Introductory Statistics

About this book

Introductory Statistics by Douglas S. Shafer and Zhiyi Zhang develops statistical reasoning from data description through probability, sampling, estimation, tests, and regression. The linked edition has eleven substantive chapters followed by a twelfth chapter of reference tables. This guide emphasizes the relationship between a research question, the data-producing process, the selected method, its assumptions, and the scope of the conclusion.

Chapter and section guide

1. Introduction

The opening chapter distinguishes a population from a sample and a parameter from a statistic. These distinctions explain why statistics is more than arithmetic on a spreadsheet. Descriptive work summarizes observed data, while inference uses data to learn about something beyond those observations. The way observations are obtained affects what can be concluded. Students should identify the target population and the unit measured before discussing an average or proportion. A large sample does not automatically repair a biased sampling process.

Study application: Describe a fictional study’s population, sampling method, measurement, parameter, and statistic. Identify one possible selection bias and explain why increasing sample size alone may not remove it.

2. Descriptive Statistics

Graphs, measures of center, and measures of spread summarize different aspects of data. A mean can conceal important variation or respond strongly to unusual observations; a median answers a different question. Displays should respect the variable’s type and make scale choices visible. The chapter develops both numerical calculation and interpretation. Students should inspect distributional shape before reducing a dataset to one number. A correct summary also identifies units, missing information, and whether a comparison is between groups or within a group.

Study application: Construct a small original dataset containing one extreme value. Compare mean and median, then explain which description is more informative for a specific question rather than declaring one universally better.

3. Probability

Probability supplies a mathematical language for uncertainty. Sample spaces, events, complements, unions, intersections, and conditional probabilities describe different questions. Independence is a specific relationship, not simply a lack of visible connection. Mutually exclusive events generally should not be confused with independent events. Students should translate verbal conditions into an event before selecting a rule. A diagram or table can reveal whether a calculation counts the same outcome twice or uses the wrong reference group.

Study application: Create a two-way table and calculate one joint probability and one conditional probability. Explain how the denominator changes and why reversing the condition can change the answer.

4. Discrete Random Variables

A discrete random variable assigns numerical values to outcomes, and its distribution describes their probabilities. Expected value summarizes a long-run weighted average rather than necessarily naming an outcome that can occur. Variance measures dispersion around that average. The binomial model requires a particular trial structure, so counting successes does not automatically justify it. Students should verify the assumptions before calculating and interpret a tail probability in the context of the event originally asked about.

Study application: Define a binomial model with stated trial count and success probability. Identify a realistic change that would violate independence or constant probability and explain why the model would then need reconsideration.

5. Continuous Random Variables

For continuous variables, probability corresponds to area under a density rather than the height of the curve at a point. The normal model provides a widely used example, with standardization connecting different means and scales. Students should distinguish an individual observation’s z-score from the distribution of a sample statistic. A model is a proposed approximation to a population, not a consequence of drawing a smooth graph. Tail direction and the event’s units should remain visible throughout the calculation.

Study application: Sketch a normal curve and shade a specified interval before standardizing. Explain why the probability of one exact value is zero in the continuous model even when the density there is positive.

6. Sampling Distributions

Repeated samples produce variation in statistics such as means and proportions. A sampling distribution describes that variation and differs from the distribution of individual observations. The central limit theorem supports useful approximations under conditions; it is not a claim that every original dataset becomes normal. Standard error measures uncertainty associated with the statistic, while standard deviation can describe variability among observations. This distinction underpins both confidence intervals and hypothesis tests.

Study application: Compare individual measurements with means from many repeated samples. Explain what changes as sample size increases and which population variability does not thereby disappear.

7. Estimation

Point estimates provide a single proposed value, while intervals express uncertainty through a procedure with a specified coverage property. A confidence interval depends on the sampling design, model, and method, not only the observed mean. Margin of error reflects some sources of uncertainty but does not automatically include bias or measurement problems. Students should interpret the interval in terms of the parameter and procedure rather than assign a new probability to a fixed parameter after calculation.

