Moreover, the videos provide nice short segments of “active” instruction that are clear and concise. Despite being a mathematical text, the text is not overly burdened by formulas and numbers but rather has “readable feel”.

This terminology and symbol use are consistent throughout the text and with common use in the field. The pdf text and online version are also consistent by content, but with the online e-book offering much greater functionality.

The chapters and topics may be used in a selective manner. Certain chapters have no pre-requisite chapter and in all cases, those required are listed at the beginning of each module. It would be straightforward to select portions of the text and reorganize as needed. The online version is highly modular offering students both ease of navigation and selection of topics.

Chapter topics are arranged appropriately. In an introductory statistics course, there is a logical flow given the buildup to the normal distribution, concept of sampling distributions, confidence intervals, hypothesis testing, regression and additional parametric and non-parametric tests. The normal distribution is central to an introductory course. Necessary precursor topics are covered in this text, while its use in significance and hypothesis testing follow, and thereafter more advanced topics, including multi-factor ANOVA.

Each chapter is structured with several modules, each beginning with pre-requisite chapter(s), learning objectives and concluding with Statistical Literacy sections providing a self-check question addressing the core concept, along with answer, followed by an extensive problem set. The clear and concise learning objectives will be of benefit to students and the course instructor. No solutions or answer key is provided to students. An instructor’s manual is available by request.

The on-line interface works well. In fact, I was pleasantly surprised by its options and functionality. The pdf appears somewhat sparse by comparison to publisher texts, lacking pictures, colored boxes, etc. But the on-line version has many active links providing definitions and graphic illustrations for key terms and topics. This can really facilitate learning as making such “refreshers” integral to the new material. Most sections also have short videos that are professionally done, with narration and smooth graphics. In this way, the text is interactive and flexible, offering varied tools for students. To note is that the interactive e-book works for both IOS and OS X.

The text in pdf form appeared to free of grammatical errors, as did the on-line version, text, graphics and videos.

This text contains no culturally insensitive or offensive content. The focus of the text is on concepts and explanation.

The text would be a great resource for students. The full content would be ambitious for a 1-semester course, such use would be unlikely. The text is clearly geared towards students with no statistics background nor calculus. The text could be used in two styles of course. For 1st year students early chapters on graphs and distributions would be the starting point, omitting later chapters on Chi-square, transformations, distribution-free and size effect chapters. Alternatively, for upper level students the introductory chapters could be bypassed with the latter chapters then covered to completion.

This text adopts a descriptive style of presentation with topics well and fully explained, much like the “Dummy series”. For this, it may seem a bit “wordy”, but this can well serve students and notably it complements powerpoint slides that are generally sparse on written content. This text could be used as the primary text, for regular lectures, or as reference for a “flipped” class. The e-book videos are an enabling tool if this approach is adopted.

Reviewed by David jabon, Associate Professor, DePaul University on 8/15/17

This text covers all the standard topics in a semester long introductory course in statistics. It is particularly well indexed and very easy to navigate. There is comprehensive hyperlinked glossary. read more

This text covers all the standard topics in a semester long introductory course in statistics. It is particularly well indexed and very easy to navigate. There is comprehensive hyperlinked glossary.

The material is completely accurate. There are no errors. The terminology is standard with one exception: the book calls what most people call the interquartile range, the H-spread in a number of places. Ideally, the term "interquartile range" would be used in place of every reference to "H-spread." "Interquartile range" is simply a better, more descriptive term of the concept that it describes. It is also more commonly used nowadays.

This book came out a number of years ago, but the material is still up to date. Some more recent case studies have been added.

The writing is very clear. There are also videos for almost every section. The section on boxplots uses a lot of technical terms that I don't find are very helpful for my students (hinge, H-spread, upper adjacent value).

The text is internally consistent with one exception that I noted (the use of the synonymous words "H-spread" and "interquartile range").

The text book is brokenly into very short sections, almost to a fault. Each section is at most two pages long. However at the end of each of these sections there are a few multiple choice questions to test yourself. These questions are a very appealing feature of the text.

The organization, in particular the ordering of the topics, is rather standard with a few exceptions. Boxplots are introduced in Chapter II before the discussion of measures of center and dispersion. Most books introduce them as part of discussion of summaries of data using measure of center and dispersion. Some statistics instructors may not like the way the text lumps all of the sampling distributions in a single chapter (sampling distribution of mean, sampling distribution for the difference of means, sampling distribution of a proportion, sampling distribution of r). I have tried this approach, and I now like this approach. But it is a very challenging chapter for students.

The book's interface has no features that distracted me. Overall the text is very clean and spare, with no additional distracting visual elements.

The book contains no grammatical errors.

The book's cultural relevance comes out in the case studies. As of this writing there are 33 such case studies, and they cover a wide range of issues from health to racial, ethnic, and gender disparity.

Each chapter as a nice set of exercises with selected answers. The thirty three case studies are excellent and can be supplement with some other online case studies. An instructor's manual and PowerPoint slides can be obtained by emailing the author. There are direct links to online simulations within the text. This text is very high quality textbook in every way.

Table of Contents

• 1. Introduction
• 2. Graphing Distributions
• 3. Summarizing Distributions
• 4. Describing Bivariate Data
• 5. Probability
• 6. Research Design
• 7. Normal Distributions
• 8. Advanced Graphs
• 9. Sampling Distributions
• 10. Estimation
• 11. Logic of Hypothesis Testing
• 12. Testing Means
• 14. Regression
• 15. Analysis of Variance
• 16. Transformations
• 17. Chi Square
• 18. Distribution-Free Tests
• 19. Effect Size
• 20. Case Studies
• 21. Glossary

Ancillary Material

• Ancillary materials are available by contacting the author or publisher .

About the Book

Introduction to Statistics is a resource for learning and teaching introductory statistics. This work is in the public domain. Therefore, it can be copied and reproduced without limitation. However, we would appreciate a citation where possible. Please cite as: Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University. Instructor's manual, PowerPoint Slides, and additional questions are available.

About the Contributors

David Lane is an Associate Professor in the Departments of Psychology, Statistics, and Management at the Rice University. Lane is the principal developer of this resource although many others have made substantial contributions. This site was developed at Rice University, University of Houston-Clear Lake, and Tufts University.

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• Indian J Anaesth
• v.60(9); 2016 Sep

Basic statistical tools in research and data analysis

Zulfiqar ali.

Department of Anaesthesiology, Division of Neuroanaesthesiology, Sheri Kashmir Institute of Medical Sciences, Soura, Srinagar, Jammu and Kashmir, India

S Bala Bhaskar

1 Department of Anaesthesiology and Critical Care, Vijayanagar Institute of Medical Sciences, Bellary, Karnataka, India

Statistical methods involved in carrying out a study include planning, designing, collecting data, analysing, drawing meaningful interpretation and reporting of the research findings. The statistical analysis gives meaning to the meaningless numbers, thereby breathing life into a lifeless data. The results and inferences are precise only if proper statistical tests are used. This article will try to acquaint the reader with the basic research tools that are utilised while conducting various studies. The article covers a brief outline of the variables, an understanding of quantitative and qualitative variables and the measures of central tendency. An idea of the sample size estimation, power analysis and the statistical errors is given. Finally, there is a summary of parametric and non-parametric tests used for data analysis.

INTRODUCTION

Statistics is a branch of science that deals with the collection, organisation, analysis of data and drawing of inferences from the samples to the whole population.[ 1 ] This requires a proper design of the study, an appropriate selection of the study sample and choice of a suitable statistical test. An adequate knowledge of statistics is necessary for proper designing of an epidemiological study or a clinical trial. Improper statistical methods may result in erroneous conclusions which may lead to unethical practice.[ 2 ]

Variable is a characteristic that varies from one individual member of population to another individual.[ 3 ] Variables such as height and weight are measured by some type of scale, convey quantitative information and are called as quantitative variables. Sex and eye colour give qualitative information and are called as qualitative variables[ 3 ] [ Figure 1 ].

Classification of variables

Quantitative variables

Quantitative or numerical data are subdivided into discrete and continuous measurements. Discrete numerical data are recorded as a whole number such as 0, 1, 2, 3,… (integer), whereas continuous data can assume any value. Observations that can be counted constitute the discrete data and observations that can be measured constitute the continuous data. Examples of discrete data are number of episodes of respiratory arrests or the number of re-intubations in an intensive care unit. Similarly, examples of continuous data are the serial serum glucose levels, partial pressure of oxygen in arterial blood and the oesophageal temperature.

A hierarchical scale of increasing precision can be used for observing and recording the data which is based on categorical, ordinal, interval and ratio scales [ Figure 1 ].

Categorical or nominal variables are unordered. The data are merely classified into categories and cannot be arranged in any particular order. If only two categories exist (as in gender male and female), it is called as a dichotomous (or binary) data. The various causes of re-intubation in an intensive care unit due to upper airway obstruction, impaired clearance of secretions, hypoxemia, hypercapnia, pulmonary oedema and neurological impairment are examples of categorical variables.

Ordinal variables have a clear ordering between the variables. However, the ordered data may not have equal intervals. Examples are the American Society of Anesthesiologists status or Richmond agitation-sedation scale.

Interval variables are similar to an ordinal variable, except that the intervals between the values of the interval variable are equally spaced. A good example of an interval scale is the Fahrenheit degree scale used to measure temperature. With the Fahrenheit scale, the difference between 70° and 75° is equal to the difference between 80° and 85°: The units of measurement are equal throughout the full range of the scale.

Ratio scales are similar to interval scales, in that equal differences between scale values have equal quantitative meaning. However, ratio scales also have a true zero point, which gives them an additional property. For example, the system of centimetres is an example of a ratio scale. There is a true zero point and the value of 0 cm means a complete absence of length. The thyromental distance of 6 cm in an adult may be twice that of a child in whom it may be 3 cm.

STATISTICS: DESCRIPTIVE AND INFERENTIAL STATISTICS

Descriptive statistics[ 4 ] try to describe the relationship between variables in a sample or population. Descriptive statistics provide a summary of data in the form of mean, median and mode. Inferential statistics[ 4 ] use a random sample of data taken from a population to describe and make inferences about the whole population. It is valuable when it is not possible to examine each member of an entire population. The examples if descriptive and inferential statistics are illustrated in Table 1 .

Example of descriptive and inferential statistics

Descriptive statistics

The extent to which the observations cluster around a central location is described by the central tendency and the spread towards the extremes is described by the degree of dispersion.

Measures of central tendency

The measures of central tendency are mean, median and mode.[ 6 ] Mean (or the arithmetic average) is the sum of all the scores divided by the number of scores. Mean may be influenced profoundly by the extreme variables. For example, the average stay of organophosphorus poisoning patients in ICU may be influenced by a single patient who stays in ICU for around 5 months because of septicaemia. The extreme values are called outliers. The formula for the mean is

where x = each observation and n = number of observations. Median[ 6 ] is defined as the middle of a distribution in a ranked data (with half of the variables in the sample above and half below the median value) while mode is the most frequently occurring variable in a distribution. Range defines the spread, or variability, of a sample.[ 7 ] It is described by the minimum and maximum values of the variables. If we rank the data and after ranking, group the observations into percentiles, we can get better information of the pattern of spread of the variables. In percentiles, we rank the observations into 100 equal parts. We can then describe 25%, 50%, 75% or any other percentile amount. The median is the 50 th percentile. The interquartile range will be the observations in the middle 50% of the observations about the median (25 th -75 th percentile). Variance[ 7 ] is a measure of how spread out is the distribution. It gives an indication of how close an individual observation clusters about the mean value. The variance of a population is defined by the following formula:

where σ 2 is the population variance, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The variance of a sample is defined by slightly different formula:

where s 2 is the sample variance, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. The formula for the variance of a population has the value ‘ n ’ as the denominator. The expression ‘ n −1’ is known as the degrees of freedom and is one less than the number of parameters. Each observation is free to vary, except the last one which must be a defined value. The variance is measured in squared units. To make the interpretation of the data simple and to retain the basic unit of observation, the square root of variance is used. The square root of the variance is the standard deviation (SD).[ 8 ] The SD of a population is defined by the following formula:

where σ is the population SD, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The SD of a sample is defined by slightly different formula:

where s is the sample SD, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. An example for calculation of variation and SD is illustrated in Table 2 .

Example of mean, variance, standard deviation

Normal distribution or Gaussian distribution

Most of the biological variables usually cluster around a central value, with symmetrical positive and negative deviations about this point.[ 1 ] The standard normal distribution curve is a symmetrical bell-shaped. In a normal distribution curve, about 68% of the scores are within 1 SD of the mean. Around 95% of the scores are within 2 SDs of the mean and 99% within 3 SDs of the mean [ Figure 2 ].

Normal distribution curve

Skewed distribution

It is a distribution with an asymmetry of the variables about its mean. In a negatively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the right of Figure 1 . In a positively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the left of the figure leading to a longer right tail.

Curves showing negatively skewed and positively skewed distribution

Inferential statistics

In inferential statistics, data are analysed from a sample to make inferences in the larger collection of the population. The purpose is to answer or test the hypotheses. A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. Hypothesis tests are thus procedures for making rational decisions about the reality of observed effects.

Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).

In inferential statistics, the term ‘null hypothesis’ ( H 0 ‘ H-naught ,’ ‘ H-null ’) denotes that there is no relationship (difference) between the population variables in question.[ 9 ]

Alternative hypothesis ( H 1 and H a ) denotes that a statement between the variables is expected to be true.[ 9 ]

The P value (or the calculated probability) is the probability of the event occurring by chance if the null hypothesis is true. The P value is a numerical between 0 and 1 and is interpreted by researchers in deciding whether to reject or retain the null hypothesis [ Table 3 ].

P values with interpretation

If P value is less than the arbitrarily chosen value (known as α or the significance level), the null hypothesis (H0) is rejected [ Table 4 ]. However, if null hypotheses (H0) is incorrectly rejected, this is known as a Type I error.[ 11 ] Further details regarding alpha error, beta error and sample size calculation and factors influencing them are dealt with in another section of this issue by Das S et al .[ 12 ]

Illustration for null hypothesis

PARAMETRIC AND NON-PARAMETRIC TESTS

Numerical data (quantitative variables) that are normally distributed are analysed with parametric tests.[ 13 ]

Two most basic prerequisites for parametric statistical analysis are:

• The assumption of normality which specifies that the means of the sample group are normally distributed
• The assumption of equal variance which specifies that the variances of the samples and of their corresponding population are equal.

However, if the distribution of the sample is skewed towards one side or the distribution is unknown due to the small sample size, non-parametric[ 14 ] statistical techniques are used. Non-parametric tests are used to analyse ordinal and categorical data.

Parametric tests

The parametric tests assume that the data are on a quantitative (numerical) scale, with a normal distribution of the underlying population. The samples have the same variance (homogeneity of variances). The samples are randomly drawn from the population, and the observations within a group are independent of each other. The commonly used parametric tests are the Student's t -test, analysis of variance (ANOVA) and repeated measures ANOVA.

Student's t -test

Student's t -test is used to test the null hypothesis that there is no difference between the means of the two groups. It is used in three circumstances:

where X = sample mean, u = population mean and SE = standard error of mean

where X 1 − X 2 is the difference between the means of the two groups and SE denotes the standard error of the difference.

• To test if the population means estimated by two dependent samples differ significantly (the paired t -test). A usual setting for paired t -test is when measurements are made on the same subjects before and after a treatment.

The formula for paired t -test is:

where d is the mean difference and SE denotes the standard error of this difference.

The group variances can be compared using the F -test. The F -test is the ratio of variances (var l/var 2). If F differs significantly from 1.0, then it is concluded that the group variances differ significantly.

Analysis of variance

The Student's t -test cannot be used for comparison of three or more groups. The purpose of ANOVA is to test if there is any significant difference between the means of two or more groups.

In ANOVA, we study two variances – (a) between-group variability and (b) within-group variability. The within-group variability (error variance) is the variation that cannot be accounted for in the study design. It is based on random differences present in our samples.

However, the between-group (or effect variance) is the result of our treatment. These two estimates of variances are compared using the F-test.

A simplified formula for the F statistic is:

where MS b is the mean squares between the groups and MS w is the mean squares within groups.

Repeated measures analysis of variance

As with ANOVA, repeated measures ANOVA analyses the equality of means of three or more groups. However, a repeated measure ANOVA is used when all variables of a sample are measured under different conditions or at different points in time.

As the variables are measured from a sample at different points of time, the measurement of the dependent variable is repeated. Using a standard ANOVA in this case is not appropriate because it fails to model the correlation between the repeated measures: The data violate the ANOVA assumption of independence. Hence, in the measurement of repeated dependent variables, repeated measures ANOVA should be used.

Non-parametric tests

When the assumptions of normality are not met, and the sample means are not normally, distributed parametric tests can lead to erroneous results. Non-parametric tests (distribution-free test) are used in such situation as they do not require the normality assumption.[ 15 ] Non-parametric tests may fail to detect a significant difference when compared with a parametric test. That is, they usually have less power.

As is done for the parametric tests, the test statistic is compared with known values for the sampling distribution of that statistic and the null hypothesis is accepted or rejected. The types of non-parametric analysis techniques and the corresponding parametric analysis techniques are delineated in Table 5 .

Analogue of parametric and non-parametric tests

Median test for one sample: The sign test and Wilcoxon's signed rank test

The sign test and Wilcoxon's signed rank test are used for median tests of one sample. These tests examine whether one instance of sample data is greater or smaller than the median reference value.

This test examines the hypothesis about the median θ0 of a population. It tests the null hypothesis H0 = θ0. When the observed value (Xi) is greater than the reference value (θ0), it is marked as+. If the observed value is smaller than the reference value, it is marked as − sign. If the observed value is equal to the reference value (θ0), it is eliminated from the sample.

If the null hypothesis is true, there will be an equal number of + signs and − signs.

The sign test ignores the actual values of the data and only uses + or − signs. Therefore, it is useful when it is difficult to measure the values.

Wilcoxon's signed rank test

There is a major limitation of sign test as we lose the quantitative information of the given data and merely use the + or – signs. Wilcoxon's signed rank test not only examines the observed values in comparison with θ0 but also takes into consideration the relative sizes, adding more statistical power to the test. As in the sign test, if there is an observed value that is equal to the reference value θ0, this observed value is eliminated from the sample.

Wilcoxon's rank sum test ranks all data points in order, calculates the rank sum of each sample and compares the difference in the rank sums.

Mann-Whitney test

It is used to test the null hypothesis that two samples have the same median or, alternatively, whether observations in one sample tend to be larger than observations in the other.

Mann–Whitney test compares all data (xi) belonging to the X group and all data (yi) belonging to the Y group and calculates the probability of xi being greater than yi: P (xi > yi). The null hypothesis states that P (xi > yi) = P (xi < yi) =1/2 while the alternative hypothesis states that P (xi > yi) ≠1/2.

Kolmogorov-Smirnov test

The two-sample Kolmogorov-Smirnov (KS) test was designed as a generic method to test whether two random samples are drawn from the same distribution. The null hypothesis of the KS test is that both distributions are identical. The statistic of the KS test is a distance between the two empirical distributions, computed as the maximum absolute difference between their cumulative curves.

Kruskal-Wallis test

The Kruskal–Wallis test is a non-parametric test to analyse the variance.[ 14 ] It analyses if there is any difference in the median values of three or more independent samples. The data values are ranked in an increasing order, and the rank sums calculated followed by calculation of the test statistic.

Jonckheere test

In contrast to Kruskal–Wallis test, in Jonckheere test, there is an a priori ordering that gives it a more statistical power than the Kruskal–Wallis test.[ 14 ]

Friedman test

The Friedman test is a non-parametric test for testing the difference between several related samples. The Friedman test is an alternative for repeated measures ANOVAs which is used when the same parameter has been measured under different conditions on the same subjects.[ 13 ]

Tests to analyse the categorical data

Chi-square test, Fischer's exact test and McNemar's test are used to analyse the categorical or nominal variables. The Chi-square test compares the frequencies and tests whether the observed data differ significantly from that of the expected data if there were no differences between groups (i.e., the null hypothesis). It is calculated by the sum of the squared difference between observed ( O ) and the expected ( E ) data (or the deviation, d ) divided by the expected data by the following formula:

A Yates correction factor is used when the sample size is small. Fischer's exact test is used to determine if there are non-random associations between two categorical variables. It does not assume random sampling, and instead of referring a calculated statistic to a sampling distribution, it calculates an exact probability. McNemar's test is used for paired nominal data. It is applied to 2 × 2 table with paired-dependent samples. It is used to determine whether the row and column frequencies are equal (that is, whether there is ‘marginal homogeneity’). The null hypothesis is that the paired proportions are equal. The Mantel-Haenszel Chi-square test is a multivariate test as it analyses multiple grouping variables. It stratifies according to the nominated confounding variables and identifies any that affects the primary outcome variable. If the outcome variable is dichotomous, then logistic regression is used.

SOFTWARES AVAILABLE FOR STATISTICS, SAMPLE SIZE CALCULATION AND POWER ANALYSIS

Numerous statistical software systems are available currently. The commonly used software systems are Statistical Package for the Social Sciences (SPSS – manufactured by IBM corporation), Statistical Analysis System ((SAS – developed by SAS Institute North Carolina, United States of America), R (designed by Ross Ihaka and Robert Gentleman from R core team), Minitab (developed by Minitab Inc), Stata (developed by StataCorp) and the MS Excel (developed by Microsoft).

There are a number of web resources which are related to statistical power analyses. A few are:

• StatPages.net – provides links to a number of online power calculators
• G-Power – provides a downloadable power analysis program that runs under DOS
• Power analysis for ANOVA designs an interactive site that calculates power or sample size needed to attain a given power for one effect in a factorial ANOVA design
• SPSS makes a program called SamplePower. It gives an output of a complete report on the computer screen which can be cut and paste into another document.

It is important that a researcher knows the concepts of the basic statistical methods used for conduct of a research study. This will help to conduct an appropriately well-designed study leading to valid and reliable results. Inappropriate use of statistical techniques may lead to faulty conclusions, inducing errors and undermining the significance of the article. Bad statistics may lead to bad research, and bad research may lead to unethical practice. Hence, an adequate knowledge of statistics and the appropriate use of statistical tests are important. An appropriate knowledge about the basic statistical methods will go a long way in improving the research designs and producing quality medical research which can be utilised for formulating the evidence-based guidelines.

Financial support and sponsorship

Conflicts of interest.

There are no conflicts of interest.

Research Papers on Statistics

Statistics is often termed as a science which requires learning from a set of collected data. Research papers on statistics involve great focus on analysis, proportional inference and interpretation. This section of Researchomatic is therefore focused on bringing to its customers some of the most carefully selected and compiled research papers on statistics that will help students to better understand the subject.

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How to Write Statistics Research Paper | Easy Guide

A statistics research paper is an academic document presenting original findings or analyses derived from the data’s collection, organization, analysis, and interpretation. It addresses research questions or hypotheses within the field of statistics.

As a rule, college students get such papers assigned during a semester to assess their knowledge of statistics. However, any statistician specialist can also write research papers and publish them in academic journals, thus developing and promoting this field.

Want to master the art of statistics research paper writing?

We’ve got expert tips from a professional research paper writing service on crafting such studies. In this article, you’ll find a step-by-step guide on writing a statistics research paper that your educators, colleagues, or clients will approve.

How to write a statistics research paper: Steps

Table of Contents

State the problem

Collect the data, write an introductory paragraph, craft an abstract, describe your methodology, present your findings: evaluate and illustrate, revise and proofread.

Research papers aren’t about describing the existing knowledge on the topic. You should state your intellectual concern with it, indicating why it’s worth studying. When choosing the problem you’ll research in the paper, emphasize its ongoing nature:

What have other researchers already studied about it? Cite at least one previous publication related to your research and provide your statistical motivation to continue researching the topic. (You’ll refer to those researches in footnotes or within the text of your paper.)

Once you have the topic (problem) to research in your paper, it’s time to collect sources you’ll use as evidence and references. For statistics papers, consider the following:

• Published research from experts in Statistics (academic journals, newspapers, books, online publications, etc.)
• Statistical data from reliable sources (Google’s Public Data and Scholar, FedStats, and others)
• Your personal hypothesis, experiments, and info-gathering activities

The last one is a must-have! Your statistics research paper requires new information gathered by you as a researcher and not previously published anywhere. The massive block of your research paper will be about the data collection methods you used to investigate the problem and come to the conclusions you’ll provide.

Some underestimate the introductory paragraphs of research papers , but they are wrong. The introduction is the first thing a reader sees to understand if your research is worth their attention and time. With that in mind, ensure your introductory paragraph is intriguing yet informative enough for the audience to continue reading.

Start with a writing hook, a sentence grabbing a reader’s attention. Also, an intro needs background information: your topic and the scientific motivation for the new research methods. (What’s wrong with existing ones? Or, what do they miss?) Finally, move on to your thesis statement: 1-2 sentences summarizing the primary idea behind your research paper.

It’s an overview of your statistics research paper where you establish notation and outline the methods and the results. Abstracts are integral for all academic studies and research, giving readers enough details to decide whether your paper is relevant to them.

What do you include in an abstract?

Introduce your topic and explain why it’s significant in your field. State the gap present in the research at the moment and reveal the aim of your paper. Then, briefly describe your research methods and approach, summarize your findings, and explain their contribution to the field.

The methods section is the most extensive one in your research paper. Here, you provide sufficient information about how you collected data for your research, what methodologies you used, and how you evaluated the results.

Be specific; describe everything so the audience can repeat your research (experiments) and reconstruct your results. It’s the value your paper brings to the academic community.

Further paragraphs of your research paper present your findings. Try to stick to one idea per paragraph to make it clear and easy for readers to consume.

Prepare and add supporting materials that will help you illustrate findings: graphs, diagrams, charts, tables, etc. You can use them in a paper’s body or add them as an appendix; these elements will support your points and serve as extra proof for your findings.

