Introductory Statistics PDF Download – Illowsky & Dean (OpenStax, 2nd Edition)

Students taking a one-semester Introductory Statistics course can download the complete textbook “Introductory Statistics 2e” by Barbara Illowsky and Susan Dean, free as a PDF from OpenStax. It is an algebra-based statistics text — no calculus is assumed anywhere in the book — built around worked examples, large end-of-section practice sets, and short in-class “collaborative exercises” that use real survey and experiment data to walk through each new idea.

This is OpenStax’s own general-purpose statistics text, distinct from OpenIntro Statistics and OpenStax’s own Introductory Business Statistics, both separate textbooks on this site. It covers the standard one-semester syllabus end to end — sampling and descriptive statistics, probability and random variables, the normal distribution and the Central Limit Theorem, confidence intervals, hypothesis testing with one and two samples, chi-square tests, linear regression, and one-way ANOVA — making it a solid fit for BS Statistics, social science, and any program with a single required statistics course.

Book Overview

Chapter List

Chapter 1: Sampling and Data

Difficulty: Easy · Key topics: population vs. sample, parameter vs. statistic, sampling methods, quantitative vs. qualitative data, levels of measurement, frequency tables

Chapter 1 opens the book by building the vocabulary statistics runs on before any calculation begins — the difference between a population and a sample, and between a parameter (describes the population) and a statistic (describes the sample). It works through the main sampling methods used to collect data without bias — simple random, stratified, cluster, systematic, and convenience sampling — and distinguishes quantitative data (numbers that can be measured) from qualitative data (categories), and discrete from continuous values. It closes by using frequency tables to organize raw data into something readable, and by naming the four levels of measurement that determine which later statistical methods a given variable can use.

Key Points:

• A population is every member of the group being studied; a sample is the subset actually measured.
• A parameter describes a population (fixed, usually unknown); a statistic describes a sample (calculated from data, used to estimate the parameter).
• Simple random sampling gives every member of the population an equal chance of selection; stratified, cluster, and systematic sampling each divide or order the population differently before sampling.
• Convenience sampling is the easiest to carry out and the most likely to be biased — it should be used cautiously in real research.
• Quantitative data can be discrete (counted, e.g. number of pets) or continuous (measured, e.g. height); qualitative data describes categories (e.g. hair color).
• The four levels of measurement — nominal, ordinal, interval, and ratio — determine which statistics (e.g. a true zero and ratios) are meaningful for a given variable.

Memory Tip: Remember NOIR for the four levels of measurement in order of increasing information: Nominal, Ordinal, Interval, Ratio — each level keeps everything the previous one could do and adds one more.

Common Mistake: Treating a convenience sample as if it were a random sample. A convenience sample (surveying whoever is easiest to reach) can look just like real data in a spreadsheet, but conclusions drawn from it do not generalize to the full population the way a properly randomized sample’s do.

Important Questions:

• What is the difference between a parameter and a statistic? A parameter is a fixed numerical fact about an entire population (usually unknown); a statistic is a number calculated from a sample and used to estimate the parameter.
• A researcher surveys the first 50 shoppers who walk into a store on a Monday morning. What sampling method is this, and what is its main weakness? This is convenience sampling; its weakness is that it is not random, so shoppers at other times or on other days may differ systematically from the ones surveyed, biasing the results.

Chapter 2: Descriptive Statistics

Difficulty: Easy · Key topics: stem-and-leaf plots, histograms, box plots, quartiles and percentiles, mean/median/mode, skewness, variance and standard deviation

Chapter 2 turns raw, collected data into numbers and pictures a reader can actually interpret. It opens with graphical tools — stem-and-leaf plots, histograms, frequency polygons, and box plots — that reveal a data set’s shape before any formula is applied, then moves to quartiles and percentiles as ways of locating a value’s position within the data. The chapter works through the three measures of center (mean, median, mode) and explains how a comparison between them reveals skewness, before finishing with the measures of spread — range, variance, and standard deviation — that describe how tightly or loosely data clusters around its center.

