Exploring One-Variable Data and Collecting Data

Redesigned for 2026-27 — read this first

AP Statistics was redesigned for 2026-27. The material below follows the previous nine-unit framework.

·The course was reorganized from nine units into five.
·Some topics were removed, and the old Unit 9 (inference for slopes) is not a unit of its own in the new framework.
·The exam moves fully digital for May 2027, with 42 multiple-choice questions of four options each.

Our 36 lessons, 181 flashcards and 6 free-response prompts are still organized against the nine-unit version. The statistics itself has not changed — a confidence interval is a confidence interval — so the material is still worth studying. What is out of date is how it is grouped, and which topics are still examinable. We are not rebuilding it from secondary sources that disagree with each other; it will be rewritten against the official framework.

We have not been able to read the official framework directly, so treat the summary above as our best understanding and confirm it yourself — AP Central is the authority.

20–30% of the exam7 lessons · 95 min95 terms

Cram this unit Flip through the cards Sit a unit test Math practice

What this unit covers

The topics below follow the published Statistics course framework for Unit 1. This unit is worth 20–30% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Displaying and describing distributionsSummary statisticsNormal distributionSampling methodsExperimental design

Lessons in this unit

Formulas in Unit 1

Sample standard deviation

s = sqrt( Σ(x_i − x-bar)² / (n − 1) )

Find each deviation from the mean, square it, average the squares using n − 1 (not n), then take the square root. Dividing by n − 1 corrects the tendency of a sample to underestimate the population spread.

Mean vs. median under skew

skewed right: mean > median · skewed left: mean < median · symmetric: mean ≈ median

The mean chases the tail. Whichever side the long tail is on, that is the side the mean sits relative to the median.

z-score (standardizing)

z = (x − μ) / σ

Subtract the mean, divide by the standard deviation. Positive z is above the mean, negative z is below. z has no units.

Empirical (68–95–99.7) rule

μ ± 1σ ≈ 68% · μ ± 2σ ≈ 95% · μ ± 3σ ≈ 99.7%

For a Normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

Outlier fences (1.5 × IQR rule)

lower fence = Q1 − 1.5·IQR · upper fence = Q3 + 1.5·IQR · IQR = Q3 − Q1

A data value is an outlier if it is below the lower fence or above the upper fence. The fences themselves are not necessarily data values.

Linear transformation y = a·x + b

mean_y = a·mean_x + b median_y = a·median_x + b SD_y = |a|·SD_x IQR_y = |a|·IQR_x variance_y = a²·variance_x

The additive constant b appears in the centers and never in the spreads. This one line answers every transformation question on the exam.