Last but not least:

Write a concluding paragraph for your statistics research paper. Repeat your thesis, summarize your findings, and conclude whether they have proved or contradicted your initial theory (hypothesis). Also, you can make suggestions for further research in the same area.

Re-read your paper several times before publishing or submitting it for review. Ensure all the information is logical and coherent, all the terms are correct, and all the elements are present and accurately placed.

Also, proofread your final draft: Spelling, grammar, and punctuation mistakes are a no-no here! Re-check the list of references again; ensure you follow the required citation style and use the proper format.

So, now you know seven easy steps for writing a statistical research paper. Whether you’re a college student or a statistician willing to make a scientific contribution to a niche, follow them to craft a professionally structured academic document:

State a problem, choose methods of analyzing it, evaluate your findings, and illustrate them to engage the audience in discussion.

If you still need clarification or have questions about writing a statistics paper, don’t hesitate to ask for assistance. Professional writers with experience in statistics are ready to help you improve your writing skills.

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Statistics Research Paper

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1. Introduction

Statistics is a body of quantitative methods associated with empirical observation. A primary goal of these methods is coping with uncertainty. Most formal statistical methods rely on probability theory to express this uncertainty and to provide a formal mathematical basis for data description and for analysis. The notion of variability associated with data, expressed through probability, plays a fundamental role in this theory. As a consequence, much statistical effort is focused on how to control and measure variability and/or how to assign it to its sources.

Almost all characterizations of statistics as a field include the following elements:

(a) Designing experiments, surveys, and other systematic forms of empirical study.

(b) Summarizing and extracting information from data.

(c) Drawing formal inferences from empirical data through the use of probability.

(d) Communicating the results of statistical investigations to others, including scientists, policy makers, and the public.

This research paper describes a number of these elements, and the historical context out of which they grew. It provides a broad overview of the field, that can serve as a starting point to many of the other statistical entries in this encyclopedia.

2. The Origins Of The Field of Statistics

The word ‘statistics’ is related to the word ‘state’ and the original activity that was labeled as statistics was social in nature and related to elements of society through the organization of economic, demographic, and political facts. Paralleling this work to some extent was the development of the probability calculus and the theory of errors, typically associated with the physical sciences. These traditions came together in the nineteenth century and led to the notion of statistics as a collection of methods for the analysis of scientific data and the drawing of inferences therefrom.

As Hacking (1990) has noted: ‘By the end of the century chance had attained the respectability of a Victorian valet, ready to be the logical servant of the natural, biological and social sciences’ ( p. 2). At the beginning of the twentieth century, we see the emergence of statistics as a field under the leadership of Karl Pearson, George Udny Yule, Francis Y. Edgeworth, and others of the ‘English’ statistical school. As Stigler (1986) suggests:

Before 1900 we see many scientists of different fields developing and using techniques we now recognize as belonging to modern statistics. After 1900 we begin to see identifiable statisticians developing such techniques into a unified logic of empirical science that goes far beyond its component parts. There was no sharp moment of birth; but with Pearson and Yule and the growing number of students in Pearson’s laboratory, the infant discipline may be said to have arrived. (p. 361)

Pearson’s laboratory at University College, London quickly became the first statistics department in the world and it was to influence subsequent developments in a profound fashion for the next three decades. Pearson and his colleagues founded the first methodologically-oriented statistics journal, Biometrika, and they stimulated the development of new approaches to statistical methods. What remained before statistics could legitimately take on the mantle of a field of inquiry, separate from mathematics or the use of statistical approaches in other fields, was the development of the formal foundations of theories of inference from observations, rooted in an axiomatic theory of probability.

Beginning at least with the Rev. Thomas Bayes and Pierre Simon Laplace in the eighteenth century, most early efforts at statistical inference used what was known as the method of inverse probability to update a prior probability using the observed data in what we now refer to as Bayes’ Theorem. (For a discussion of who really invented Bayes’ Theorem, see Stigler 1999, Chap. 15). Inverse probability came under challenge in the nineteenth century, but viable alternative approaches gained little currency. It was only with the work of R. A. Fisher on statistical models, estimation, and significance tests, and Jerzy Neyman and Egon Pearson, in the 1920s and 1930s, on tests of hypotheses, that alternative approaches were fully articulated and given a formal foundation. Neyman’s advocacy of the role of probability in the structuring of a frequency-based approach to sample surveys in 1934 and his development of confidence intervals further consolidated this effort at the development of a foundation for inference (cf. Statistical Methods, History of: Post- 1900 and the discussion of ‘The inference experts’ in Gigerenzer et al. 1989).

At about the same time Kolmogorov presented his famous axiomatic treatment of probability, and thus by the end of the 1930s, all of the requisite elements were finally in place for the identification of statistics as a field. Not coincidentally, the first statistical society devoted to the mathematical underpinnings of the field, The Institute of Mathematical Statistics, was created in the United States in the mid-1930s. It was during this same period that departments of statistics and statistical laboratories and groups were first formed in universities in the United States.

3. Emergence Of Statistics As A Field

3.1 the role of world war ii.

Perhaps the greatest catalysts to the emergence of statistics as a field were two major social events: the Great Depression of the 1930s and World War II. In the United States, one of the responses to the depression was the development of large-scale probability-based surveys to measure employment and unemployment. This was followed by the institutionalization of sampling as part of the 1940 US decennial census. But with World War II raging in Europe and in Asia, mathematicians and statisticians were drawn into the war effort, and as a consequence they turned their attention to a broad array of new problems. In particular, multiple statistical groups were established in both England and the US specifically to develop new methods and to provide consulting. (See Wallis 1980, on statistical groups in the US; Barnard and Plackett 1985, for related efforts in the United Kingdom; and Fienberg 1985). These groups not only created imaginative new techniques such as sequential analysis and statistical decision theory, but they also developed a shared research agenda. That agenda led to a blossoming of statistics after the war, and in the 1950s and 1960s to the creation of departments of statistics at universities—from coast to coast in the US, and to a lesser extent in England and elsewhere.

3.2 The Neo-Bayesian Revival

Although inverse probability came under challenge in the 1920s and 1930s, it was not totally abandoned. John Maynard Keynes (1921) wrote A Treatise on Probability that was rooted in this tradition, and Frank Ramsey (1926) provided an early effort at justifying the subjective nature of prior distributions and suggested the importance of utility functions as an adjunct to statistical inference. Bruno de Finetti provided further development of these ideas in the 1930s, while Harold Jeffreys (1938) created a separate ‘objective’ development of these and other statistical ideas on inverse probability.

Yet as statistics flourished in the post-World War II era, it was largely based on the developments of Fisher, Neyman and Pearson, as well as the decision theory methods of Abraham Wald (1950). L. J. Savage revived interest in the inverse probability approach with The Foundations of Statistics (1954) in which he attempted to provide the axiomatic foundation from the subjective perspective. In an essentially independent effort, Raiffa and Schlaifer (1961) attempted to provide inverse probability counterparts to many of the then existing frequentist tools, referring to these alternatives as ‘Bayesian.’ By 1960, the term ‘Bayesian inference’ had become standard usage in the statistical literature, the theoretical interest in the development of Bayesian approaches began to take hold, and the neo-Bayesian revival was underway. But the movement from Bayesian theory to statistical practice was slow, in large part because the computations associated with posterior distributions were an overwhelming stumbling block for those who were interested in the methods. Only in the 1980s and 1990s did new computational approaches revolutionize both Bayesian methods, and the interest in them, in a broad array of areas of application.

3.3 The Role Of Computation In Statistics

From the days of Pearson and Fisher, computation played a crucial role in the development and application of statistics. Pearson’s laboratory employed dozens of women who used mechanical devices to carry out the careful and painstaking calculations required to tabulate values from various probability distributions. This effort ultimately led to the creation of the Biometrika Tables for Statisticians that were so widely used by others applying tools such as chisquare tests and the like. Similarly, Fisher also developed his own set of statistical tables with Frank Yates when he worked at Rothamsted Experiment Station in the 1920s and 1930s. One of the most famous pictures of Fisher shows him seated at Whittingehame Lodge, working at his desk calculator (see Box 1978).

The development of the modern computer revolutionized statistical calculation and practice, beginning with the creation of the first statistical packages in the 1960s—such as the BMDP package for biological and medical applications, and Datatext for statistical work in the social sciences. Other packages soon followed—such as SAS and SPSS for both data management and production-like statistical analyses, and MINITAB for the teaching of statistics. In 2001, in the era of the desktop personal computer, almost everyone has easy access to interactive statistical programs that can implement complex statistical procedures and produce publication-quality graphics. And there is a new generation of statistical tools that rely upon statistical simulation such as the bootstrap and Markov Chain Monte Carlo methods. Complementing the traditional production-like packages for statistical analysis are more methodologically oriented languages such as S and S-PLUS, and symbolic and algebraic calculation packages. Statistical journals and those in various fields of application devote considerable space to descriptions of such tools.

4. Statistics At The End Of The Twentieth Century

It is widely recognized that any statistical analysis can only be as good as the underlying data. Consequently, statisticians take great care in the the design of methods for data collection and in their actual implementation. Some of the most important modes of statistical data collection include censuses, experiments, observational studies, and sample Surveys, all of which are discussed elsewhere in this encyclopedia. Statistical experiments gain their strength and validity both through the random assignment of treatments to units and through the control of nontreatment variables. Similarly sample surveys gain their validity for generalization through the careful design of survey questionnaires and probability methods used for the selection of the sample units. Approaches to cope with the failure to fully implement randomization in experiments or random selection in sample surveys are discussed in Experimental Design: Compliance and Nonsampling Errors.

Data in some statistical studies are collected essentially at a single point in time (cross-sectional studies), while in others they are collected repeatedly at several time points or even continuously, while in yet others observations are collected sequentially, until sufficient information is available for inferential purposes. Different entries discuss these options and their strengths and weaknesses.

After a century of formal development, statistics as a field has developed a number of different approaches that rely on probability theory as a mathematical basis for description, analysis, and statistical inference. We provide an overview of some of these in the remainder of this section and provide some links to other entries in this encyclopedia.

4.1 Data Analysis

The least formal approach to inference is often the first employed. Its name stems from a famous article by John Tukey (1962), but it is rooted in the more traditional forms of descriptive statistical methods used for centuries.

Today, data analysis relies heavily on graphical methods and there are different traditions, such as those associated with

(a) The ‘exploratory data analysis’ methods suggested by Tukey and others.

(b) The more stylized correspondence analysis techniques of Benzecri and the French school.

(c) The alphabet soup of computer-based multivariate methods that have emerged over the past decade such as ACE, MARS, CART, etc.

No matter which ‘school’ of data analysis someone adheres to, the spirit of the methods is typically to encourage the data to ‘speak for themselves.’ While no theory of data analysis has emerged, and perhaps none is to be expected, the flexibility of thought and method embodied in the data analytic ideas have influenced all of the other approaches.

4.2 Frequentism

The name of this group of methods refers to a hypothetical infinite sequence of data sets generated as was the data set in question. Inferences are to be made with respect to this hypothetical infinite sequence. (For details, see Frequentist Inference).

One of the leading frequentist methods is significance testing, formalized initially by R. A. Fisher (1925) and subsequently elaborated upon and extended by Neyman and Pearson and others (see below). Here a null hypothesis is chosen, for example, that the mean, µ, of a normally distributed set of observations is 0. Fisher suggested the choice of a test statistic, e.g., based on the sample mean, x, and the calculation of the likelihood of observing an outcome as or more extreme as x is from µ 0, a quantity usually labeled as the p-value. When p is small (e.g., less than 5 percent), either a rare event has occurred or the null hypothesis is false. Within this theory, no probability can be given for which of these two conclusions is the case.

A related set of methods is testing hypotheses, as proposed by Neyman and Pearson (1928, 1932). In this approach, procedures are sought having the property that, for an infinite sequence of such sets, in only (say) 5 percent for would the null hypothesis be rejected if the null hypothesis were true. Often the infinite sequence is restricted to sets having the same sample size, but this is unnecessary. Here, in addition to the null hypothesis, an alternative hypothesis is specified. This permits the definition of a power curve, reflecting the frequency of rejecting the null hypothesis when the specified alternative is the case. But, as with the Fisherian approach, no probability can be given to either the null or the alternative hypotheses.

The construction of confidence intervals, following the proposal of Neyman (1934), is intimately related to testing hypotheses; indeed a 95 percent confidence interval may be regarded as the set of null hypotheses which, had they been tested at the 5 percent level of significance, would not have been rejected. A confidence interval is a random interval, having the property that the specified proportion (say 95 percent) of the infinite sequence, of random intervals would have covered the true value. For example, an interval that 95 percent of the time (by auxiliary randomization) is the whole real line, and 5 percent of the time is the empty set, is a valid 95 percent confidence interval.

Estimation of parameters—i.e., choosing a single value of the parameters that is in some sense best—is also an important frequentist method. Many methods have been proposed, both for particular models and as general approaches regardless of model, and their frequentist properties explored. These methods usually extended to intervals of values through inversion of test statistics or via other related devices. The resulting confidence intervals share many of the frequentist theoretical properties of the corresponding test procedures.