Key Points:

• A box plot is built from the five-number summary (minimum, Q1, median, Q3, maximum) and visually flags skew and potential outliers.
• Mean is pulled toward outliers and skew; median is not — a right-skewed distribution typically has mean > median.
• Standard deviation (σ for a population, s for a sample) measures the typical distance of a data point from the mean, in the data’s own units.
• Variance is standard deviation squared (σ² or s²); it is in squared units, which is why standard deviation is reported instead for interpretation.
• The kth percentile is the value below which k% of the data fall; Q1, the median, and Q3 are the 25th, 50th, and 75th percentiles.
• A larger standard deviation means data is more spread out from the mean, regardless of what the mean itself is.

Memory Tip: To read skew from a box plot or histogram quickly, remember “skew follows the tail” — if the long tail points right, the distribution is right- (positively) skewed, and the mean sits to the right of the median.

Common Mistake: Confusing variance and standard deviation, or reporting variance as if it were in the same units as the data. Variance is in squared units (e.g. dollars²), which is almost never the useful number to report — standard deviation, in the original units, is what should be quoted and interpreted.

Important Questions:

• A data set has mean 50 and median 42. Is the distribution more likely left-skewed or right-skewed? Right-skewed — the mean is pulled above the median by a long right tail of high values.
• What are the five numbers in a box plot’s five-number summary? The minimum, the first quartile (Q1), the median, the third quartile (Q3), and the maximum.

Chapter 3: Probability Topics

Difficulty: Medium · Key topics: sample space, independent vs. mutually exclusive events, Addition and Multiplication Rules, conditional probability, contingency tables, tree and Venn diagrams

Chapter 3 builds the probability vocabulary and rules that every later inferential chapter depends on. It defines the terminology — sample space, event, outcome — and draws the critical distinction between independent events (one does not affect the other) and mutually exclusive events (they cannot happen together), which students often conflate. From there it develops the Addition Rule and Multiplication Rule for combining probabilities, and introduces conditional probability (P(A|B)) as probability updated by new information. The chapter closes by using contingency tables, tree diagrams, and Venn diagrams as three different visual tools for organizing and solving the same kinds of probability problems.

Key Points:

• Two events are mutually exclusive if they cannot both occur (P(A AND B) = 0); they are independent if one occurring does not change the probability of the other.
• The Addition Rule: P(A OR B) = P(A) + P(B) − P(A AND B); the subtracted term is dropped only when A and B are mutually exclusive.
• The Multiplication Rule: P(A AND B) = P(A)·P(B|A); this simplifies to P(A)·P(B) only when A and B are independent.
• Conditional probability P(A|B) = P(A AND B)/P(B) restricts the sample space to the cases where B already happened.
• A contingency table cross-tabulates two categorical variables and makes conditional and joint probabilities easy to read off directly.
• Mutually exclusive events are not automatically independent, and independent events are not automatically mutually exclusive — the two ideas answer different questions.

Memory Tip: “Exclusive means EITHER, independent means IGNORES” — mutually exclusive events can only go one way or the other (never both); independent events ignore each other entirely (neither affects the other’s probability).

Common Mistake: Assuming mutually exclusive events are automatically independent (or vice versa). Two events can be mutually exclusive but still fail every test of independence, because knowing one occurred (P=1) tells you the other’s probability dropped to exactly zero — about as dependent as two events can be.

Important Questions:

• If P(A) = 0.3, P(B) = 0.4, and A and B are independent, what is P(A AND B)? Since independent events multiply directly, P(A AND B) = 0.3 × 0.4 = 0.12.
• In a deck of cards, are drawing a heart and drawing a face card mutually exclusive? No — a card can be both a heart and a face card (e.g. the king of hearts), so the two events can occur together and are not mutually exclusive.

Chapter 4: Discrete Random Variables

Difficulty: Medium · Key topics: probability distribution function, expected value, binomial distribution, geometric distribution, hypergeometric distribution, Poisson distribution

Chapter 4 introduces the random variable — a rule that assigns a number to each outcome of a random process — and focuses on the discrete case, where outcomes can be counted. It defines a probability distribution function (PDF) as the full list of a discrete variable’s possible values paired with their probabilities, and defines expected value as the long-run average outcome. The bulk of the chapter works through four named discrete distributions in turn — binomial (fixed number of independent trials, two outcomes), geometric (trials until the first success), hypergeometric (sampling without replacement), and Poisson (counts of rare events over a fixed interval) — and teaches how to recognize which situation calls for which distribution.