Every term in Unit 1

All 95 terms we publish for Exploring One-Variable Data and Collecting Data, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Observational study vs experiment: An experiment imposes a treatment; an observational study only records. Only an experiment can establish causation.
1.5 × IQR outlier rule: A point is an outlier if it lies below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).
z-score: z = (value − mean)/SD: how many standard deviations a value sits from the mean; unitless, so it compares across distributions.
Blocking: Grouping units that are similar on a known variable, then randomizing treatments within each block to reduce variability — the design analog of stratifying.
Categorical vs quantitative variables: Categorical places individuals in groups; quantitative takes numerical values you can average. Zip codes are numbers but categorical.
Describing a distribution: Shape, Outliers, Center, Spread — and always in context. Omitting context is the commonest lost point in the whole course.
Skewness direction: Skewed right has a long right tail and mean above median; skewed left is the reverse. The tail names the skew, not the bulk of the data.
Mean vs median resistance: The median resists outliers; the mean does not. Report the median and IQR for skewed data, mean and standard deviation for roughly symmetric data.
Standard deviation: The typical distance of a value from the mean. It is zero only when every value is identical, and is never negative.
Interquartile range: IQR = Q3 − Q1, the spread of the middle half. Resistant to outliers, unlike the range.
Boxplot limitations: Shows the five-number summary but hides multimodality — two very different distributions can produce identical boxplots.
Effect of transformations: Adding a constant shifts center but not spread; multiplying scales both center and spread by that constant.
Normal distribution and the empirical rule: About 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean.
Percentile: The percentage of observations at or below a value. The 90th percentile is not the same as a score of 90.
Density curve: A smooth model with total area 1 under it. Area corresponds to proportion, which is why probability is an area.
Comparing distributions: Compare shape, center and spread explicitly with comparative language — "the median for group A is higher than for group B" — not two separate descriptions.
Population vs sample: The population is everyone you want to describe; the sample is who you actually measure. Inference generalises from one to the other.
Parameter vs statistic: A parameter describes a population (μ, p, σ); a statistic describes a sample (x̄, p̂, s). Statistics estimate parameters.
Simple random sample: Every group of the given size has an equal chance of being chosen — a stronger condition than every individual having an equal chance.
Stratified random sample: Divide the population into similar groups and sample within each. Reduces variability when strata differ from each other.
Cluster sample: Divide into groups, then randomly select whole groups. Cheaper than stratifying, and used when clusters resemble the population.
Systematic sample: Select every kth individual after a random start. Valid unless there is a periodic pattern matching the interval.
Convenience and voluntary response samples: Both are biased — voluntary response over-represents people with strong opinions. Neither supports inference.
Undercoverage: Some part of the population has no chance of being selected, so the sample cannot represent it however large it is.
Nonresponse bias: Selected individuals who do not respond may differ systematically from those who do.
Response bias: The way a question is asked, or who asks it, changes the answers given. Distinct from nonresponse.
Confounding variable: A variable associated with both the explanatory variable and the response, so their effects cannot be separated.
Three principles of experimental design: Comparison with a control group, random assignment of treatments, and replication with enough experimental units.
Random assignment vs random selection: Random assignment permits causal conclusions; random selection permits generalization to the population. They answer different questions.
Blinding and placebo effect: Single-blind hides treatment from subjects, double-blind from subjects and those measuring. Controls for expectation, which is a real physiological effect.
Matched pairs design: Each subject receives both treatments, or subjects are paired by a similar characteristic. Analyzed with a one-sample t procedure on the differences.
Why context is scored: Every description must name the variable and its units. "The distribution is skewed right" is incomplete; "the distribution of commute times is skewed right" is not.
Choosing a graph type: Bar charts and pie charts for categorical data; dotplots, stemplots, histograms and boxplots for quantitative. Using a bar chart for quantitative data loses the point.
Histogram vs bar chart: Histogram bars touch because the variable is continuous; bar chart bars are separated because the categories are distinct.
Effect of an outlier on mean and median: The mean is pulled toward the outlier; the median barely moves. This is what "resistant" means.
Standardizing and comparing: Converting to z-scores lets you compare values from different distributions, such as an SAT score against an ACT score.
Normal probability calculations: Sketch, shade, standardize, then find the area. The sketch is worth doing because it catches the reversed-tail error.
Cumulative relative frequency graph: Read percentiles directly: the height at a value is the proportion at or below it, so the median is where the curve crosses 0.5.
Describing a sampling procedure fully: Name the population, how individuals are numbered, how the random mechanism is applied, and what to do about repeats.
Why a large biased sample is worse than a small random one: Bias does not shrink with sample size; only variability does. A million-person voluntary poll is still unrepresentative.
Scope of inference: Random selection lets you generalize to the population; random assignment lets you claim causation. Both together permit both conclusions.
Experimental units vs subjects: Subjects are human experimental units. Naming them correctly matters when describing replication.
Completely randomized vs block design: Completely randomized assigns treatments at random to all units; blocking groups similar units first and randomises within each block.
Why blocking is not the same as stratifying: Blocking happens in an experiment to control a nuisance variable; stratifying happens in a sample to reduce variability of an estimate.
Placebo and control group: A control group provides a comparison baseline; a placebo controls for the psychological effect of receiving a treatment. They are not synonyms.
Dotplot: One dot per observation above its value. Best for small data sets, because it shows every individual value, gaps and clusters — information a histogram hides inside its bars.
Stemplot: Splits each value into a stem and a leaf, so the shape is visible while every original value is preserved. Back-to-back stemplots are a compact way to compare two small groups.
Frequency vs relative frequency: Frequency is a count; relative frequency is that count divided by the total. Relative frequency is required when comparing groups of different sizes — comparing raw counts across unequal groups is a standard error.
Modality: The number of distinct peaks: unimodal, bimodal, multimodal. Bimodality often signals two subgroups mixed together, so it is worth mentioning because it suggests the data should be split.
Uniform distribution shape: Roughly equal frequencies across all values, giving a flat histogram. It has no peak, so describing its shape means saying "approximately uniform" rather than naming a center of concentration.
Gaps and clusters: A gap is an interval with no observations; a cluster is a group separated from the rest. Both belong under "unusual features" in a description and often indicate a mixture of populations.
Range as a measure of spread: Maximum minus minimum — a single number using only two observations, so it is maximally sensitive to outliers. Report it as context, not as the primary measure of spread.
Why variance uses squares: Deviations from the mean sum to zero, so they must be made positive before averaging. Squaring does that while weighting large deviations more heavily, and it makes variances add for independent variables.
Why the sample standard deviation divides by n − 1: Dividing by n underestimates the population spread, because deviations are measured from the sample mean rather than the true mean. Using n − 1 corrects the bias; the quantity is the degrees of freedom.