Frequentist statisticians have explored a number of general properties thought to be desirable in a procedure, such as invariance, unbiasedness, sufficiency, conditioning on ancillary statistics, etc. While each of these properties has examples in which it appears to produce satisfactory recommendations, there are others in which it does not. Additionally, these properties can conflict with each other. No general frequentist theory has emerged that proposes a hierarchy of desirable properties, leaving a frequentist without guidance in facing a new problem.

4.3 Likelihood Methods

The likelihood function (first studied systematically by R. A. Fisher) is the probability density of the data, viewed as a function of the parameters. It occupies an interesting middle ground in the philosophical debate, as it is used both by frequentists (as in maximum likelihood estimation) and by Bayesians in the transition from prior distributions to posterior distributions. A small group of scholars (among them G. A. Barnard, A. W. F. Edwards, R. Royall, D. Sprott) have proposed the likelihood function as an independent basis for inference. The issue of nuisance parameters has perplexed this group, since maximization, as would be consistent with maximum likelihood estimation, leads to different results in general than does integration, which would be consistent with Bayesian ideas.

4.4 Bayesian Methods

Both frequentists and Bayesians accept Bayes’ Theorem as correct, but Bayesians use it far more heavily. Bayesian analysis proceeds from the idea that probability is personal or subjective, reflecting the views of a particular person at a particular point in time. These views are summarized in the prior distribution over the parameter space. Together the prior distribution and the likelihood function define the joint distribution of the parameters and the data. This joint distribution can alternatively be factored as the product of the posterior distribution of the parameter given the data times the predictive distribution of the data.

In the past, Bayesian methods were deemed to be controversial because of the avowedly subjective nature of the prior distribution. But the controversy surrounding their use has lessened as recognition of the subjective nature of the likelihood has spread. Unlike frequentist methods, Bayesian methods are, in principle, free of the paradoxes and counterexamples that make classical statistics so perplexing. The development of hierarchical modeling and Markov Chain Monte Carlo (MCMC) methods have further added to the current popularity of the Bayesian approach, as they allow analyses of models that would otherwise be intractable.

Bayesian decision theory, which interacts closely with Bayesian statistical methods, is a useful way of modeling and addressing decision problems of experimental designs and data analysis and inference. It introduces the notion of utilities and the optimum decision combines probabilities of events with utilities by the calculation of expected utility and maximizing the latter (e.g., see the discussion in Lindley 2000).

Current research is attempting to use the Bayesian approach to hypothesis testing to provide tests and pvalues with good frequentist properties (see Bayarri and Berger 2000).

4.5 Broad Models: Nonparametrics And Semiparametrics

These models include parameter spaces of infinite dimensions, whether addressed in a frequentist or Bayesian manner. In a sense, these models put more inferential weight on the assumption of conditional independence than does an ordinary parametric model.

4.6 Some Cross-Cutting Themes

Often different fields of application of statistics need to address similar issues. For example, dimensionality of the parameter space is often a problem. As more parameters are added, the model will in general fit better (at least no worse). Is the apparent gain in accuracy worth the reduction in parsimony? There are many different ways to address this question in the various applied areas of statistics.

Another common theme, in some sense the obverse of the previous one, is the question of model selection and goodness of fit. In what sense can one say that a set of observations is well-approximated by a particular distribution? (cf. Goodness of Fit: Overview). All statistical theory relies at some level on the use of formal models, and the appropriateness of those models and their detailed specification are of concern to users of statistical methods, no matter which school of statistical inference they choose to work within.

5. Statistics In The Twenty-first Century

5.1 adapting and generalizing methodology.

Statistics as a field provides scientists with the basis for dealing with uncertainty, and, among other things, for generalizing from a sample to a population. There is a parallel sense in which statistics provides a basis for generalization: when similar tools are developed within specific substantive fields, such as experimental design methodology in agriculture and medicine, and sample surveys in economics and sociology. Statisticians have long recognized the common elements of such methodologies and have sought to develop generalized tools and theories to deal with these separate approaches (see e.g., Fienberg and Tanur 1989).

One hallmark of modern statistical science is the development of general frameworks that unify methodology. Thus the tools of Generalized Linear Models draw together methods for linear regression and analysis of various models with normal errors and those log-linear and logistic models for categorical data, in a broader and richer framework. Similarly, graphical models developed in the 1970s and 1980s use concepts of independence to integrate work in covariance section, decomposable log-linear models, and Markov random field models, and produce new methodology as a consequence. And the latent variable approaches from psychometrics and sociology have been tied with simultaneous equation and measurement error models from econometrics into a broader theory of covariance analysis and structural equations models.

Another hallmark of modern statistical science is the borrowing of methods in one field for application in another. One example is provided by Markov Chain Monte Carlo methods, now used widely in Bayesian statistics, which were first used in physics. Survival analysis, used in biostatistics to model the disease-free time or time-to-mortality of medical patients, and analyzed as reliability in quality control studies, are now used in econometrics to measure the time until an unemployed person gets a job. We anticipate that this trend of methodological borrowing will continue across fields of application.

5.2 Where Will New Statistical Developments Be Focused ?

In the issues of its year 2000 volume, the Journal of the American Statistical Association explored both the state of the art of statistics in diverse areas of application, and that of theory and methods, through a series of vignettes or short articles. These essays provide an excellent supplement to the entries of this encyclopedia on a wide range of topics, not only presenting a snapshot of the current state of play in selected areas of the field but also affecting some speculation on the next generation of developments. In an afterword to the last set of these vignettes, Casella (2000) summarizes five overarching themes that he observed in reading through the entire collection:

(a) Large datasets.

(b) High-dimensional/nonparametric models.

(c) Accessible computing.

(d) Bayes/frequentist/who cares?

(e) Theory/applied/why differentiate?

Not surprisingly, these themes fit well those that one can read into the statistical entries in this encyclopedia. The coming together of Bayesian and frequentist methods, for example, is illustrated by the movement of frequentists towards the use of hierarchical models and the regular consideration of frequentist properties of Bayesian procedures (e.g., Bayarri and Berger 2000). Similarly, MCMC methods are being widely used in non-Bayesian settings and, because they focus on long-run sequences of dependent draws from multivariate probability distributions, there are frequentist elements that are brought to bear in the study of the convergence of MCMC procedures. Thus the oft-made distinction between the different schools of statistical inference (suggested in the preceding section) is not always clear in the context of real applications.

5.3 The Growing Importance Of Statistics Across The Social And Behavioral Sciences

Statistics touches on an increasing number of fields of application, in the social sciences as in other areas of scholarship. Historically, the closest links have been with economics; together these fields share parentage of econometrics. There are now vigorous interactions with political science, law, sociology, psychology, anthropology, archeology, history, and many others.

In some fields, the development of statistical methods has not been universally welcomed. Using these methods well and knowledgeably requires an understanding both of the substantive field and of statistical methods. Sometimes this combination of skills has been difficult to develop.

Statistical methods are having increasing success in addressing questions throughout the social and behavioral sciences. Data are being collected and analyzed on an increasing variety of subjects, and the analyses are becoming increasingly sharply focused on the issues of interest.

We do not anticipate, nor would we find desirable, a future in which only statistical evidence was accepted in the social and behavioral sciences. There is room for, and need for, many different approaches. Nonetheless, we expect the excellent progress made in statistical methods in the social and behavioral sciences in recent decades to continue and intensify.

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effective use of statistics in research - methods and tools for data analysis

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Statistics plays a crucial role in research, providing the means to analyze, interpret, and draw meaningful conclusions from data. In this comprehensive article, we will delve into the effective use of statistics in research, focusing on the methods and tools available for data analysis. Whether you are a seasoned researcher or just starting in the field, understanding the principles of statistical analysis is essential for producing credible and valuable research outcomes

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• Published: 26 March 2024

Predicting and improving complex beer flavor through machine learning

• Michiel Schreurs   ORCID: orcid.org/0000-0002-9449-5619 1 , 2 , 3   na1 ,
• Supinya Piampongsant 1 , 2 , 3   na1 ,
• Miguel Roncoroni   ORCID: orcid.org/0000-0001-7461-1427 1 , 2 , 3   na1 ,
• Lloyd Cool   ORCID: orcid.org/0000-0001-9936-3124 1 , 2 , 3 , 4 ,
• Beatriz Herrera-Malaver   ORCID: orcid.org/0000-0002-5096-9974 1 , 2 , 3 ,
• Christophe Vanderaa   ORCID: orcid.org/0000-0001-7443-5427 4 ,
• Florian A. Theßeling 1 , 2 , 3 ,
• Łukasz Kreft   ORCID: orcid.org/0000-0001-7620-4657 5 ,
• Alexander Botzki   ORCID: orcid.org/0000-0001-6691-4233 5 ,
• Philippe Malcorps 6 ,
• Luk Daenen 6 ,
• Tom Wenseleers   ORCID: orcid.org/0000-0002-1434-861X 4 &
• Kevin J. Verstrepen   ORCID: orcid.org/0000-0002-3077-6219 1 , 2 , 3  

Nature Communications volume  15 , Article number:  2368 ( 2024 ) Cite this article

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The perception and appreciation of food flavor depends on many interacting chemical compounds and external factors, and therefore proves challenging to understand and predict. Here, we combine extensive chemical and sensory analyses of 250 different beers to train machine learning models that allow predicting flavor and consumer appreciation. For each beer, we measure over 200 chemical properties, perform quantitative descriptive sensory analysis with a trained tasting panel and map data from over 180,000 consumer reviews to train 10 different machine learning models. The best-performing algorithm, Gradient Boosting, yields models that significantly outperform predictions based on conventional statistics and accurately predict complex food features and consumer appreciation from chemical profiles. Model dissection allows identifying specific and unexpected compounds as drivers of beer flavor and appreciation. Adding these compounds results in variants of commercial alcoholic and non-alcoholic beers with improved consumer appreciation. Together, our study reveals how big data and machine learning uncover complex links between food chemistry, flavor and consumer perception, and lays the foundation to develop novel, tailored foods with superior flavors.

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Introduction

Predicting and understanding food perception and appreciation is one of the major challenges in food science. Accurate modeling of food flavor and appreciation could yield important opportunities for both producers and consumers, including quality control, product fingerprinting, counterfeit detection, spoilage detection, and the development of new products and product combinations (food pairing) 1 , 2 , 3 , 4 , 5 , 6 . Accurate models for flavor and consumer appreciation would contribute greatly to our scientific understanding of how humans perceive and appreciate flavor. Moreover, accurate predictive models would also facilitate and standardize existing food assessment methods and could supplement or replace assessments by trained and consumer tasting panels, which are variable, expensive and time-consuming 7 , 8 , 9 . Lastly, apart from providing objective, quantitative, accurate and contextual information that can help producers, models can also guide consumers in understanding their personal preferences 10 .

Despite the myriad of applications, predicting food flavor and appreciation from its chemical properties remains a largely elusive goal in sensory science, especially for complex food and beverages 11 , 12 . A key obstacle is the immense number of flavor-active chemicals underlying food flavor. Flavor compounds can vary widely in chemical structure and concentration, making them technically challenging and labor-intensive to quantify, even in the face of innovations in metabolomics, such as non-targeted metabolic fingerprinting 13 , 14 . Moreover, sensory analysis is perhaps even more complicated. Flavor perception is highly complex, resulting from hundreds of different molecules interacting at the physiochemical and sensorial level. Sensory perception is often non-linear, characterized by complex and concentration-dependent synergistic and antagonistic effects 15 , 16 , 17 , 18 , 19 , 20 , 21 that are further convoluted by the genetics, environment, culture and psychology of consumers 22 , 23 , 24 . Perceived flavor is therefore difficult to measure, with problems of sensitivity, accuracy, and reproducibility that can only be resolved by gathering sufficiently large datasets 25 . Trained tasting panels are considered the prime source of quality sensory data, but require meticulous training, are low throughput and high cost. Public databases containing consumer reviews of food products could provide a valuable alternative, especially for studying appreciation scores, which do not require formal training 25 . Public databases offer the advantage of amassing large amounts of data, increasing the statistical power to identify potential drivers of appreciation. However, public datasets suffer from biases, including a bias in the volunteers that contribute to the database, as well as confounding factors such as price, cult status and psychological conformity towards previous ratings of the product.

Classical multivariate statistics and machine learning methods have been used to predict flavor of specific compounds by, for example, linking structural properties of a compound to its potential biological activities or linking concentrations of specific compounds to sensory profiles 1 , 26 . Importantly, most previous studies focused on predicting organoleptic properties of single compounds (often based on their chemical structure) 27 , 28 , 29 , 30 , 31 , 32 , 33 , thus ignoring the fact that these compounds are present in a complex matrix in food or beverages and excluding complex interactions between compounds. Moreover, the classical statistics commonly used in sensory science 34 , 35 , 36 , 37 , 38 , 39 require a large sample size and sufficient variance amongst predictors to create accurate models. They are not fit for studying an extensive set of hundreds of interacting flavor compounds, since they are sensitive to outliers, have a high tendency to overfit and are less suited for non-linear and discontinuous relationships 40 .