Key Points:

• A discrete probability distribution lists every possible value of X with its probability, and the probabilities must sum to exactly 1.
• Expected value E(X) = Σ[x·P(x)] is the long-run average value of X over many repetitions, not necessarily a value X can actually take.
• The binomial distribution applies to a fixed number of independent trials, each with the same two possible outcomes (success/failure) and the same probability of success.
• The geometric distribution counts the number of trials needed to get the first success, with trials continuing until it happens.
• The hypergeometric distribution is used when sampling without replacement from a finite population, so each draw changes the probabilities for the next.
• The Poisson distribution models the number of times a rare event occurs in a fixed interval of time or space, given a known average rate.

Memory Tip: To choose the right discrete distribution fast, ask in order: fixed number of trials? (binomial), counting until first success? (geometric), sampling without replacement? (hypergeometric) — if none fit and you’re counting rare events over an interval, it’s Poisson.

Common Mistake: Using the binomial distribution when sampling is done without replacement from a small population. Binomial assumes independence between trials; sampling without replacement changes the probabilities as items are removed, which calls for the hypergeometric distribution instead — binomial is only a safe approximation when the population is much larger than the sample.

Important Questions:

• A fair coin is flipped 10 times. What distribution models the number of heads, and why? Binomial — there are a fixed number of trials (10), each independent, with two outcomes (heads/tails) and the same probability of success each time.
• What does it mean for the probabilities in a discrete probability distribution to “sum to 1”? It means the random variable is guaranteed to take on one of its listed values — the distribution accounts for every possible outcome with no gaps or overlaps.

Chapter 5: Continuous Random Variables

Difficulty: Medium · Key topics: probability density function, cumulative distribution function, uniform distribution, exponential distribution

Chapter 5 extends the random-variable idea from Chapter 4 to variables that are measured rather than counted — heights, times, distances — where any value in a range is possible. It introduces the probability density function (pdf), whose graph is a curve rather than a list of spikes, and establishes the key rule that probability is read as area under that curve, never as the height of a single point. The chapter then works through two named continuous distributions: the uniform distribution, where every value in an interval is equally likely, and the exponential distribution, commonly used to model waiting times between events.

Key Points:

• For a continuous random variable, probability is area under the density curve, not a single point’s height — P(X = c) = 0 for any exact value c.
• The entire area under a probability density function, above the x-axis, always equals exactly 1.
• P(c < X < d) equals P(c ≤ X ≤ d) for a continuous variable, since a single boundary point contributes zero area.
• The uniform distribution assigns equal probability density to every value in an interval [a, b] — its graph is a flat rectangle.
• The exponential distribution models the waiting time until a random event occurs, and is closely linked to the Poisson distribution from Chapter 4 (Poisson counts events; exponential times the gaps between them).
• Finding exact areas under most continuous density curves formally requires calculus, but this book finds them with geometry, formulas, or technology instead.

Memory Tip: Picture a continuous distribution as a solid shape with total area 1 — any single probability question is really “how much of that area is shaded between these two boundaries,” never a height on the curve.

Common Mistake: Trying to compute P(X = c) for a continuous variable as a nonzero number. Because area under a single point has zero width, P(X = c) is always 0 for any continuous distribution — only an interval, like P(c < X < d), can have a nonzero probability.

Important Questions:

• A city bus arrives at a stop at a uniformly random time between 0 and 20 minutes after the hour. What is the probability it arrives in the first 5 minutes? Since the distribution is uniform on [0, 20], the probability is the proportion of the interval: 5/20 = 0.25.
• Why is P(X = c) always 0 for a continuous random variable, but not for a discrete one? A continuous variable’s probability is area under a curve, and a single exact value c has zero width and therefore zero area; a discrete variable instead assigns actual probability mass to each individual value.

Chapter 6: The Normal Distribution

Difficulty: Medium · Key topics: standard normal distribution, z-scores, empirical rule, normal probability calculations

Chapter 6 covers the single most important distribution in the book — the bell-shaped normal distribution, defined completely by its mean (μ) and standard deviation (σ). It shows that every normal distribution can be converted to the standard normal distribution (mean 0, standard deviation 1) by computing a z-score, which measures how many standard deviations a value sits from its mean — this is what makes one set of probability tables or one calculator function work for every possible normal distribution. The chapter works through reading and computing normal probabilities in both directions: finding a probability given a value, and finding a value given a probability (a percentile).