Five-number summary: Minimum, Q1, median, Q3, maximum — the basis of a boxplot. It describes center and spread with resistant statistics and requires no assumption about shape.
Locating the quartiles: Q1 is the median of the lower half of the ordered data and Q3 the median of the upper half, with the overall median excluded from both halves when n is odd.
Modified boxplot: Plots outliers individually as separate points and draws the whiskers only to the most extreme non-outliers. It is the version to draw, because a plain boxplot conceals outliers inside the whisker.
Side-by-side boxplots: The standard tool for comparing several groups on one axis. They show center, spread and outliers cleanly and hide shape almost entirely — in particular they cannot reveal bimodality.
Empirical rule: For an approximately Normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. It applies only when the distribution is roughly Normal.
Working backward from a percentile: Find the z with the required area below it, then convert with x = μ + zσ. The 90th percentile corresponds to z ≈ 1.28 and the 95th to z ≈ 1.645.
Normal probability plot: Plots observed values against expected Normal scores. A roughly straight line supports Normality; systematic curvature indicates skew, and an S-shape indicates heavy or light tails.
Comparing values on different scales: Convert each to a z-score. A score 1.8 standard deviations above the mean on one test beats one 1.2 above on another, whatever the raw numbers or maximum possible scores.
Effect of an outlier on the standard deviation: Large, because the deviation is squared before averaging — a single extreme value can inflate the standard deviation substantially while barely moving the IQR.
Resistant measures: Statistics barely affected by extreme values: median, IQR, quartiles. The mean, standard deviation, range and correlation are all non-resistant, which is why shape decides which to report.
The mean as a balance point: The point at which the deviations above and below exactly cancel. This is why a long tail drags the mean toward it, while the median only counts how many observations lie on each side.
Position of the median: At the (n + 1)/2 ordered position: the middle value when n is odd, the average of the two middle values when n is even. Note this is a position, not a value.
Naming the direction of skew: Skew is named for the tail, not the peak. A right-skewed distribution has its long tail toward the high values and its bulk on the left, and its mean exceeds its median.
Bin width in a histogram: Too wide hides structure; too narrow makes random noise look like structure. The same data can appear unimodal or bimodal depending on the choice, which is why a dotplot is safer for small data sets.
Describing shape in context: Name the shape and say what it means for the variable: "the distribution of repair times is right-skewed, so a few jobs take much longer than typical." Shape alone earns less than shape interpreted.
Cumulative relative frequency plot: Rises from 0 to 1 and never falls. Read a percentile by entering on the vertical axis; the curve is steepest where observations are densest.
Standardizing does not normalize: A z-score can be computed for any distribution and always yields mean 0 and standard deviation 1. Converting that z to a percentile with a Normal table requires the distribution to be approximately Normal.
Segmented bar chart: Shows the conditional distribution of one categorical variable within each level of another. Comparing the segments across bars is how association between two categorical variables is judged visually.
Marginal distribution: The row or column totals of a two-way table expressed as proportions — the distribution of one variable ignoring the other. It answers a different question from a conditional distribution.
Why the same data can be described several ways: Mean and standard deviation, median and IQR, and the five-number summary are all legitimate; the shape determines which is honest. Reporting a mean for strongly skewed data is a poor choice rather than an arithmetic error.
Sampling frame: The list from which the sample is actually drawn. If it omits part of the population — an unlisted-number problem, an out-of-date register — undercoverage results no matter how good the randomization is.
Census: An attempt to measure every member of the population. It removes sampling variability but is expensive, slow and often less accurate than a good sample because of nonresponse and measurement error at scale.
Multistage sampling: Sampling in successive stages — states, then districts, then households. Most large national surveys use it because a simple random sample of a whole country is logistically impossible.
Replication: Applying each treatment to enough experimental units that a real effect can be distinguished from chance variation. It is one of the three principles of experimental design, alongside control and randomization.
Purpose of a control group: Provides a baseline against which the treatment is compared, so that any change can be attributed to the treatment rather than to the passage of time or the act of being studied.
Double-blind: Neither the subjects nor those measuring the response know the assignment. This blocks both the placebo effect and unconscious bias in how the outcome is assessed.
Single-blind: Only one party — usually the subjects — is unaware of the assignment. Weaker than double-blind, because the assessor may still influence or interpret the outcome differently by group.
Placebo effect: A response produced by the expectation of treatment rather than the treatment itself. Its existence is the reason a control group receiving an inert treatment is preferred to no treatment at all.
Factor, level, treatment: A factor is an explanatory variable; its levels are its possible values; a treatment is one combination of levels across all factors. Two factors with three and two levels give six treatments.
Randomized block design: Subjects are grouped into blocks of similar individuals and randomization occurs separately within each block. It removes the blocking variable as a source of variation, increasing sensitivity.
Why random assignment permits causal claims: It makes the treatment groups similar in every respect, measured and unmeasured, so a significant difference in response can be attributed to the treatment rather than to a pre-existing difference.
Why random selection permits generalization: It makes the sample representative of the population it was drawn from, so the results can be extended to that population. It is a separate question from causation and requires a separate design feature.
The scope-of-inference grid: Random assignment plus random selection allows a causal conclusion generalized to the population; random assignment alone allows causation for the subjects studied; random selection alone allows generalized association only; neither allows neither.
Question wording effects: Small changes in phrasing produce large changes in response. Leading or emotionally loaded wording is a source of response bias, and it is not fixed by increasing the sample size.
Social desirability bias: Respondents shade answers toward what they believe is socially approved, particularly on sensitive topics. Anonymity and self-administration reduce it; a face-to-face interviewer increases it.
Nonresponse versus undercoverage: Undercoverage means part of the population could never be selected; nonresponse means selected individuals did not answer. Both bias results, and they call for different remedies.
Retrospective versus prospective: A retrospective study looks backward at records or recall; a prospective study follows subjects forward. Prospective designs suffer less recall bias but take much longer and cost more.
Limitations of observational studies: They can establish association and cannot establish causation, because groups differ in ways other than the variable of interest. Statistical adjustment helps but cannot rule out unmeasured confounders.
Describing a randomization procedure: Say how you label the units, how you generate the random numbers, what you do about repeats, and when you stop. A description a reader could not follow exactly does not earn the point.
Statistically significant in an experiment: A difference in response too large to be plausibly explained by the random assignment alone. With random assignment, that supports a causal conclusion for the subjects in the study.
Why a large biased sample is not better: Sample size controls variability, not bias. A large voluntary-response poll estimates the wrong quantity precisely, which is more dangerous than estimating it imprecisely.