In this study, we combine extensive chemical analyses and sensory data of a set of different commercial beers with machine learning approaches to develop models that predict taste, smell, mouthfeel and appreciation from compound concentrations. Beer is particularly suited to model the relationship between chemistry, flavor and appreciation. First, beer is a complex product, consisting of thousands of flavor compounds that partake in complex sensory interactions 41 , 42 , 43 . This chemical diversity arises from the raw materials (malt, yeast, hops, water and spices) and biochemical conversions during the brewing process (kilning, mashing, boiling, fermentation, maturation and aging) 44 , 45 . Second, the advent of the internet saw beer consumers embrace online review platforms, such as RateBeer (ZX Ventures, Anheuser-Busch InBev SA/NV) and BeerAdvocate (Next Glass, inc.). In this way, the beer community provides massive data sets of beer flavor and appreciation scores, creating extraordinarily large sensory databases to complement the analyses of our professional sensory panel. Specifically, we characterize over 200 chemical properties of 250 commercial beers, spread across 22 beer styles, and link these to the descriptive sensory profiling data of a 16-person in-house trained tasting panel and data acquired from over 180,000 public consumer reviews. These unique and extensive datasets enable us to train a suite of machine learning models to predict flavor and appreciation from a beer’s chemical profile. Dissection of the best-performing models allows us to pinpoint specific compounds as potential drivers of beer flavor and appreciation. Follow-up experiments confirm the importance of these compounds and ultimately allow us to significantly improve the flavor and appreciation of selected commercial beers. Together, our study represents a significant step towards understanding complex flavors and reinforces the value of machine learning to develop and refine complex foods. In this way, it represents a stepping stone for further computer-aided food engineering applications 46 .

To generate a comprehensive dataset on beer flavor, we selected 250 commercial Belgian beers across 22 different beer styles (Supplementary Fig.  S1 ). Beers with ≤ 4.2% alcohol by volume (ABV) were classified as non-alcoholic and low-alcoholic. Blonds and Tripels constitute a significant portion of the dataset (12.4% and 11.2%, respectively) reflecting their presence on the Belgian beer market and the heterogeneity of beers within these styles. By contrast, lager beers are less diverse and dominated by a handful of brands. Rare styles such as Brut or Faro make up only a small fraction of the dataset (2% and 1%, respectively) because fewer of these beers are produced and because they are dominated by distinct characteristics in terms of flavor and chemical composition.

Extensive analysis identifies relationships between chemical compounds in beer

For each beer, we measured 226 different chemical properties, including common brewing parameters such as alcohol content, iso-alpha acids, pH, sugar concentration 47 , and over 200 flavor compounds (Methods, Supplementary Table  S1 ). A large portion (37.2%) are terpenoids arising from hopping, responsible for herbal and fruity flavors 16 , 48 . A second major category are yeast metabolites, such as esters and alcohols, that result in fruity and solvent notes 48 , 49 , 50 . Other measured compounds are primarily derived from malt, or other microbes such as non- Saccharomyces yeasts and bacteria (‘wild flora’). Compounds that arise from spices or staling are labeled under ‘Others’. Five attributes (caloric value, total acids and total ester, hop aroma and sulfur compounds) are calculated from multiple individually measured compounds.

As a first step in identifying relationships between chemical properties, we determined correlations between the concentrations of the compounds (Fig.  1 , upper panel, Supplementary Data  1 and 2 , and Supplementary Fig.  S2 . For the sake of clarity, only a subset of the measured compounds is shown in Fig.  1 ). Compounds of the same origin typically show a positive correlation, while absence of correlation hints at parameters varying independently. For example, the hop aroma compounds citronellol, and alpha-terpineol show moderate correlations with each other (Spearman’s rho=0.39 and 0.57), but not with the bittering hop component iso-alpha acids (Spearman’s rho=0.16 and −0.07). This illustrates how brewers can independently modify hop aroma and bitterness by selecting hop varieties and dosage time. If hops are added early in the boiling phase, chemical conversions increase bitterness while aromas evaporate, conversely, late addition of hops preserves aroma but limits bitterness 51 . Similarly, hop-derived iso-alpha acids show a strong anti-correlation with lactic acid and acetic acid, likely reflecting growth inhibition of lactic acid and acetic acid bacteria, or the consequent use of fewer hops in sour beer styles, such as West Flanders ales and Fruit beers, that rely on these bacteria for their distinct flavors 52 . Finally, yeast-derived esters (ethyl acetate, ethyl decanoate, ethyl hexanoate, ethyl octanoate) and alcohols (ethanol, isoamyl alcohol, isobutanol, and glycerol), correlate with Spearman coefficients above 0.5, suggesting that these secondary metabolites are correlated with the yeast genetic background and/or fermentation parameters and may be difficult to influence individually, although the choice of yeast strain may offer some control 53 .

Spearman rank correlations are shown. Descriptors are grouped according to their origin (malt (blue), hops (green), yeast (red), wild flora (yellow), Others (black)), and sensory aspect (aroma, taste, palate, and overall appreciation). Please note that for the chemical compounds, for the sake of clarity, only a subset of the total number of measured compounds is shown, with an emphasis on the key compounds for each source. For more details, see the main text and Methods section. Chemical data can be found in Supplementary Data  1 , correlations between all chemical compounds are depicted in Supplementary Fig.  S2 and correlation values can be found in Supplementary Data  2 . See Supplementary Data  4 for sensory panel assessments and Supplementary Data  5 for correlation values between all sensory descriptors.

Interestingly, different beer styles show distinct patterns for some flavor compounds (Supplementary Fig.  S3 ). These observations agree with expectations for key beer styles, and serve as a control for our measurements. For instance, Stouts generally show high values for color (darker), while hoppy beers contain elevated levels of iso-alpha acids, compounds associated with bitter hop taste. Acetic and lactic acid are not prevalent in most beers, with notable exceptions such as Kriek, Lambic, Faro, West Flanders ales and Flanders Old Brown, which use acid-producing bacteria ( Lactobacillus and Pediococcus ) or unconventional yeast ( Brettanomyces ) 54 , 55 . Glycerol, ethanol and esters show similar distributions across all beer styles, reflecting their common origin as products of yeast metabolism during fermentation 45 , 53 . Finally, low/no-alcohol beers contain low concentrations of glycerol and esters. This is in line with the production process for most of the low/no-alcohol beers in our dataset, which are produced through limiting fermentation or by stripping away alcohol via evaporation or dialysis, with both methods having the unintended side-effect of reducing the amount of flavor compounds in the final beer 56 , 57 .

Besides expected associations, our data also reveals less trivial associations between beer styles and specific parameters. For example, geraniol and citronellol, two monoterpenoids responsible for citrus, floral and rose flavors and characteristic of Citra hops, are found in relatively high amounts in Christmas, Saison, and Brett/co-fermented beers, where they may originate from terpenoid-rich spices such as coriander seeds instead of hops 58 .

Tasting panel assessments reveal sensorial relationships in beer

To assess the sensory profile of each beer, a trained tasting panel evaluated each of the 250 beers for 50 sensory attributes, including different hop, malt and yeast flavors, off-flavors and spices. Panelists used a tasting sheet (Supplementary Data  3 ) to score the different attributes. Panel consistency was evaluated by repeating 12 samples across different sessions and performing ANOVA. In 95% of cases no significant difference was found across sessions ( p  > 0.05), indicating good panel consistency (Supplementary Table  S2 ).

Aroma and taste perception reported by the trained panel are often linked (Fig.  1 , bottom left panel and Supplementary Data  4 and 5 ), with high correlations between hops aroma and taste (Spearman’s rho=0.83). Bitter taste was found to correlate with hop aroma and taste in general (Spearman’s rho=0.80 and 0.69), and particularly with “grassy” noble hops (Spearman’s rho=0.75). Barnyard flavor, most often associated with sour beers, is identified together with stale hops (Spearman’s rho=0.97) that are used in these beers. Lactic and acetic acid, which often co-occur, are correlated (Spearman’s rho=0.66). Interestingly, sweetness and bitterness are anti-correlated (Spearman’s rho = −0.48), confirming the hypothesis that they mask each other 59 , 60 . Beer body is highly correlated with alcohol (Spearman’s rho = 0.79), and overall appreciation is found to correlate with multiple aspects that describe beer mouthfeel (alcohol, carbonation; Spearman’s rho= 0.32, 0.39), as well as with hop and ester aroma intensity (Spearman’s rho=0.39 and 0.35).

Similar to the chemical analyses, sensorial analyses confirmed typical features of specific beer styles (Supplementary Fig.  S4 ). For example, sour beers (Faro, Flanders Old Brown, Fruit beer, Kriek, Lambic, West Flanders ale) were rated acidic, with flavors of both acetic and lactic acid. Hoppy beers were found to be bitter and showed hop-associated aromas like citrus and tropical fruit. Malt taste is most detected among scotch, stout/porters, and strong ales, while low/no-alcohol beers, which often have a reputation for being ‘worty’ (reminiscent of unfermented, sweet malt extract) appear in the middle. Unsurprisingly, hop aromas are most strongly detected among hoppy beers. Like its chemical counterpart (Supplementary Fig.  S3 ), acidity shows a right-skewed distribution, with the most acidic beers being Krieks, Lambics, and West Flanders ales.

Tasting panel assessments of specific flavors correlate with chemical composition

We find that the concentrations of several chemical compounds strongly correlate with specific aroma or taste, as evaluated by the tasting panel (Fig.  2 , Supplementary Fig.  S5 , Supplementary Data  6 ). In some cases, these correlations confirm expectations and serve as a useful control for data quality. For example, iso-alpha acids, the bittering compounds in hops, strongly correlate with bitterness (Spearman’s rho=0.68), while ethanol and glycerol correlate with tasters’ perceptions of alcohol and body, the mouthfeel sensation of fullness (Spearman’s rho=0.82/0.62 and 0.72/0.57 respectively) and darker color from roasted malts is a good indication of malt perception (Spearman’s rho=0.54).

Heatmap colors indicate Spearman’s Rho. Axes are organized according to sensory categories (aroma, taste, mouthfeel, overall), chemical categories and chemical sources in beer (malt (blue), hops (green), yeast (red), wild flora (yellow), Others (black)). See Supplementary Data  6 for all correlation values.

Interestingly, for some relationships between chemical compounds and perceived flavor, correlations are weaker than expected. For example, the rose-smelling phenethyl acetate only weakly correlates with floral aroma. This hints at more complex relationships and interactions between compounds and suggests a need for a more complex model than simple correlations. Lastly, we uncovered unexpected correlations. For instance, the esters ethyl decanoate and ethyl octanoate appear to correlate slightly with hop perception and bitterness, possibly due to their fruity flavor. Iron is anti-correlated with hop aromas and bitterness, most likely because it is also anti-correlated with iso-alpha acids. This could be a sign of metal chelation of hop acids 61 , given that our analyses measure unbound hop acids and total iron content, or could result from the higher iron content in dark and Fruit beers, which typically have less hoppy and bitter flavors 62 .

Public consumer reviews complement expert panel data

To complement and expand the sensory data of our trained tasting panel, we collected 180,000 reviews of our 250 beers from the online consumer review platform RateBeer. This provided numerical scores for beer appearance, aroma, taste, palate, overall quality as well as the average overall score.

Public datasets are known to suffer from biases, such as price, cult status and psychological conformity towards previous ratings of a product. For example, prices correlate with appreciation scores for these online consumer reviews (rho=0.49, Supplementary Fig.  S6 ), but not for our trained tasting panel (rho=0.19). This suggests that prices affect consumer appreciation, which has been reported in wine 63 , while blind tastings are unaffected. Moreover, we observe that some beer styles, like lagers and non-alcoholic beers, generally receive lower scores, reflecting that online reviewers are mostly beer aficionados with a preference for specialty beers over lager beers. In general, we find a modest correlation between our trained panel’s overall appreciation score and the online consumer appreciation scores (Fig.  3 , rho=0.29). Apart from the aforementioned biases in the online datasets, serving temperature, sample freshness and surroundings, which are all tightly controlled during the tasting panel sessions, can vary tremendously across online consumers and can further contribute to (among others, appreciation) differences between the two categories of tasters. Importantly, in contrast to the overall appreciation scores, for many sensory aspects the results from the professional panel correlated well with results obtained from RateBeer reviews. Correlations were highest for features that are relatively easy to recognize even for untrained tasters, like bitterness, sweetness, alcohol and malt aroma (Fig.  3 and below).

RateBeer text mining results can be found in Supplementary Data  7 . Rho values shown are Spearman correlation values, with asterisks indicating significant correlations ( p  < 0.05, two-sided). All p values were smaller than 0.001, except for Esters aroma (0.0553), Esters taste (0.3275), Esters aroma—banana (0.0019), Coriander (0.0508) and Diacetyl (0.0134).