Key Points:

• A normal distribution is fully determined by two parameters: its mean μ (center) and standard deviation σ (spread).
• The z-score z = (x − μ)/σ converts any normal value into the equivalent position on the standard normal distribution (μ = 0, σ = 1).
• The normal curve is symmetric about μ, so the mean, median, and mode of a normal distribution are all equal.
• The empirical rule: roughly 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
• Changing μ shifts the curve left or right without changing its shape; changing σ makes the curve wider (flatter) or narrower (taller).
• A z-score’s sign tells direction: negative means below the mean, positive means above it.

Memory Tip: A z-score measures how many steps, in standard deviations, a value is from home (the mean) — a z-score of 2 means a value two full standard deviations above the mean, regardless of what the original units were.

Common Mistake: Looking up or reporting a raw x-value’s probability as if it were already a z-score, or forgetting to convert back to the original units after working in z-scores. Always convert x to z before using the standard normal distribution, and convert the answer back to x’s original units if the question asked for a value, not a z-score.

Important Questions:

• A data set is normally distributed with μ = 100 and σ = 15. What is the z-score for x = 130? z = (130 − 100)/15 = 2 — the value 130 is exactly 2 standard deviations above the mean.
• Using the empirical rule, approximately what percentage of values in a normal distribution fall within 2 standard deviations of the mean? Approximately 95%.

Chapter 7: The Central Limit Theorem

Difficulty: Hard · Key topics: sampling distribution of the sample mean, sampling distribution of the sum, standard error, sample size conditions

Chapter 7 covers the theorem the rest of inferential statistics is built on: no matter what shape the original population has, the distribution of sample means (or sample sums) taken from it becomes approximately normal once the sample size is large enough — generally n ≥ 30, or any size if the population itself is already normal. The chapter develops two parallel versions of the theorem, one for sample means and one for sums, and introduces the standard error (σ/√n) as the sample mean’s own standard deviation, which shrinks as the sample size grows. This is the mechanical justification behind nearly every confidence interval and hypothesis test later in the book.

Key Points:

• The Central Limit Theorem (CLT) says the sampling distribution of the sample mean approaches a normal distribution as sample size n increases, regardless of the shape of the original population.
• The sampling distribution of the sample mean has the same mean μ as the population, but a smaller standard deviation: the standard error σ/√n.
• A sample size of n ≥ 30 is generally treated as “large enough” for the CLT to apply when the population’s own distribution is unknown or non-normal.
• If the original population is already normal, the sampling distribution of the mean is normal for any sample size, even n = 1.
• The CLT also has a version for sample sums, whose sampling distribution has mean nμ and standard deviation σ√n.
• Larger samples produce a tighter sampling distribution (smaller standard error), which is why bigger samples give more reliable estimates of the population mean.

Memory Tip: The standard error shrinks by √n, not n — quadrupling the sample size only halves the standard error, it does not cut it to a quarter.

Common Mistake: Confusing the population standard deviation σ with the standard error of the mean (σ/√n), and using the wrong one in a calculation. σ describes the spread of individual data points; the standard error describes the spread of sample means across repeated samples, and the two are almost never the same number.

Important Questions:

• A population has σ = 20. For samples of size n = 25, what is the standard error of the sample mean? Standard error = σ/√n = 20/√25 = 20/5 = 4.
• Why does the Central Limit Theorem matter even when the original population is not normally distributed? Because it guarantees that the sampling distribution of the sample mean becomes approximately normal once n is large enough, which lets normal-based methods (confidence intervals, hypothesis tests) be used on sample means even when nothing is known about the shape of the underlying population.

Chapter 8: Confidence Intervals

Difficulty: Hard · Key topics: point estimate, confidence interval for a mean, Student’s t-distribution, confidence interval for a proportion, sample size calculation

Chapter 8 moves from describing samples to making inferences about the population they came from. It introduces the point estimate (a single best guess, like the sample mean) and then builds a confidence interval around it — a range of plausible values for the true population parameter, together with a stated confidence level such as 95%. The chapter teaches when to use the normal distribution versus the Student’s t-distribution for a population mean (t is used whenever the population standard deviation is unknown and must be estimated from the sample), covers confidence intervals for a population proportion, and ends by showing how to calculate the sample size needed in advance to achieve a target margin of error.

