Introductory Statistics 2e

Citation

Illowsky, B., Dean, S., et al. (2023). Introductory Statistics 2e. OpenStax. Free official edition: https://openstax.org/details/books/introductory-statistics-2e

Chapter Summary

Chapter 1: Sampling and Data

The opening chapter introduces the language of the course—population, sample, statistic, parameter, variable—and distinguishes descriptive from inferential statistics, with a brief look at probability as long-run relative frequency. It classifies data as qualitative or quantitative (discrete or continuous), compares simple random, stratified, cluster, systematic and convenience sampling, and asks students to evaluate samples critically for bias and size. Frequency tables, a rounding rule and the four levels of measurement follow. A section on experimental design (lurking variables, random assignment, control groups, placebos, blinding) ends with research ethics, including a data-fraud case and the role of Institutional Review Boards.

Chapter 2: Descriptive Statistics

This chapter turns raw data into pictures and summary numbers: stem-and-leaf plots, line and bar graphs, histograms, frequency polygons and time-series graphs, percentiles, quartiles, the interquartile range (with the 1.5 × IQR outlier rule) and box plots. Measures of centre (mean, median, mode, and the mean of grouped data) are linked to the shape of a distribution through skewness. The chapter finishes with variance and standard deviation, z-scores for comparing values from different data sets, and Chebyshev's rule and the empirical rule for how much data lie near the mean.

Chapter 3: Probability Topics

Here the book builds the probability needed for inference: sample spaces, events, equally likely outcomes, conditional probability and complements. It distinguishes independent from mutually exclusive events, then gives the multiplication and addition rules. Most of the chapter is about organising problems—contingency tables, tree diagrams and Venn diagrams—with many examples drawn from surveys and everyday situations.

Chapter 4: Discrete Random Variables

Random variables are introduced (X defined in words, x as a value), followed by probability distribution tables, expected value and standard deviation (linked to the law of large numbers), and then four named distributions: binomial, geometric, hypergeometric and Poisson, each with its defining conditions, mean and standard deviation. It stresses recognising which distribution a word problem describes, and includes the Poisson approximation to the binomial.

Chapter 5: Continuous Random Variables

The shift from counting to measuring is made through probability as area under a density curve and the cumulative distribution function. Two distributions are studied in detail: the uniform distribution, where probabilities reduce to rectangle areas, and the exponential distribution, including its memoryless property and its connection to the Poisson distribution for counts of events.

Chapter 6: The Normal Distribution

This chapter introduces the normal distribution through its two parameters and the standard normal distribution of z-scores. It shows how to move between values and z-scores, how to find probabilities and percentiles with a calculator (the book notes that tables are now largely unnecessary), and how to apply the empirical rule. Labs use lap times and finger-length measurements to check whether real data look normal.

Chapter 7: The Central Limit Theorem

The central limit theorem is presented in two forms, for sample means and for sums: for large enough samples, the distribution of the mean (or sum) is approximately normal, with standard deviation given by the standard error. Section 7.3 applies the theorem to probability and percentile problems and uses it to illustrate the law of large numbers, and the chapter objectives include classifying word problems by the distribution they require. Two labs—one sampling classmates' pocket change, one using a supplied data set on how long cookie recipes lasted—show the theorem at work.

Chapter 8: Confidence Intervals

Inference begins with interval estimation: the chapter builds a confidence interval for a mean when the population standard deviation is known, explains the error bound and how confidence level and sample size affect it, and shows how to choose a sample size. It then introduces Student's t-distribution for the more realistic case of unknown σ, and intervals for a population proportion, including the "plus four" adjustment for small samples. The text stresses that the interval, not the parameter, is random.

Chapter 9: Hypothesis Testing with One Sample

Hypothesis testing is set up as a decision procedure: state null and alternative hypotheses, understand Type I and Type II errors and power, choose the distribution (normal or t for a mean, normal for a proportion), and compare a p-value with a preset significance level. Section 9.5 collects full worked examples—left-, right- and two-tailed—each ending with a written conclusion in context, and encourages judgment about p-values rather than mechanical rules.

Chapter 10: Hypothesis Testing with Two Samples

The same five-step procedure is extended to comparisons of two groups: two means with unknown standard deviations (the unpooled Aspin–Welch t-test), two means with known standard deviations, two independent proportions (using a pooled proportion), and matched or paired samples. The chapter also introduces Cohen's d as a measure of effect size. Degrees of freedom and test statistics are left to a calculator or computer.

Chapter 11: The Chi-Square Distribution

After describing the chi-square distribution and how its shape depends on degrees of freedom, the chapter covers three tests built on observed and expected counts—goodness of fit, independence and homogeneity—and then compares them so students can tell them apart. It also covers a test for a single variance. Two labs carry out the goodness-of-fit and independence tests on data students collect themselves (grocery receipt totals and favourite snacks).

