Statistics

Statistics by Barbara Illowsky, Susan Dean, and contributors is the OpenStax high-school text developed from material associated with the Texas Education Agency. Thirteen chapters connect sampling, descriptive methods, probability, distributions, estimation, testing, regression, and analysis of variance. This guide follows that specific source and emphasizes explaining a statistical conclusion in context, checking assumptions, and distinguishing a calculation from a defensible inference. Projects and practice resources in the appendices support application beyond isolated formulas.

Chapter-wise summaries and study tasks

Chapter 1 Sampling and Data

Statistical reasoning begins with a question, a population, and a plan for obtaining observations. Samples, variables, data types, and levels of measurement affect what can be concluded. Sampling bias and nonresponse can undermine a large dataset, while experimental assignment and population sampling answer different concerns. Ethics and clear definitions belong at the planning stage.

Identify what one recorded value represents and who could be excluded. A convenient sample is not automatically representative, and an observational association does not become causal simply through careful arithmetic.

Study task. Plan a fictional survey or experiment with a target population, variables, and collection method. Identify a likely bias and state the strongest conclusion the design could support.

Chapter 2 Descriptive Statistics

Descriptive statistics organizes the observed data through tables, graphs, center, position, and variability. Mean and median respond differently to skew or outliers, while spread distinguishes datasets with similar centers. Grouped summaries and visual choices can conceal features that remain visible in the underlying records.

Choose a summary for the question rather than report every available number. Explain units, sample size, and unusual observations. A graph should support a fair comparison with clear axes and scale, not create an impression unsupported by the measurements.

Study task. Construct two datasets with a similar center and different spread. Compare mean, median, and graphs, then explain which summary best answers a stated question and which feature it omits.

Chapter 3 Probability Topics

Probability describes uncertainty through outcomes, events, and rules for combining them. Conditional probability changes the reference set, while independence states a specific relationship rather than a general lack of relevance. Mutually exclusive events differ from independent events. Tables, diagrams, and trees help organize counting and conditions.

State what is known before calculating. Reversing a conditional can produce a persuasive but incorrect conclusion, especially when base rates differ. Check whether probabilities cover the intended possibilities without overlap or omission.

Study task. Use a fictional two-way table to compare two reversed conditional probabilities. Explain the denominators and distinguish an independence claim from a mutually exclusive event relationship.

Chapter 4 Discrete Random Variables

A discrete random variable assigns numerical values to outcomes, and its distribution specifies probabilities over those values. Expected value summarizes a weighted average, not a result guaranteed in one trial. Variance and standard deviation describe variability. Named models require conditions such as a fixed trial structure or specified event mechanism.

Choose the distribution by its assumptions, not only the appearance of integer data. Distinguish a probability at one value from a cumulative probability and verify that the complete distribution has valid totals.

Study task. Build a small discrete distribution, calculate its expectation and spread, and compare them with simulated repetitions. Justify or reject a named model using the underlying trial conditions.

Chapter 5 Continuous Random Variables

Continuous random variables use density and interval probability rather than assigning positive probability to each exact point. Areas under a valid density account for probability, while the height of the curve is not itself an event probability. Uniform and other models illustrate different mechanisms and support different interpretations.

Check the domain, units, and total area before calculation. A continuous model approximates measured data under assumptions; recording rounded numbers does not by itself change the underlying phenomenon into the same discrete process as a count.

Study task. Interpret interval probabilities for a simple density and explain why a single point has zero probability under the model. Compare density height with the area representing the actual event.

Chapter 6 The Normal Distribution

The normal distribution supplies a symmetric model characterized by location and scale. Standardization expresses distance from the mean in standard-deviation units, allowing probabilities or percentiles to be related to a reference distribution. A transformation does not establish that the original data are normally distributed.

Inspect the context and data before using the model. Tail probabilities and central intervals require clear event statements, while a typical-range rule is only an approximation under appropriate conditions. Separate a modeled probability from an observed sample proportion.

Study task. Translate a contextual question into a standardized event and recover a percentile in original units. Explain what evidence would be needed before applying a normal model to the data.

Chapter 7 The Central Limit Theorem

The central limit theorem concerns the sampling behavior of sums or means under conditions, not a claim that every raw dataset becomes normal as more values are collected. Sampling distributions connect population variation, sample size, and the variability of a statistic. Standard error describes that sampling variability rather than the spread of individual observations.

Sample-size effects depend on independence and the underlying distribution. Distinguish one large sample from many hypothetical repetitions used to define a sampling distribution. This distinction supports later estimation and testing.

Study task. Simulate repeated sample means from a nonnormal population at several sample sizes. Compare their center and spread with individual observations and explain what the theorem does not change about the raw population.