Key Points:

• A confidence interval is a range of values, built around a point estimate, meant to capture the true population parameter with a stated level of confidence (e.g. 95%).
• A 95% confidence interval does not mean “95% probability the parameter is in this interval” — it means 95% of intervals built this way, over repeated sampling, would contain the true parameter.
• Use the normal distribution for a confidence interval on the mean when the population standard deviation σ is known; use the Student’s t-distribution when it is unknown and estimated by the sample standard deviation s.
• The t-distribution is flatter and has heavier tails than the normal distribution, especially for small samples, but approaches the normal distribution as sample size grows.
• A confidence interval for a population proportion uses the sample proportion (p-hat) and its own standard error formula, not the mean’s.
• A higher confidence level (e.g. 99% instead of 95%) widens the interval; a larger sample size narrows it — the two work in opposite directions.

Memory Tip: Unknown σ means use t — whenever the population standard deviation has to be estimated from the sample itself, the extra uncertainty is captured by switching from the normal distribution to the wider-tailed Student’s t-distribution.

Common Mistake: Interpreting a 95% confidence interval as “there’s a 95% chance the true mean is in this specific interval.” The true population mean is a fixed number, not a random variable — once a single interval is calculated, it either does or does not contain it; the 95% describes the long-run success rate of the method, not this one interval.

Important Questions:

• A sample of n = 10 is used to estimate a population mean, and the population standard deviation is unknown. Should the normal or the Student’s t-distribution be used? The Student’s t-distribution — the population standard deviation is unknown and must be estimated from the small sample, which is exactly when t applies instead of the normal distribution.
• What happens to the width of a confidence interval if the confidence level is increased from 90% to 99%, with everything else unchanged? The interval becomes wider — a higher confidence level requires capturing more of the sampling distribution, which demands a larger margin of error.

Chapter 9: Hypothesis Testing with One Sample

Difficulty: Hard · Key topics: null and alternative hypotheses, Type I and Type II errors, p-value, hypothesis test for a single mean, hypothesis test for a single proportion

Chapter 9 formalizes how a statistician decides whether a claim about a population is supported by sample data. It sets up two contradictory hypotheses — the null hypothesis (rejected or not) and the alternative hypothesis (what is being tested for) — and defines the two ways a decision can go wrong: a Type I error (rejecting a true null hypothesis) and a Type II error (failing to reject a false one). The chapter builds the full five-step hypothesis-testing procedure around the p-value, the probability of seeing a result at least as extreme as the sample’s if the null hypothesis were actually true, and applies it to tests of a single population mean and a single population proportion.

Key Points:

• The null hypothesis is the claim assumed true by default (usually “no effect” or “no difference”); the alternative hypothesis is what the test is trying to find evidence for.
• A Type I error rejects a true null hypothesis (a “false positive”); a Type II error fails to reject a false null hypothesis (a “false negative”).
• The significance level (commonly 0.05, written α) is the Type I error rate the researcher is willing to accept, chosen before collecting data.
• The p-value is the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true — a small p-value is evidence against it.
• Decision rule: reject the null hypothesis when the p-value is less than α; otherwise, fail to reject it (never “accept” it outright).
• The same five-step structure (hypotheses, distribution, test statistic/p-value, decision, conclusion) applies whether the test concerns a single mean or a single proportion.

Memory Tip: Small p, strong evidence against the null — the smaller the p-value, the less plausible the null hypothesis is given what the sample actually showed.

Common Mistake: Saying a hypothesis test “proves” or “accepts” the null hypothesis when the p-value is large. Failing to reject the null hypothesis only means the sample did not provide enough evidence against it — it is never proof that it is true, since a larger sample might still find a significant effect.

Important Questions:

• A hypothesis test produces a p-value of 0.03 at a significance level of α = 0.05. What is the decision? Reject the null hypothesis, since the p-value (0.03) is less than α (0.05).
• What is the difference between a Type I and a Type II error? A Type I error rejects a null hypothesis that is actually true; a Type II error fails to reject a null hypothesis that is actually false.