Study application: Write an interval conclusion that names the population parameter and assumptions. Explain how a narrower interval can still be misleading if the sample systematically excludes part of the target population.

8. Hypothesis Tests

Testing begins with a claim about a population and a model of how evidence behaves under the null hypothesis. A p-value is calculated conditionally on that model; it is not the probability that the null is true. Significance level, test direction, and errors should be specified and understood. Rejecting a null differs from proving a theory, and not rejecting it differs from demonstrating exact equality. The chapter connects numerical decisions with the size and practical meaning of an effect.

Study application: Write null and alternative hypotheses for an original question. Describe a Type I error and a Type II error in plain language, then explain why statistical significance alone does not establish practical importance.

9. Two-Sample Problems

Comparing groups introduces questions about independence, pairing, population characteristics, and the quantity being compared. Matched observations require different reasoning from two independent samples. A difference in sample averages must be evaluated relative to its uncertainty and the study design. Students should identify the parameter, method, and assumptions before substituting values. Even a well-calculated comparison does not establish causation when the design leaves important alternative explanations unresolved.

Study application: Describe one paired study and one independent-groups study. Explain the appropriate unit of analysis and why ignoring pairing can discard useful information or distort the uncertainty calculation.

10. Correlation and Regression

Correlation describes a form of association, while regression models a response in relation to an explanatory variable. The fitted line, residuals, and uncertainty each supply different information. A strong association does not establish a causal mechanism, and prediction outside the observed range adds unsupported assumptions. Students should inspect plots, unusual observations, and the plausibility of the linear form. A slope has units and a conditional interpretation that should be explained in the setting of the data.

Study application: Fit or sketch an original simple relationship and interpret its slope with units. Identify one possible confounder and explain why the regression calculation alone cannot eliminate it.

11. Chi-Square and F Tests

The final teaching chapter extends inference to questions involving categorical counts and comparisons using additional distributions. Expected counts, degrees of freedom, assumptions, and the structure of the null hypothesis matter to choosing a procedure. A statistically unusual pattern does not by itself identify its cause. Students should distinguish observed frequencies from expected frequencies and a test statistic from the substantive conclusion. The chapter broadens the set of tools while retaining the same disciplined sequence of question, model, evidence, and interpretation.

Study application: For a small fictional contingency table, explain how expected counts under independence are constructed. State what a rejection would indicate and what it would not establish about causation.

12. Statistical Tables

The appendix supplies reference distributions and values used in earlier chapters. A table is useful only when the parameter, degrees of freedom, and tail convention are correctly identified. Different tables may report cumulative or tail areas, so copying a number without reading the heading can invert a calculation. Students should connect the table lookup with a shaded sketch and the event being calculated. Software output should be interpreted with the same attention to conventions.

Study application: Take one probability calculation and verify it with both a table and an independent method available to you. Explain any difference caused by rounding or table conventions.

Study instructions

1. State the population, parameter, sampling process, and question before choosing a statistical method.
2. Draw the relevant distribution or data display and label the event or comparison.
3. Check assumptions explicitly, including independence, pairing, variable type, and model suitability.
4. Report an estimate or effect size alongside inferential results and distinguish uncertainty from bias.
5. Write conclusions in the context of the study, preserving the limits imposed by its design.

Practice and review task

Design a small fictional comparison study. Specify the sampling process and parameter, choose an appropriate summary and inferential approach, and write a conclusion that distinguishes association, uncertainty, and any unsupported causal claim.

Questions for reflection

Why can a very precise estimate be biased? How do a p-value, a confidence interval, and a practically meaningful effect answer different questions?

Sources and coverage

This original study guide follows Linked Shafer–Zhang textbook PDF, eleven teaching chapters and statistical-table appendix. It draws on the source contents, chapter introductions, selected explanations and examples, and review material. It condenses the resource rather than replacing the full text; study applications are original exercises.

Reviewed source text; Book and cover catalog record.