Chapter 8 Confidence Intervals

Confidence intervals combine an estimate with uncertainty under a stated procedure and assumptions. Confidence level describes the long-run performance of that procedure rather than the probability that a fixed parameter lies in one already computed interval. The appropriate distribution and standard error depend on the parameter and available information.

Interpret the interval in context, including sampling limitations. A narrow interval does not correct bias, and greater nominal confidence generally trades against precision. Sample-size planning requires a target and assumptions rather than a universal number of observations.

Study task. Construct a fictional interval and explain its confidence meaning accurately. Compare effects of sample size and confidence level, identifying a bias the interval calculation would not repair.

Chapter 9 Hypothesis Testing with One Sample

One-sample hypothesis testing compares observations with a defined reference claim. Null and alternative hypotheses determine the event and direction being assessed, while the test statistic and p-value depend on the model. Type I and Type II errors describe different decision risks. Statistical significance does not establish a large or important effect.

A p-value is not the probability that the null hypothesis is true. State assumptions and distinguish rejection from failure to reject. Report the estimate and context alongside the decision rather than replace reasoning with a threshold.

Study task. Write hypotheses for a fictional question before examining results. Interpret a supplied p-value, estimate, and interval, then explain what a failure to reject would and would not establish.

Chapter 10 Hypothesis Testing with Two Samples

Two-sample inference compares populations or conditions while respecting whether observations are independent or paired. A paired design analyzes within-pair differences, not two unrelated samples. Means and proportions require different methods and assumptions. Unequal sample sizes or variability affect calculations and interpretation.

Study design remains central: a detected difference does not alone identify its cause. Consider practical magnitude and uncertainty as well as significance. Define the comparison parameter so the sign and units of the result remain meaningful.

Study task. Compare a paired and an independent version of a fictional study. Specify the parameter, suitable analysis, and assumptions, then explain why treating matched observations as unrelated would discard useful design information.

Chapter 11 The Chi-Square Distribution

Chi-square methods compare observed counts with expectations under a specified categorical model. Goodness-of-fit and independence questions use different structures even though both can involve chi-square statistics. Expected counts and degrees of freedom follow from the question and model constraints. The table contains counts, not arbitrary percentages treated as new observations.

A significant association does not establish causation or identify every category responsible. Check conditions and examine the pattern of departures. A large sample can make a modest departure detectable, reinforcing the need for contextual interpretation.

Study task. Set up expected counts for a fictional goodness-of-fit or independence question. Explain degrees of freedom and identify which departures contribute most, without converting association into a causal claim.

Chapter 12 Linear Regression and Correlation

Correlation and linear regression describe relationships between quantitative variables through different summaries. A fitted line predicts an outcome under a model, while residuals reveal departures from its structure. Outliers, influential observations, restricted ranges, and nonlinear patterns can undermine a simple coefficient-based interpretation.

Association does not settle causal direction or confounding. Extrapolation beyond observed conditions requires justification. Check a scatterplot and residual behavior before treating a fitted equation as dependable, and preserve units when interpreting slope.

Study task. Plot a fictional dataset, fit a line, and inspect residuals. Explain slope in context, compare correlation with prediction, and identify an extrapolation or causal claim the data do not support.

Chapter 13 F Distribution and One-way Anova

The F distribution and one-way ANOVA support comparison of several group means through between-group and within-group variation under assumptions. A significant overall result indicates that the full equal-means model is inconsistent with the evidence; it does not by itself identify every differing pair. Independence, variance conditions, and distributional assumptions require consideration.

Grouping and study design determine the population comparison. Several unplanned separate tests are not an equivalent replacement for a coherent analysis. Explain effect size or group patterns alongside an overall decision and retain uncertainty about specific contrasts.

Study task. Analyze fictional group summaries through the ANOVA logic. State the overall hypotheses, assumptions, and limits of a significant result, then identify what additional analysis would be needed for a particular pairwise claim.

How to study this book

1. Start with the question, population, variable type, and study design before selecting a method.
2. Use the source’s datasets, group projects, review exercises, and practice tests for mixed application.
3. Explain each result in words with units, assumptions, uncertainty, and limits.
4. Check graphs and raw records before relying on a summary statistic or automated output.
5. Use calculator guidance as support for the reasoning, verifying the selected procedure and the meaning of its inputs.

Questions for reflection

• Which uncertainty does the method quantify, and which bias remains outside it?
• What changes when observations are paired rather than independent?
• Does the conclusion concern description, prediction, or causation?

Source edition and coverage

This guide covers all thirteen chapters of the linked OpenStax high-school Statistics text. It is distinct from Introductory Statistics 2e and the business-statistics editions. Project materials, datasets, calculator notes, tables, and practice assessments remain in the complete source. These are original study explanations and learning tasks. Read the complete book at the official source.