Besides collecting consumer appreciation from these online reviews, we developed automated text analysis tools to gather additional data from review texts (Supplementary Data  7 ). Processing review texts on the RateBeer database yielded comparable results to the scores given by the trained panel for many common sensory aspects, including acidity, bitterness, sweetness, alcohol, malt, and hop tastes (Fig.  3 ). This is in line with what would be expected, since these attributes require less training for accurate assessment and are less influenced by environmental factors such as temperature, serving glass and odors in the environment. Consumer reviews also correlate well with our trained panel for 4-vinyl guaiacol, a compound associated with a very characteristic aroma. By contrast, correlations for more specific aromas like ester, coriander or diacetyl are underrepresented in the online reviews, underscoring the importance of using a trained tasting panel and standardized tasting sheets with explicit factors to be scored for evaluating specific aspects of a beer. Taken together, our results suggest that public reviews are trustworthy for some, but not all, flavor features and can complement or substitute taste panel data for these sensory aspects.

Models can predict beer sensory profiles from chemical data

The rich datasets of chemical analyses, tasting panel assessments and public reviews gathered in the first part of this study provided us with a unique opportunity to develop predictive models that link chemical data to sensorial features. Given the complexity of beer flavor, basic statistical tools such as correlations or linear regression may not always be the most suitable for making accurate predictions. Instead, we applied different machine learning models that can model both simple linear and complex interactive relationships. Specifically, we constructed a set of regression models to predict (a) trained panel scores for beer flavor and quality and (b) public reviews’ appreciation scores from beer chemical profiles. We trained and tested 10 different models (Methods), 3 linear regression-based models (simple linear regression with first-order interactions (LR), lasso regression with first-order interactions (Lasso), partial least squares regressor (PLSR)), 5 decision tree models (AdaBoost regressor (ABR), extra trees (ET), gradient boosting regressor (GBR), random forest (RF) and XGBoost regressor (XGBR)), 1 support vector regression (SVR), and 1 artificial neural network (ANN) model.

To compare the performance of our machine learning models, the dataset was randomly split into a training and test set, stratified by beer style. After a model was trained on data in the training set, its performance was evaluated on its ability to predict the test dataset obtained from multi-output models (based on the coefficient of determination, see Methods). Additionally, individual-attribute models were ranked per descriptor and the average rank was calculated, as proposed by Korneva et al. 64 . Importantly, both ways of evaluating the models’ performance agreed in general. Performance of the different models varied (Table  1 ). It should be noted that all models perform better at predicting RateBeer results than results from our trained tasting panel. One reason could be that sensory data is inherently variable, and this variability is averaged out with the large number of public reviews from RateBeer. Additionally, all tree-based models perform better at predicting taste than aroma. Linear models (LR) performed particularly poorly, with negative R 2 values, due to severe overfitting (training set R 2  = 1). Overfitting is a common issue in linear models with many parameters and limited samples, especially with interaction terms further amplifying the number of parameters. L1 regularization (Lasso) successfully overcomes this overfitting, out-competing multiple tree-based models on the RateBeer dataset. Similarly, the dimensionality reduction of PLSR avoids overfitting and improves performance, to some extent. Still, tree-based models (ABR, ET, GBR, RF and XGBR) show the best performance, out-competing the linear models (LR, Lasso, PLSR) commonly used in sensory science 65 .

GBR models showed the best overall performance in predicting sensory responses from chemical information, with R 2 values up to 0.75 depending on the predicted sensory feature (Supplementary Table  S4 ). The GBR models predict consumer appreciation (RateBeer) better than our trained panel’s appreciation (R 2 value of 0.67 compared to R 2 value of 0.09) (Supplementary Table  S3 and Supplementary Table  S4 ). ANN models showed intermediate performance, likely because neural networks typically perform best with larger datasets 66 . The SVR shows intermediate performance, mostly due to the weak predictions of specific attributes that lower the overall performance (Supplementary Table  S4 ).

Model dissection identifies specific, unexpected compounds as drivers of consumer appreciation

Next, we leveraged our models to infer important contributors to sensory perception and consumer appreciation. Consumer preference is a crucial sensory aspects, because a product that shows low consumer appreciation scores often does not succeed commercially 25 . Additionally, the requirement for a large number of representative evaluators makes consumer trials one of the more costly and time-consuming aspects of product development. Hence, a model for predicting chemical drivers of overall appreciation would be a welcome addition to the available toolbox for food development and optimization.

Since GBR models on our RateBeer dataset showed the best overall performance, we focused on these models. Specifically, we used two approaches to identify important contributors. First, rankings of the most important predictors for each sensorial trait in the GBR models were obtained based on impurity-based feature importance (mean decrease in impurity). High-ranked parameters were hypothesized to be either the true causal chemical properties underlying the trait, to correlate with the actual causal properties, or to take part in sensory interactions affecting the trait 67 (Fig.  4A ). In a second approach, we used SHAP 68 to determine which parameters contributed most to the model for making predictions of consumer appreciation (Fig.  4B ). SHAP calculates parameter contributions to model predictions on a per-sample basis, which can be aggregated into an importance score.

A The impurity-based feature importance (mean deviance in impurity, MDI) calculated from the Gradient Boosting Regression (GBR) model predicting RateBeer appreciation scores. The top 15 highest ranked chemical properties are shown. B SHAP summary plot for the top 15 parameters contributing to our GBR model. Each point on the graph represents a sample from our dataset. The color represents the concentration of that parameter, with bluer colors representing low values and redder colors representing higher values. Greater absolute values on the horizontal axis indicate a higher impact of the parameter on the prediction of the model. C Spearman correlations between the 15 most important chemical properties and consumer overall appreciation. Numbers indicate the Spearman Rho correlation coefficient, and the rank of this correlation compared to all other correlations. The top 15 important compounds were determined using SHAP (panel B).

Both approaches identified ethyl acetate as the most predictive parameter for beer appreciation (Fig.  4 ). Ethyl acetate is the most abundant ester in beer with a typical ‘fruity’, ‘solvent’ and ‘alcoholic’ flavor, but is often considered less important than other esters like isoamyl acetate. The second most important parameter identified by SHAP is ethanol, the most abundant beer compound after water. Apart from directly contributing to beer flavor and mouthfeel, ethanol drastically influences the physical properties of beer, dictating how easily volatile compounds escape the beer matrix to contribute to beer aroma 69 . Importantly, it should also be noted that the importance of ethanol for appreciation is likely inflated by the very low appreciation scores of non-alcoholic beers (Supplementary Fig.  S4 ). Despite not often being considered a driver of beer appreciation, protein level also ranks highly in both approaches, possibly due to its effect on mouthfeel and body 70 . Lactic acid, which contributes to the tart taste of sour beers, is the fourth most important parameter identified by SHAP, possibly due to the generally high appreciation of sour beers in our dataset.

Interestingly, some of the most important predictive parameters for our model are not well-established as beer flavors or are even commonly regarded as being negative for beer quality. For example, our models identify methanethiol and ethyl phenyl acetate, an ester commonly linked to beer staling 71 , as a key factor contributing to beer appreciation. Although there is no doubt that high concentrations of these compounds are considered unpleasant, the positive effects of modest concentrations are not yet known 72 , 73 .

To compare our approach to conventional statistics, we evaluated how well the 15 most important SHAP-derived parameters correlate with consumer appreciation (Fig.  4C ). Interestingly, only 6 of the properties derived by SHAP rank amongst the top 15 most correlated parameters. For some chemical compounds, the correlations are so low that they would have likely been considered unimportant. For example, lactic acid, the fourth most important parameter, shows a bimodal distribution for appreciation, with sour beers forming a separate cluster, that is missed entirely by the Spearman correlation. Additionally, the correlation plots reveal outliers, emphasizing the need for robust analysis tools. Together, this highlights the need for alternative models, like the Gradient Boosting model, that better grasp the complexity of (beer) flavor.

Finally, to observe the relationships between these chemical properties and their predicted targets, partial dependence plots were constructed for the six most important predictors of consumer appreciation 74 , 75 , 76 (Supplementary Fig.  S7 ). One-way partial dependence plots show how a change in concentration affects the predicted appreciation. These plots reveal an important limitation of our models: appreciation predictions remain constant at ever-increasing concentrations. This implies that once a threshold concentration is reached, further increasing the concentration does not affect appreciation. This is false, as it is well-documented that certain compounds become unpleasant at high concentrations, including ethyl acetate (‘nail polish’) 77 and methanethiol (‘sulfury’ and ‘rotten cabbage’) 78 . The inability of our models to grasp that flavor compounds have optimal levels, above which they become negative, is a consequence of working with commercial beer brands where (off-)flavors are rarely too high to negatively impact the product. The two-way partial dependence plots show how changing the concentration of two compounds influences predicted appreciation, visualizing their interactions (Supplementary Fig.  S7 ). In our case, the top 5 parameters are dominated by additive or synergistic interactions, with high concentrations for both compounds resulting in the highest predicted appreciation.

To assess the robustness of our best-performing models and model predictions, we performed 100 iterations of the GBR, RF and ET models. In general, all iterations of the models yielded similar performance (Supplementary Fig.  S8 ). Moreover, the main predictors (including the top predictors ethanol and ethyl acetate) remained virtually the same, especially for GBR and RF. For the iterations of the ET model, we did observe more variation in the top predictors, which is likely a consequence of the model’s inherent random architecture in combination with co-correlations between certain predictors. However, even in this case, several of the top predictors (ethanol and ethyl acetate) remain unchanged, although their rank in importance changes (Supplementary Fig.  S8 ).

Next, we investigated if a combination of RateBeer and trained panel data into one consolidated dataset would lead to stronger models, under the hypothesis that such a model would suffer less from bias in the datasets. A GBR model was trained to predict appreciation on the combined dataset. This model underperformed compared to the RateBeer model, both in the native case and when including a dataset identifier (R 2  = 0.67, 0.26 and 0.42 respectively). For the latter, the dataset identifier is the most important feature (Supplementary Fig.  S9 ), while most of the feature importance remains unchanged, with ethyl acetate and ethanol ranking highest, like in the original model trained only on RateBeer data. It seems that the large variation in the panel dataset introduces noise, weakening the models’ performances and reliability. In addition, it seems reasonable to assume that both datasets are fundamentally different, with the panel dataset obtained by blind tastings by a trained professional panel.

Lastly, we evaluated whether beer style identifiers would further enhance the model’s performance. A GBR model was trained with parameters that explicitly encoded the styles of the samples. This did not improve model performance (R2 = 0.66 with style information vs R2 = 0.67). The most important chemical features are consistent with the model trained without style information (eg. ethanol and ethyl acetate), and with the exception of the most preferred (strong ale) and least preferred (low/no-alcohol) styles, none of the styles were among the most important features (Supplementary Fig.  S9 , Supplementary Table  S5 and S6 ). This is likely due to a combination of style-specific chemical signatures, such as iso-alpha acids and lactic acid, that implicitly convey style information to the original models, as well as the low number of samples belonging to some styles, making it difficult for the model to learn style-specific patterns. Moreover, beer styles are not rigorously defined, with some styles overlapping in features and some beers being misattributed to a specific style, all of which leads to more noise in models that use style parameters.

Model validation

To test if our predictive models give insight into beer appreciation, we set up experiments aimed at improving existing commercial beers. We specifically selected overall appreciation as the trait to be examined because of its complexity and commercial relevance. Beer flavor comprises a complex bouquet rather than single aromas and tastes 53 . Hence, adding a single compound to the extent that a difference is noticeable may lead to an unbalanced, artificial flavor. Therefore, we evaluated the effect of combinations of compounds. Because Blond beers represent the most extensive style in our dataset, we selected a beer from this style as the starting material for these experiments (Beer 64 in Supplementary Data  1 ).

In the first set of experiments, we adjusted the concentrations of compounds that made up the most important predictors of overall appreciation (ethyl acetate, ethanol, lactic acid, ethyl phenyl acetate) together with correlated compounds (ethyl hexanoate, isoamyl acetate, glycerol), bringing them up to 95 th percentile ethanol-normalized concentrations (Methods) within the Blond group (‘Spiked’ concentration in Fig.  5A ). Compared to controls, the spiked beers were found to have significantly improved overall appreciation among trained panelists, with panelist noting increased intensity of ester flavors, sweetness, alcohol, and body fullness (Fig.  5B ). To disentangle the contribution of ethanol to these results, a second experiment was performed without the addition of ethanol. This resulted in a similar outcome, including increased perception of alcohol and overall appreciation.

Adding the top chemical compounds, identified as best predictors of appreciation by our model, into poorly appreciated beers results in increased appreciation from our trained panel. Results of sensory tests between base beers and those spiked with compounds identified as the best predictors by the model. A Blond and Non/Low-alcohol (0.0% ABV) base beers were brought up to 95th-percentile ethanol-normalized concentrations within each style. B For each sensory attribute, tasters indicated the more intense sample and selected the sample they preferred. The numbers above the bars correspond to the p values that indicate significant changes in perceived flavor (two-sided binomial test: alpha 0.05, n  = 20 or 13).

In a last experiment, we tested whether using the model’s predictions can boost the appreciation of a non-alcoholic beer (beer 223 in Supplementary Data  1 ). Again, the addition of a mixture of predicted compounds (omitting ethanol, in this case) resulted in a significant increase in appreciation, body, ester flavor and sweetness.