Chapter 12: Linear Regression and Correlation

Starting from linear equations and scatter plots, this chapter develops the least-squares regression line, residuals and residual plots, the correlation coefficient r and the coefficient of determination r². It tests whether the population correlation is zero (by a t-test or a table of critical values), uses the line for prediction while warning against extrapolation, and gives rules for identifying outliers and influential points. Three labs collect bivariate data (distance from school, textbook pages and cost, fuel efficiency).

Chapter 13: F Distribution and One-Way ANOVA

The last chapter extends comparisons to more than two groups through one-way analysis of variance, stating its assumptions and building the F-ratio from variation between and within groups, summarised in an ANOVA table. It describes the shape of the F distribution and closes with the F test for two variances, noting that it depends heavily on normality.

Key Concepts

Chapter 1: Sampling and Data

• Statistic versus parameter: a statistic is a number computed from a sample, used to estimate a parameter, the corresponding number for the whole population.
• Random sampling methods: simple random, stratified, cluster and systematic sampling give every member a known chance of selection, unlike convenience sampling, which tends to produce bias.
• Experimental design: random assignment, a control group, placebos and blinding are used so that differences between groups can be attributed to the treatment.

Chapter 2: Descriptive Statistics

• Quartiles and IQR: the quartiles split ordered data into four parts, and the interquartile range (Q3 − Q1) measures spread and flags possible outliers.
• Mean, median and skewness: in skewed data the mean is pulled toward the tail, so comparing mean and median reveals the shape of the distribution.

Chapter 3: Probability Topics

• Conditional probability: the probability of one event given that another has occurred, P(A|B).
• Independent versus mutually exclusive: independent events do not affect each other's probabilities, while mutually exclusive events cannot happen together.

Chapter 4: Discrete Random Variables

• Binomial distribution: the number of successes in a fixed number of independent trials with the same success probability.
• Poisson distribution: the number of events in a fixed interval when events occur independently at a known average rate.

Chapter 5: Continuous Random Variables

• Probability density function: a curve whose area between two values gives the probability of falling in that range; total area is one.
• Memoryless property: for the exponential distribution, the chance of waiting a further time does not depend on how long one has already waited.

Chapter 6: The Normal Distribution

• z-score: the number of standard deviations a value lies above or below the mean, used to compare and standardise values.
• Empirical rule: for normal data, about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.

Chapter 7: The Central Limit Theorem

• Standard error: the standard deviation of the sample mean, equal to σ divided by the square root of n.
• Central limit theorem: for large enough samples, sample means and sums are approximately normally distributed whatever the shape of the population.

Chapter 8: Confidence Intervals

• Error bound (margin of error): the distance added to and subtracted from a point estimate to form an interval at a given confidence level.
• Student's t-distribution: the distribution used instead of the normal when σ is unknown and estimated by s, with n − 1 degrees of freedom.

Chapter 9: Hypothesis Testing with One Sample

• Null and alternative hypotheses: two contradictory statements about a parameter, with the null containing equality and the alternative written with <, > or ≠.
• p-value: the probability, assuming the null hypothesis is true, of a sample result at least as extreme as the one observed.
• Type I and Type II errors: rejecting a true null hypothesis (probability α) and failing to reject a false one (probability β); power is 1 − β.

Chapter 10: Hypothesis Testing with Two Samples

• Independent versus matched samples: independent groups are compared through the difference of their means or proportions, paired data through the mean of the differences.
• Cohen's d: an effect-size measure that divides the difference between two means by a pooled standard deviation.

Chapter 11: The Chi-Square Distribution

• Goodness-of-fit test: compares observed counts with those expected under a proposed distribution.
• Test of independence: uses a contingency table to test whether two categorical variables are related.

Chapter 12: Linear Regression and Correlation

• Least-squares line: the line that minimises the sum of squared residuals; its slope is interpreted as the average change in y per unit change in x.
• Correlation coefficient r: a number between −1 and 1 measuring the strength and direction of a linear relationship; r² gives the share of variation in y explained by the line.

Chapter 13: F Distribution and One-Way ANOVA

• One-way ANOVA: a test of whether several population means are equal, comparing variation between group means with variation within groups.
• F-ratio: the test statistic of ANOVA, with numerator and denominator degrees of freedom; large values point to differences among means.