What examiners penalize here

When a prompt says "skewed" or mentions outliers, default to **median and IQR**. When it says "roughly symmetric" or "bell-shaped," the **mean and standard deviation** are appropriate. Matching the summary to the shape is a routine AP decision point.
Show the fence arithmetic explicitly on free response: state IQR = Q3 − Q1, multiply by 1.5, then add to Q3 and subtract from Q1. Graders want to see the boundary values, not just the word "outlier."
A reliable checklist: **shape, center, spread, unusual — in context, comparatively**. Four elements, two requirements. Say the variable name and units at least once. If the distributions are skewed, say so *and* switch to median and IQR in the same breath.
Keep three quantities distinct. The **raw score** is the measurement in its own units. The **z-score** says how many standard deviations from the mean it lies and is unitless. The **percentile** says what proportion of the distribution lies below it. They are three different answers to three different questions, and a question asking for one earns nothing for another.

Practice Statistics

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

84 practice questionsEvery one with a written explanation FRQ trainerWritten answers scored against the rubric Statistics study guideAll units, weights and score data in one place All 5 Statistics unitsThe full course

Questions about this unit

How much of the AP Statistics exam is Unit 1?

Unit 1, Exploring One-Variable Data and Collecting Data, is worth 20–30% of the Statistics multiple-choice section according to the published course framework. Across all 5 units that makes it one of the heaviest units on the exam, and worth front-loading.

What topics are covered in Statistics Unit 1?

Exploring One-Variable Data and Collecting Data covers Displaying and describing distributions, Summary statistics, Normal distribution, Sampling methods and Experimental design. We publish 95 terms with definitions for this unit, all of them on this page.

How should I study Statistics Unit 1?

Read the 7 lessons below first — about 95 minutes — then drill the 95 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.

All 5 units of AP Statistics

Unit names, topics and exam weights follow the published College Board course framework for AP Statistics. AP® is a trademark registered by the College Board, which does not endorse this site.