Predicting flavor and consumer appreciation from chemical composition is one of the ultimate goals of sensory science. A reliable, systematic and unbiased way to link chemical profiles to flavor and food appreciation would be a significant asset to the food and beverage industry. Such tools would substantially aid in quality control and recipe development, offer an efficient and cost-effective alternative to pilot studies and consumer trials and would ultimately allow food manufacturers to produce superior, tailor-made products that better meet the demands of specific consumer groups more efficiently.

A limited set of studies have previously tried, to varying degrees of success, to predict beer flavor and beer popularity based on (a limited set of) chemical compounds and flavors 79 , 80 . Current sensitive, high-throughput technologies allow measuring an unprecedented number of chemical compounds and properties in a large set of samples, yielding a dataset that can train models that help close the gaps between chemistry and flavor, even for a complex natural product like beer. To our knowledge, no previous research gathered data at this scale (250 samples, 226 chemical parameters, 50 sensory attributes and 5 consumer scores) to disentangle and validate the chemical aspects driving beer preference using various machine-learning techniques. We find that modern machine learning models outperform conventional statistical tools, such as correlations and linear models, and can successfully predict flavor appreciation from chemical composition. This could be attributed to the natural incorporation of interactions and non-linear or discontinuous effects in machine learning models, which are not easily grasped by the linear model architecture. While linear models and partial least squares regression represent the most widespread statistical approaches in sensory science, in part because they allow interpretation 65 , 81 , 82 , modern machine learning methods allow for building better predictive models while preserving the possibility to dissect and exploit the underlying patterns. Of the 10 different models we trained, tree-based models, such as our best performing GBR, showed the best overall performance in predicting sensory responses from chemical information, outcompeting artificial neural networks. This agrees with previous reports for models trained on tabular data 83 . Our results are in line with the findings of Colantonio et al. who also identified the gradient boosting architecture as performing best at predicting appreciation and flavor (of tomatoes and blueberries, in their specific study) 26 . Importantly, besides our larger experimental scale, we were able to directly confirm our models’ predictions in vivo.

Our study confirms that flavor compound concentration does not always correlate with perception, suggesting complex interactions that are often missed by more conventional statistics and simple models. Specifically, we find that tree-based algorithms may perform best in developing models that link complex food chemistry with aroma. Furthermore, we show that massive datasets of untrained consumer reviews provide a valuable source of data, that can complement or even replace trained tasting panels, especially for appreciation and basic flavors, such as sweetness and bitterness. This holds despite biases that are known to occur in such datasets, such as price or conformity bias. Moreover, GBR models predict taste better than aroma. This is likely because taste (e.g. bitterness) often directly relates to the corresponding chemical measurements (e.g., iso-alpha acids), whereas such a link is less clear for aromas, which often result from the interplay between multiple volatile compounds. We also find that our models are best at predicting acidity and alcohol, likely because there is a direct relation between the measured chemical compounds (acids and ethanol) and the corresponding perceived sensorial attribute (acidity and alcohol), and because even untrained consumers are generally able to recognize these flavors and aromas.

The predictions of our final models, trained on review data, hold even for blind tastings with small groups of trained tasters, as demonstrated by our ability to validate specific compounds as drivers of beer flavor and appreciation. Since adding a single compound to the extent of a noticeable difference may result in an unbalanced flavor profile, we specifically tested our identified key drivers as a combination of compounds. While this approach does not allow us to validate if a particular single compound would affect flavor and/or appreciation, our experiments do show that this combination of compounds increases consumer appreciation.

It is important to stress that, while it represents an important step forward, our approach still has several major limitations. A key weakness of the GBR model architecture is that amongst co-correlating variables, the largest main effect is consistently preferred for model building. As a result, co-correlating variables often have artificially low importance scores, both for impurity and SHAP-based methods, like we observed in the comparison to the more randomized Extra Trees models. This implies that chemicals identified as key drivers of a specific sensory feature by GBR might not be the true causative compounds, but rather co-correlate with the actual causative chemical. For example, the high importance of ethyl acetate could be (partially) attributed to the total ester content, ethanol or ethyl hexanoate (rho=0.77, rho=0.72 and rho=0.68), while ethyl phenylacetate could hide the importance of prenyl isobutyrate and ethyl benzoate (rho=0.77 and rho=0.76). Expanding our GBR model to include beer style as a parameter did not yield additional power or insight. This is likely due to style-specific chemical signatures, such as iso-alpha acids and lactic acid, that implicitly convey style information to the original model, as well as the smaller sample size per style, limiting the power to uncover style-specific patterns. This can be partly attributed to the curse of dimensionality, where the high number of parameters results in the models mainly incorporating single parameter effects, rather than complex interactions such as style-dependent effects 67 . A larger number of samples may overcome some of these limitations and offer more insight into style-specific effects. On the other hand, beer style is not a rigid scientific classification, and beers within one style often differ a lot, which further complicates the analysis of style as a model factor.

Our study is limited to beers from Belgian breweries. Although these beers cover a large portion of the beer styles available globally, some beer styles and consumer patterns may be missing, while other features might be overrepresented. For example, many Belgian ales exhibit yeast-driven flavor profiles, which is reflected in the chemical drivers of appreciation discovered by this study. In future work, expanding the scope to include diverse markets and beer styles could lead to the identification of even more drivers of appreciation and better models for special niche products that were not present in our beer set.

In addition to inherent limitations of GBR models, there are also some limitations associated with studying food aroma. Even if our chemical analyses measured most of the known aroma compounds, the total number of flavor compounds in complex foods like beer is still larger than the subset we were able to measure in this study. For example, hop-derived thiols, that influence flavor at very low concentrations, are notoriously difficult to measure in a high-throughput experiment. Moreover, consumer perception remains subjective and prone to biases that are difficult to avoid. It is also important to stress that the models are still immature and that more extensive datasets will be crucial for developing more complete models in the future. Besides more samples and parameters, our dataset does not include any demographic information about the tasters. Including such data could lead to better models that grasp external factors like age and culture. Another limitation is that our set of beers consists of high-quality end-products and lacks beers that are unfit for sale, which limits the current model in accurately predicting products that are appreciated very badly. Finally, while models could be readily applied in quality control, their use in sensory science and product development is restrained by their inability to discern causal relationships. Given that the models cannot distinguish compounds that genuinely drive consumer perception from those that merely correlate, validation experiments are essential to identify true causative compounds.

Despite the inherent limitations, dissection of our models enabled us to pinpoint specific molecules as potential drivers of beer aroma and consumer appreciation, including compounds that were unexpected and would not have been identified using standard approaches. Important drivers of beer appreciation uncovered by our models include protein levels, ethyl acetate, ethyl phenyl acetate and lactic acid. Currently, many brewers already use lactic acid to acidify their brewing water and ensure optimal pH for enzymatic activity during the mashing process. Our results suggest that adding lactic acid can also improve beer appreciation, although its individual effect remains to be tested. Interestingly, ethanol appears to be unnecessary to improve beer appreciation, both for blond beer and alcohol-free beer. Given the growing consumer interest in alcohol-free beer, with a predicted annual market growth of >7% 84 , it is relevant for brewers to know what compounds can further increase consumer appreciation of these beers. Hence, our model may readily provide avenues to further improve the flavor and consumer appreciation of both alcoholic and non-alcoholic beers, which is generally considered one of the key challenges for future beer production.

Whereas we see a direct implementation of our results for the development of superior alcohol-free beverages and other food products, our study can also serve as a stepping stone for the development of novel alcohol-containing beverages. We want to echo the growing body of scientific evidence for the negative effects of alcohol consumption, both on the individual level by the mutagenic, teratogenic and carcinogenic effects of ethanol 85 , 86 , as well as the burden on society caused by alcohol abuse and addiction. We encourage the use of our results for the production of healthier, tastier products, including novel and improved beverages with lower alcohol contents. Furthermore, we strongly discourage the use of these technologies to improve the appreciation or addictive properties of harmful substances.

The present work demonstrates that despite some important remaining hurdles, combining the latest developments in chemical analyses, sensory analysis and modern machine learning methods offers exciting avenues for food chemistry and engineering. Soon, these tools may provide solutions in quality control and recipe development, as well as new approaches to sensory science and flavor research.

Beer selection

250 commercial Belgian beers were selected to cover the broad diversity of beer styles and corresponding diversity in chemical composition and aroma. See Supplementary Fig.  S1 .

Chemical dataset

Sample preparation.

Beers within their expiration date were purchased from commercial retailers. Samples were prepared in biological duplicates at room temperature, unless explicitly stated otherwise. Bottle pressure was measured with a manual pressure device (Steinfurth Mess-Systeme GmbH) and used to calculate CO 2 concentration. The beer was poured through two filter papers (Macherey-Nagel, 500713032 MN 713 ¼) to remove carbon dioxide and prevent spontaneous foaming. Samples were then prepared for measurements by targeted Headspace-Gas Chromatography-Flame Ionization Detector/Flame Photometric Detector (HS-GC-FID/FPD), Headspace-Solid Phase Microextraction-Gas Chromatography-Mass Spectrometry (HS-SPME-GC-MS), colorimetric analysis, enzymatic analysis, Near-Infrared (NIR) analysis, as described in the sections below. The mean values of biological duplicates are reported for each compound.

HS-GC-FID/FPD

HS-GC-FID/FPD (Shimadzu GC 2010 Plus) was used to measure higher alcohols, acetaldehyde, esters, 4-vinyl guaicol, and sulfur compounds. Each measurement comprised 5 ml of sample pipetted into a 20 ml glass vial containing 1.75 g NaCl (VWR, 27810.295). 100 µl of 2-heptanol (Sigma-Aldrich, H3003) (internal standard) solution in ethanol (Fisher Chemical, E/0650DF/C17) was added for a final concentration of 2.44 mg/L. Samples were flushed with nitrogen for 10 s, sealed with a silicone septum, stored at −80 °C and analyzed in batches of 20.

The GC was equipped with a DB-WAXetr column (length, 30 m; internal diameter, 0.32 mm; layer thickness, 0.50 µm; Agilent Technologies, Santa Clara, CA, USA) to the FID and an HP-5 column (length, 30 m; internal diameter, 0.25 mm; layer thickness, 0.25 µm; Agilent Technologies, Santa Clara, CA, USA) to the FPD. N 2 was used as the carrier gas. Samples were incubated for 20 min at 70 °C in the headspace autosampler (Flow rate, 35 cm/s; Injection volume, 1000 µL; Injection mode, split; Combi PAL autosampler, CTC analytics, Switzerland). The injector, FID and FPD temperatures were kept at 250 °C. The GC oven temperature was first held at 50 °C for 5 min and then allowed to rise to 80 °C at a rate of 5 °C/min, followed by a second ramp of 4 °C/min until 200 °C kept for 3 min and a final ramp of (4 °C/min) until 230 °C for 1 min. Results were analyzed with the GCSolution software version 2.4 (Shimadzu, Kyoto, Japan). The GC was calibrated with a 5% EtOH solution (VWR International) containing the volatiles under study (Supplementary Table  S7 ).

HS-SPME-GC-MS

HS-SPME-GC-MS (Shimadzu GCMS-QP-2010 Ultra) was used to measure additional volatile compounds, mainly comprising terpenoids and esters. Samples were analyzed by HS-SPME using a triphase DVB/Carboxen/PDMS 50/30 μm SPME fiber (Supelco Co., Bellefonte, PA, USA) followed by gas chromatography (Thermo Fisher Scientific Trace 1300 series, USA) coupled to a mass spectrometer (Thermo Fisher Scientific ISQ series MS) equipped with a TriPlus RSH autosampler. 5 ml of degassed beer sample was placed in 20 ml vials containing 1.75 g NaCl (VWR, 27810.295). 5 µl internal standard mix was added, containing 2-heptanol (1 g/L) (Sigma-Aldrich, H3003), 4-fluorobenzaldehyde (1 g/L) (Sigma-Aldrich, 128376), 2,3-hexanedione (1 g/L) (Sigma-Aldrich, 144169) and guaiacol (1 g/L) (Sigma-Aldrich, W253200) in ethanol (Fisher Chemical, E/0650DF/C17). Each sample was incubated at 60 °C in the autosampler oven with constant agitation. After 5 min equilibration, the SPME fiber was exposed to the sample headspace for 30 min. The compounds trapped on the fiber were thermally desorbed in the injection port of the chromatograph by heating the fiber for 15 min at 270 °C.

The GC-MS was equipped with a low polarity RXi-5Sil MS column (length, 20 m; internal diameter, 0.18 mm; layer thickness, 0.18 µm; Restek, Bellefonte, PA, USA). Injection was performed in splitless mode at 320 °C, a split flow of 9 ml/min, a purge flow of 5 ml/min and an open valve time of 3 min. To obtain a pulsed injection, a programmed gas flow was used whereby the helium gas flow was set at 2.7 mL/min for 0.1 min, followed by a decrease in flow of 20 ml/min to the normal 0.9 mL/min. The temperature was first held at 30 °C for 3 min and then allowed to rise to 80 °C at a rate of 7 °C/min, followed by a second ramp of 2 °C/min till 125 °C and a final ramp of 8 °C/min with a final temperature of 270 °C.