Critical Analysis

Strengths

• Built for active learning. Collaborative exercises that generate class data recur throughout, and every chapter ends with "Stats Lab" sections (such as the pocket-change central limit theorem lab and the textbook-cost regression lab) in which students collect, analyse and interpret their own data. Appendix D adds group projects that forbid convenience sampling.
• Consistent, step-by-step method. Worked examples follow the same pattern—define the random variable in words, choose the distribution, compute, draw a picture, decide and write a conclusion in a sentence. Full examples in Chapter 9, such as the swimmer-and-goggles test, also spell out what Type I and Type II errors would mean in context.
• Plenty of graded practice. Each chapter has Try It problems after examples, Practice and Homework sets, and often "Bringing It Together" problems mixing topics; appendices add review exercises for Chapters 3–13, four practice tests and final exams, data sets and fill-in solution sheets for hypothesis tests. Answers to selected problems are printed at the end of each chapter.
• Attention to data quality and ethics. Chapter 1 spends real time on sampling bias, critical evaluation of studies, experimental design and research misconduct, so the later inferential methods rest on a discussion of where good data come from.

Limitations

• Tied to TI graphing calculators. Step-by-step technology instructions are written almost entirely for TI-83/84 calculators, with spreadsheets and other software mentioned only in passing. Students using R, Python or a spreadsheet will need separate guidance.
• Little theory or derivation. As the preface says, the book favours application over theory and assumes only intermediate algebra. Formulas are presented rather than derived (for example, the least-squares line is stated with only a remark that calculus yields it), so students heading for mathematical statistics will need a more rigorous text.
• Limited scope beyond the standard syllabus. There is no treatment of Bayes' theorem by name, nonparametric methods, multiple regression or simulation-based inference, and one-way ANOVA is explicitly only an overview.
• Uneven finish and rigid rules of thumb. Example 9.13 writes a hypothesis about a mean with the letter p, the key-terms lists thin out in later chapters (Chapter 12's lists only "outlier"), and conventions such as always putting equality in the null hypothesis or n ≥ 30 for the central limit theorem are given as rules with little discussion.

Applications in the Book

Applications run through every chapter rather than sitting in one. Examples draw on college life (exam scores, textbook costs), health (aspirin and placebo studies), business (car sales, ice-cream sales against temperature) and sport (swim times, lap times, batting averages). The labs turn these into data-collection tasks—sampling candies to estimate probabilities, comparing finger lengths and race lap times with the normal model, and gathering bivariate data for regression—and Chapter 1 applies the material to research ethics through a published fraud case and Institutional Review Boards.

Where It Fits in the Course

Within the Probability and Statistics course list, this is the applied, algebra-level entry point. David Spiegelhalter's The Art of Statistics is generally described as a book on statistical thinking for a wide audience, so the two pair well—Spiegelhalter for the ideas, Illowsky and Dean for practice with procedures. Blitzstein and Hwang's Introduction to Probability is a calculus-based probability text and a natural next step for the theory that Chapters 3–7 here only sketch.

The calculus-based books in the list—Larsen and Marx's An Introduction to Mathematical Statistics and Its Applications, Mavrakakis and Penzer's Probability and Statistical Inference, Casella and Berger's Statistical Inference, Williams's Probability with Martingales, and Mitzenmacher and Upfal's Probability and Computing—assume calculus and mathematical maturity and develop the theory behind the methods. Read this book before them, not instead of them: it supplies the vocabulary and intuition (sampling distributions, confidence intervals, p-values, regression) that makes their derivations easier to follow.

How to Use This Book

The book targets a one-semester introductory course for students not majoring in mathematics or engineering, and assumes intermediate algebra but no calculus. It also suits self-study and revision before a research methods course.

Read Chapters 1–2 first, then 3–7 as the probability foundation, and treat Chapters 8–10 as the core of inference; Chapters 11–13 can be chosen according to the course. If correlation and regression are needed early, the preface's alternative order places Chapter 12 directly after Chapter 2. For each section, work through the examples, then try the Try It problem before checking the answer (Try It answers are on the OpenStax student resources page, and selected Practice and Homework answers are at the end of each chapter). Do at least one lab or collaborative exercise per chapter, since they are where the ideas of sampling variation become concrete, and use Appendix B's practice tests for revision. Students without a TI calculator should pick a software tool early and translate each "Using the TI-83, 83+, 84, 84+ Calculator" box into it. Appendix F's summary of symbols and formulas is handy for exams.

Edition and Sources

This guide is based on Introductory Statistics 2e (second edition, first published 2023; senior contributing authors Barbara Illowsky and Susan Dean with nine contributing authors), read in the free official PDF linked from the OpenStax book page, https://openstax.org/details/books/introductory-statistics-2e (PDF generated April 2026, CC BY-NC-SA 4.0 licence). The guide draws on the preface, the full table of contents, and for all thirteen chapters the chapter objectives and introductions, every section heading, the chapter reviews and key terms, and formula reviews, with Sections 1.1–1.4, 9.5 and 12.3 and selected labs and appendices (C, D, E) read in more depth.