Chapter 10: Hypothesis Testing with Two Samples

Difficulty: Hard · Key topics: independent vs. matched samples, two population means, two population proportions, paired t-test

Chapter 10 extends the one-sample hypothesis-testing machinery from Chapter 9 to comparisons between two groups, which is how most real research questions are actually framed. The chapter’s central distinction is between independent samples (two separate, unrelated groups, such as a treatment group and a placebo group) and matched or paired samples (the same subjects measured twice, such as before-and-after weights), because the two situations use different standard error formulas and different test statistics. It works through tests for the difference between two means (with standard deviations known or unknown), the difference between two proportions, and the paired t-test for matched data.

Key Points:

• Independent samples contain two groups where values in one are unrelated to values in the other; matched (paired) samples measure the same subjects twice, so the two sets of values are linked.
• A paired t-test works on the differences within each pair, reducing the two-sample problem back to a one-sample test on those differences.
• Comparing two population means with known standard deviations uses a normal-based test statistic; with unknown standard deviations, a t-based test statistic is used instead, same as in Chapter 8.
• Comparing two population proportions pools the two sample proportions under the null hypothesis that the true proportions are equal.
• The null hypothesis for two-sample tests is typically that the two population parameters are equal (no difference); the alternative states a direction or simply “not equal.”
• Choosing independent vs. paired analysis is a design decision made before data collection, not something a formula detects automatically — using the wrong one gives a wrong standard error.

Memory Tip: Paired data is never independent — if the same subjects (or matched pairs, like twins or before/after measurements) appear in both samples, the data must be analyzed as paired, because the two measurements are correlated with each other.

Common Mistake: Analyzing paired (dependent) data with an independent-samples test. Treating before-and-after measurements on the same people as if they came from two unrelated groups ignores the correlation between the pairs and typically understates the true precision of the comparison.

Important Questions:

• A researcher weighs 20 patients before and after a diet program. Should this be analyzed as independent or paired samples? Paired — the same 20 patients are measured twice, so the before and after values are linked, not independent.
• What does the null hypothesis typically state in a two-sample test comparing two population means? That the two population means are equal — i.e., the difference between them is zero.

Chapter 11: The Chi-Square Distribution

Difficulty: Hard · Key topics: goodness-of-fit test, test of independence, test of homogeneity, test of a single variance

Chapter 11 introduces the chi-square distribution and its four classic hypothesis-testing applications, all built around comparing observed counts to expected counts. The goodness-of-fit test checks whether a single categorical variable follows a claimed distribution (such as whether lottery numbers are drawn evenly); the test of independence checks whether two categorical variables are related, using a contingency table; the test of homogeneity checks whether several populations share the same distribution of a categorical variable; and the test of a single variance checks a claim about a population’s spread rather than its center. All four share the same chi-square test statistic formula comparing observed to expected values.

Key Points:

• The chi-square test statistic is the sum, across all categories or cells, of (observed − expected)² / expected.
• A goodness-of-fit test checks whether one categorical variable’s observed distribution matches a specific claimed (expected) distribution.
• A test of independence uses a contingency table to check whether two categorical variables are associated, rather than independent of each other.
• A test of homogeneity checks whether multiple populations share the same proportions across categories of one variable — similar in structure to a test of independence but asking a different question.
• A test of a single variance compares a sample variance to a claimed population variance, unlike the other three tests, which work with categorical counts.
• The chi-square distribution is not symmetric; it is skewed right, and its exact shape depends on its degrees of freedom.

Memory Tip: All four chi-square tests ask the same underlying question in different costumes — how far are the observed counts from what we’d expect if the claim were true? — only the setup (one variable, two variables, several populations, or a variance) changes.

Common Mistake: Confusing a test of independence with a test of homogeneity because both use a contingency table and the same chi-square formula. A test of independence asks whether two variables measured on one population are related; a test of homogeneity asks whether several separate populations have the same distribution of one variable — the data collection design, not the formula, is what tells them apart.