Mass acquisition range was 33 to 550 amu at a scan rate of 5 scans/s. Electron impact ionization energy was 70 eV. The interface and ion source were kept at 275 °C and 250 °C, respectively. A mix of linear n-alkanes (from C7 to C40, Supelco Co.) was injected into the GC-MS under identical conditions to serve as external retention index markers. Identification and quantification of the compounds were performed using an in-house developed R script as described in Goelen et al. and Reher et al. 87 , 88 (for package information, see Supplementary Table  S8 ). Briefly, chromatograms were analyzed using AMDIS (v2.71) 89 to separate overlapping peaks and obtain pure compound spectra. The NIST MS Search software (v2.0 g) in combination with the NIST2017, FFNSC3 and Adams4 libraries were used to manually identify the empirical spectra, taking into account the expected retention time. After background subtraction and correcting for retention time shifts between samples run on different days based on alkane ladders, compound elution profiles were extracted and integrated using a file with 284 target compounds of interest, which were either recovered in our identified AMDIS list of spectra or were known to occur in beer. Compound elution profiles were estimated for every peak in every chromatogram over a time-restricted window using weighted non-negative least square analysis after which peak areas were integrated 87 , 88 . Batch effect correction was performed by normalizing against the most stable internal standard compound, 4-fluorobenzaldehyde. Out of all 284 target compounds that were analyzed, 167 were visually judged to have reliable elution profiles and were used for final analysis.

Discrete photometric and enzymatic analysis

Discrete photometric and enzymatic analysis (Thermo Scientific TM Gallery TM Plus Beermaster Discrete Analyzer) was used to measure acetic acid, ammonia, beta-glucan, iso-alpha acids, color, sugars, glycerol, iron, pH, protein, and sulfite. 2 ml of sample volume was used for the analyses. Information regarding the reagents and standard solutions used for analyses and calibrations is included in Supplementary Table  S7 and Supplementary Table  S9 .

NIR analyses

NIR analysis (Anton Paar Alcolyzer Beer ME System) was used to measure ethanol. Measurements comprised 50 ml of sample, and a 10% EtOH solution was used for calibration.

Correlation calculations

Pairwise Spearman Rank correlations were calculated between all chemical properties.

Sensory dataset

Trained panel.

Our trained tasting panel consisted of volunteers who gave prior verbal informed consent. All compounds used for the validation experiment were of food-grade quality. The tasting sessions were approved by the Social and Societal Ethics Committee of the KU Leuven (G-2022-5677-R2(MAR)). All online reviewers agreed to the Terms and Conditions of the RateBeer website.

Sensory analysis was performed according to the American Society of Brewing Chemists (ASBC) Sensory Analysis Methods 90 . 30 volunteers were screened through a series of triangle tests. The sixteen most sensitive and consistent tasters were retained as taste panel members. The resulting panel was diverse in age [22–42, mean: 29], sex [56% male] and nationality [7 different countries]. The panel developed a consensus vocabulary to describe beer aroma, taste and mouthfeel. Panelists were trained to identify and score 50 different attributes, using a 7-point scale to rate attributes’ intensity. The scoring sheet is included as Supplementary Data  3 . Sensory assessments took place between 10–12 a.m. The beers were served in black-colored glasses. Per session, between 5 and 12 beers of the same style were tasted at 12 °C to 16 °C. Two reference beers were added to each set and indicated as ‘Reference 1 & 2’, allowing panel members to calibrate their ratings. Not all panelists were present at every tasting. Scores were scaled by standard deviation and mean-centered per taster. Values are represented as z-scores and clustered by Euclidean distance. Pairwise Spearman correlations were calculated between taste and aroma sensory attributes. Panel consistency was evaluated by repeating samples on different sessions and performing ANOVA to identify differences, using the ‘stats’ package (v4.2.2) in R (for package information, see Supplementary Table  S8 ).

Online reviews from a public database

The ‘scrapy’ package in Python (v3.6) (for package information, see Supplementary Table  S8 ). was used to collect 232,288 online reviews (mean=922, min=6, max=5343) from RateBeer, an online beer review database. Each review entry comprised 5 numerical scores (appearance, aroma, taste, palate and overall quality) and an optional review text. The total number of reviews per reviewer was collected separately. Numerical scores were scaled and centered per rater, and mean scores were calculated per beer.

For the review texts, the language was estimated using the packages ‘langdetect’ and ‘langid’ in Python. Reviews that were classified as English by both packages were kept. Reviewers with fewer than 100 entries overall were discarded. 181,025 reviews from >6000 reviewers from >40 countries remained. Text processing was done using the ‘nltk’ package in Python. Texts were corrected for slang and misspellings; proper nouns and rare words that are relevant to the beer context were specified and kept as-is (‘Chimay’,’Lambic’, etc.). A dictionary of semantically similar sensorial terms, for example ‘floral’ and ‘flower’, was created and collapsed together into one term. Words were stemmed and lemmatized to avoid identifying words such as ‘acid’ and ‘acidity’ as separate terms. Numbers and punctuation were removed.

Sentences from up to 50 randomly chosen reviews per beer were manually categorized according to the aspect of beer they describe (appearance, aroma, taste, palate, overall quality—not to be confused with the 5 numerical scores described above) or flagged as irrelevant if they contained no useful information. If a beer contained fewer than 50 reviews, all reviews were manually classified. This labeled data set was used to train a model that classified the rest of the sentences for all beers 91 . Sentences describing taste and aroma were extracted, and term frequency–inverse document frequency (TFIDF) was implemented to calculate enrichment scores for sensorial words per beer.

The sex of the tasting subject was not considered when building our sensory database. Instead, results from different panelists were averaged, both for our trained panel (56% male, 44% female) and the RateBeer reviews (70% male, 30% female for RateBeer as a whole).

Beer price collection and processing

Beer prices were collected from the following stores: Colruyt, Delhaize, Total Wine, BeerHawk, The Belgian Beer Shop, The Belgian Shop, and Beer of Belgium. Where applicable, prices were converted to Euros and normalized per liter. Spearman correlations were calculated between these prices and mean overall appreciation scores from RateBeer and the taste panel, respectively.

Pairwise Spearman Rank correlations were calculated between all sensory properties.

Machine learning models

Predictive modeling of sensory profiles from chemical data.

Regression models were constructed to predict (a) trained panel scores for beer flavors and quality from beer chemical profiles and (b) public reviews’ appreciation scores from beer chemical profiles. Z-scores were used to represent sensory attributes in both data sets. Chemical properties with log-normal distributions (Shapiro-Wilk test, p  <  0.05 ) were log-transformed. Missing chemical measurements (0.1% of all data) were replaced with mean values per attribute. Observations from 250 beers were randomly separated into a training set (70%, 175 beers) and a test set (30%, 75 beers), stratified per beer style. Chemical measurements (p = 231) were normalized based on the training set average and standard deviation. In total, three linear regression-based models: linear regression with first-order interaction terms (LR), lasso regression with first-order interaction terms (Lasso) and partial least squares regression (PLSR); five decision tree models, Adaboost regressor (ABR), Extra Trees (ET), Gradient Boosting regressor (GBR), Random Forest (RF) and XGBoost regressor (XGBR); one support vector machine model (SVR) and one artificial neural network model (ANN) were trained. The models were implemented using the ‘scikit-learn’ package (v1.2.2) and ‘xgboost’ package (v1.7.3) in Python (v3.9.16). Models were trained, and hyperparameters optimized, using five-fold cross-validated grid search with the coefficient of determination (R 2 ) as the evaluation metric. The ANN (scikit-learn’s MLPRegressor) was optimized using Bayesian Tree-Structured Parzen Estimator optimization with the ‘Optuna’ Python package (v3.2.0). Individual models were trained per attribute, and a multi-output model was trained on all attributes simultaneously.

Model dissection

GBR was found to outperform other methods, resulting in models with the highest average R 2 values in both trained panel and public review data sets. Impurity-based rankings of the most important predictors for each predicted sensorial trait were obtained using the ‘scikit-learn’ package. To observe the relationships between these chemical properties and their predicted targets, partial dependence plots (PDP) were constructed for the six most important predictors of consumer appreciation 74 , 75 .

The ‘SHAP’ package in Python (v0.41.0) was implemented to provide an alternative ranking of predictor importance and to visualize the predictors’ effects as a function of their concentration 68 .

Validation of causal chemical properties

To validate the effects of the most important model features on predicted sensory attributes, beers were spiked with the chemical compounds identified by the models and descriptive sensory analyses were carried out according to the American Society of Brewing Chemists (ASBC) protocol 90 .

Compound spiking was done 30 min before tasting. Compounds were spiked into fresh beer bottles, that were immediately resealed and inverted three times. Fresh bottles of beer were opened for the same duration, resealed, and inverted thrice, to serve as controls. Pairs of spiked samples and controls were served simultaneously, chilled and in dark glasses as outlined in the Trained panel section above. Tasters were instructed to select the glass with the higher flavor intensity for each attribute (directional difference test 92 ) and to select the glass they prefer.

The final concentration after spiking was equal to the within-style average, after normalizing by ethanol concentration. This was done to ensure balanced flavor profiles in the final spiked beer. The same methods were applied to improve a non-alcoholic beer. Compounds were the following: ethyl acetate (Merck KGaA, W241415), ethyl hexanoate (Merck KGaA, W243906), isoamyl acetate (Merck KGaA, W205508), phenethyl acetate (Merck KGaA, W285706), ethanol (96%, Colruyt), glycerol (Merck KGaA, W252506), lactic acid (Merck KGaA, 261106).

Significant differences in preference or perceived intensity were determined by performing the two-sided binomial test on each attribute.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

The data that support the findings of this work are available in the Supplementary Data files and have been deposited to Zenodo under accession code 10653704 93 . The RateBeer scores data are under restricted access, they are not publicly available as they are property of RateBeer (ZX Ventures, USA). Access can be obtained from the authors upon reasonable request and with permission of RateBeer (ZX Ventures, USA).  Source data are provided with this paper.

Code availability

The code for training the machine learning models, analyzing the models, and generating the figures has been deposited to Zenodo under accession code 10653704 93 .

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Acknowledgements

We thank all lab members for their discussions and thank all tasting panel members for their contributions. Special thanks go out to Dr. Karin Voordeckers for her tremendous help in proofreading and improving the manuscript. M.S. was supported by a Baillet-Latour fellowship, L.C. acknowledges financial support from KU Leuven (C16/17/006), F.A.T. was supported by a PhD fellowship from FWO (1S08821N). Research in the lab of K.J.V. is supported by KU Leuven, FWO, VIB, VLAIO and the Brewing Science Serves Health Fund. Research in the lab of T.W. is supported by FWO (G.0A51.15) and KU Leuven (C16/17/006).

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These authors contributed equally: Michiel Schreurs, Supinya Piampongsant, Miguel Roncoroni.

Authors and Affiliations

VIB—KU Leuven Center for Microbiology, Gaston Geenslaan 1, B-3001, Leuven, Belgium

Michiel Schreurs, Supinya Piampongsant, Miguel Roncoroni, Lloyd Cool, Beatriz Herrera-Malaver, Florian A. Theßeling & Kevin J. Verstrepen

CMPG Laboratory of Genetics and Genomics, KU Leuven, Gaston Geenslaan 1, B-3001, Leuven, Belgium

Leuven Institute for Beer Research (LIBR), Gaston Geenslaan 1, B-3001, Leuven, Belgium

Laboratory of Socioecology and Social Evolution, KU Leuven, Naamsestraat 59, B-3000, Leuven, Belgium

Lloyd Cool, Christophe Vanderaa & Tom Wenseleers

VIB Bioinformatics Core, VIB, Rijvisschestraat 120, B-9052, Ghent, Belgium

Łukasz Kreft & Alexander Botzki

AB InBev SA/NV, Brouwerijplein 1, B-3000, Leuven, Belgium

Philippe Malcorps & Luk Daenen

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Contributions

S.P., M.S. and K.J.V. conceived the experiments. S.P., M.S. and K.J.V. designed the experiments. S.P., M.S., M.R., B.H. and F.A.T. performed the experiments. S.P., M.S., L.C., C.V., L.K., A.B., P.M., L.D., T.W. and K.J.V. contributed analysis ideas. S.P., M.S., L.C., C.V., T.W. and K.J.V. analyzed the data. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to Kevin J. Verstrepen .

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Competing interests.

K.J.V. is affiliated with bar.on. The other authors declare no competing interests.

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Schreurs, M., Piampongsant, S., Roncoroni, M. et al. Predicting and improving complex beer flavor through machine learning. Nat Commun 15 , 2368 (2024). https://doi.org/10.1038/s41467-024-46346-0

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Received : 30 October 2023

Accepted : 21 February 2024

Published : 26 March 2024

DOI : https://doi.org/10.1038/s41467-024-46346-0

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