Important Questions:

• A die is rolled 60 times to test whether it is fair. Each face is expected 10 times but face “6” comes up 18 times. What type of chi-square test is this? A goodness-of-fit test — it checks whether one categorical variable’s (the die face’s) observed counts match a specific claimed (uniform) distribution.
• What does a large chi-square test statistic generally indicate? That the observed counts are far from the expected counts under the null hypothesis, which (combined with a small p-value) is evidence against that null hypothesis.

Chapter 12: Linear Regression and Correlation

Difficulty: Medium · Key topics: scatter plots, line of best fit, correlation coefficient, coefficient of determination, outliers, prediction

Chapter 12 studies bivariate data — pairs of numeric measurements on the same subjects — and asks whether the two variables move together in a predictable way. It starts with scatter plots as the visual first step, then develops the least-squares regression equation (the line of best fit) that minimizes the total squared vertical distance between the line and the data points. The chapter introduces the correlation coefficient r as a single number summarizing the strength and direction of a linear relationship, tests whether that correlation is statistically significant, and covers using the regression line for prediction while explaining how outliers can distort both the line and the correlation.

Key Points:

• The correlation coefficient r ranges from −1 to 1; values near ±1 indicate a strong linear relationship, and values near 0 indicate little to no linear relationship.
• The least-squares regression line minimizes the sum of the squared vertical distances between the data points and the line.
• The coefficient of determination r² is the proportion of variation in y that is explained by the linear relationship with x.
• A statistically significant correlation requires testing r against a critical value or p-value, not just looking at its size — significance depends on sample size too.
• Predicting y for an x-value far outside the range of the original data (extrapolation) is unreliable, even when the line fits the observed data well.
• A single extreme outlier can noticeably shift both the slope of the regression line and the value of r, especially in a small data set.

Memory Tip: r talks about direction and strength, r² talks about explanatory power — r close to 1 or −1 means a strong linear pattern, while r² converts that into “percent of the variation in y accounted for by x.”

Common Mistake: Using the regression equation to predict y for an x-value well outside the range of the data that was actually collected. A line fit to data between, say, x = 10 and x = 50 gives no real guarantee about what happens at x = 200 — the underlying relationship could easily be different outside the observed range.

Important Questions:

• A regression analysis gives r = 0.85. What does r² tell you that r alone does not? r² = 0.7225, meaning about 72.25% of the variation in y is explained by the linear relationship with x — it converts the correlation into a direct measure of explanatory power.
• Why can a single outlier be especially dangerous in a small data set’s regression line? Because the least-squares method minimizes squared vertical distances, a single far-away point contributes a disproportionately large squared distance, pulling the line’s slope toward itself much more than it would in a larger data set.

Chapter 13: F Distribution and One-Way ANOVA

Difficulty: Hard · Key topics: one-way ANOVA, F distribution, F-ratio, test of two variances

Chapter 13 closes the book with Analysis of Variance (ANOVA), the method for comparing the means of more than two groups at once without inflating the error rate that would come from running many separate two-sample t-tests. One-way ANOVA tests whether at least one group’s mean differs from the others by comparing the variance between group means to the variance within each group, using the F-ratio as its test statistic; a large F-ratio suggests the between-group differences are too large to be explained by chance alone. The chapter also covers the F distribution’s own properties and its second major use in this book: testing whether two populations have equal variances.

Key Points:

• One-way ANOVA tests whether the means of three or more groups are equal, using a single overall test rather than many pairwise t-tests.
• The F-ratio compares the variance between group means to the variance within groups; it is larger when the groups’ means differ more relative to their internal spread.
• A significant ANOVA result (small p-value) says at least one group mean differs from the others — it does not by itself say which group or groups differ.
• The F distribution is not symmetric, is always non-negative, and its exact shape depends on two separate degrees-of-freedom values.
• The null hypothesis for one-way ANOVA is that all group population means are equal; the alternative is simply that not all of them are equal.
• The F distribution is also used to test whether two populations have equal variances, a different null hypothesis from ANOVA’s but the same underlying distribution.

Memory Tip: ANOVA splits variance into between and within — the F-ratio is literally (variance between group means) ÷ (variance within groups), so a bigger gap between group averages relative to their own spread drives F up.

Common Mistake: Running three separate two-sample t-tests instead of one ANOVA when comparing three or more group means. Each additional t-test inflates the overall chance of a false positive (Type I error); one-way ANOVA tests all groups at once while keeping the overall error rate at the stated significance level.

Important Questions:

• A one-way ANOVA comparing four groups produces a p-value of 0.002. What can be concluded? At least one of the four group means is significantly different from the others — the test does not identify which group(s) without a follow-up comparison.
• Why is one-way ANOVA preferred over running multiple two-sample t-tests when comparing more than two group means? Because running many pairwise t-tests compounds the Type I error rate across tests, while one-way ANOVA tests the overall hypothesis that all means are equal in a single test, controlling the error rate properly.

Download Introductory Statistics PDF (Free)

This book is free from its official source, OpenStax. Click below to download the complete PDF — a free web-based reader edition (with per-section, linkable pages) is also available on the OpenStax site if you’d rather read online.

How to Study This Book

Start with Chapters 1 and 2 (Sampling and Data, Descriptive Statistics) as foundation — the vocabulary and graphical tools introduced here (population vs. sample, mean/median/mode, standard deviation) are assumed without re-explanation in every later chapter.

Chapters 3 through 5 build the probability machinery the rest of the course depends on: Chapter 3 gives the general rules of probability, Chapter 4 covers discrete random variables (binomial, Poisson, and the others), and Chapter 5 covers continuous random variables (uniform, exponential). Work through these three in order — Chapter 6 assumes all of them.

Chapter 6 (The Normal Distribution) and Chapter 7 (The Central Limit Theorem) are the hinge of the entire course. Z-scores from Chapter 6 and the standard error from Chapter 7 are used in literally every chapter that follows — do not move on to Chapter 8 until both are comfortable.

Chapters 8 through 10 (Confidence Intervals, Hypothesis Testing with One Sample, Hypothesis Testing with Two Samples) form the core of inferential statistics and should be read in order, since each chapter’s procedure builds directly on the last one’s five-step structure.

Chapters 11 through 13 (The Chi-Square Distribution, Linear Regression and Correlation, F Distribution and One-Way ANOVA) are largely independent of one another and can be read in whatever order your syllabus assigns them — many one-semester courses cover only two of the three. Check your course outline before assuming all three are required.

The appendices are reference and practice material, not additional syllabus chapters: Appendix E’s solution sheets are meant to be photocopied and used for the hypothesis-test homework in Chapters 9 and 10, Appendix H holds the probability tables needed if you are not using a calculator or statistical software, and Appendices A–D and G hold review exercises, practice tests, data sets for projects, and TI-83/84 calculator instructions.

Used In These Programs

This book is used for the Introductory Statistics course in BS Statistics and other BS Social Science programs with a one-semester statistics requirement. Browse all Statistics books or all Statistics category books.

Who Should Read This

Introductory Statistics 2e is written for a student taking their required one-semester Introductory Statistics course in a BS Statistics, social science, or any general BS program that requires a single statistics course — it assumes only intermediate algebra, never calculus, and works from basic sampling and descriptive statistics all the way through linear regression and one-way ANOVA. Its large practice-and-homework sets after every section, plus its many real-data collaborative exercises and simulation labs, suit a student who learns by working through problems repeatedly rather than one who wants a narrative-first text. Students needing a business-context version of the same course should instead look for OpenStax’s Introductory Business Statistics, and students in a more data-science-leaning program may prefer OpenIntro Statistics’s approach — both are separate textbooks, to be linked here once published on this site.

Applicable Universities

This book is useful for students at Pakistani universities offering BS Statistics, BS Social Sciences, BBA, or any program with a one-semester Introductory Statistics requirement, including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

FAQs

Related Books

• Discrete Mathematics: An Open Introduction – Oscar Levin (4th Edition)
• Calculus Volume 1 – Strang & Herman (OpenStax)

Introductory Statistics 2e is OpenStax’s broadest, most general entry point into a one-semester statistics course, covering everything from sampling to ANOVA in a single algebra-based text. Browse more Statistics books for the rest of your semester.

Introductory Statistics 2e, by Barbara Illowsky and Susan Dean, with contributing authors. OpenStax, Rice University. Free under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence. Access for free at https://openstax.org/details/books/introductory-statistics-2e