Review AP Statistics Unit 1 to build the foundation for the entire course: describing and comparing distributions of one variable, calculating summary statistics, and understanding how data are collected through sampling and experimental design. Every inference procedure in later units depends on the vocabulary and reasoning skills introduced here.
Use this page to review all 13 topics, from identifying variables and reading graphs to evaluating sampling methods and experimental designs.
AP Statistics Unit 1 asks two big questions: What does the data show, and where did the data come from? Answering the first question requires graphical and numerical tools for describing one-variable distributions. Answering the second requires understanding how sampling and experimental design affect what conclusions are valid.
Topics 1.1 through 1.9 build the skill of describing data. You classify variables as categorical or quantitative, represent them with appropriate graphs (bar charts, pie charts, histograms, dotplots, stem-and-leaf plots, boxplots), describe distributions using shape, center, variability, and unusual features, and calculate statistics like mean, median, IQR, standard deviation, and z-scores.
A key judgment in Unit 1 is knowing when to use resistant measures (median and IQR) versus nonresistant ones (mean, range, standard deviation). Outliers pull the mean and standard deviation but barely affect the median and IQR. Skewed distributions call for median and IQR; roughly symmetric distributions can be summarized with mean and standard deviation.
Topics 1.10 through 1.13 shift to how data are produced. Random sampling allows generalization to a population. Random assignment of treatments in a well-designed experiment allows cause-and-effect conclusions. Observational studies, even with random samples, cannot establish causation because confounding variables remain possible.
Every topic in Unit 1 connects to a single cycle: pose an investigative question, collect data using a sound method, describe and analyze the data, and draw conclusions that are appropriate for the study design. A question about a population requires a representative sample. A cause-and-effect claim requires random assignment. Describing data requires matching the graph and statistic to the variable type and distribution shape. Getting this cycle right is what makes statistical reasoning valid.
Identify the components of a statistical study: population, sample, investigative question, and the role of variation and uncertainty in drawing conclusions from data.
Classify variables as categorical or quantitative, and quantitative variables as discrete or continuous. Distinguish parameters (population) from statistics (sample).
Build frequency and relative frequency tables for categorical data and use counts and proportions to make claims in context.
Construct and interpret bar charts and pie charts; use them to compare two or more groups on the same categorical variable.
Create histograms, dotplots, and stem-and-leaf plots to display the distribution of a quantitative variable and see its shape and spread.
Describe a quantitative distribution by addressing shape (symmetric, skewed, unimodal, bimodal), center, variability, and unusual features such as outliers, gaps, or clusters in context.
Calculate mean, median, quartiles, range, IQR, and standard deviation; apply the 1.5×IQR and 2-standard-deviation outlier rules; choose resistant vs. nonresistant measures appropriately.
Construct boxplots from the five-number summary, identify outliers on a boxplot, and use the relative positions of the mean and median to describe skew.
Compare two or more distributions using parallel boxplots, back-to-back stem-and-leaf plots, or dotplots; calculate and interpret z-scores to compare relative positions within or across distributions.
Distinguish experiments, observational studies, surveys, and censuses; determine when random selection justifies generalization and when random assignment justifies causal conclusions.
Identify and compare simple random samples, stratified samples, cluster samples, and systematic samples; explain which method is most appropriate for a given population and question.
Recognize voluntary response bias, undercoverage bias, nonresponse bias, and response bias; explain how each type of bias affects the validity of conclusions.
Identify the four elements of a well-designed experiment (comparison, random assignment, replication, control); distinguish completely randomized, randomized block, and matched pairs designs; explain why random assignment supports cause-and-effect conclusions.
Open this guide for a closer review of the topic.
A statistical study collects data from a sample to answer an investigative question about a population. Every study has observational units (the individuals measured), variables (characteristics that vary across units), parameters (numerical summaries of the population), and statistics (numerical summaries of the sample). Variables are either categorical (group labels, like political party) or quantitative (numerical measurements or counts, like height in cm). Quantitative variables are further classified as discrete (countable values, like number of siblings) or continuous (any value in an interval, like weight).
| Concept | Population | Sample |
|---|---|---|
| Size symbol | N | n |
| Numerical summary | Parameter | Statistic |
| Example (mean) | μ (mu) | x̄ (x-bar) |
| Example (std dev) | σ (sigma) | s |
Categorical data are organized into frequency tables (counts per category) or relative frequency tables (proportions per category). Bar charts display these counts or proportions with bar height representing the frequency or relative frequency of each category. Pie charts show each category as a slice whose area equals its relative frequency, with all slices summing to 1. Both graph types can be used to compare two or more groups on the same categorical variable.
| Display | Shows counts? | Shows proportions? | Good for comparing groups? |
|---|---|---|---|
| Frequency table | Yes | No | Yes, side by side |
| Relative frequency table | No | Yes | Yes, standardizes group sizes |
| Bar chart | Either | Either | Yes |
| Pie chart | Either | Yes | Limited |
Quantitative distributions are displayed with histograms (values grouped into bins, bar height shows frequency or relative frequency), dotplots (one dot per value), and stem-and-leaf plots (retains original values, useful for small data sets). To describe any quantitative distribution, address shape, center, variability, and unusual features in context. Shape terms include symmetric, skewed right (long right tail), skewed left (long left tail), unimodal, bimodal, and uniform. Unusual features include outliers, gaps, and clusters.
Measures of center are the mean (x̄ = (1/n)Σxᵢ) and median (middle value when ordered). Measures of variability are the range (max minus min), IQR (Q3 minus Q1), and standard deviation (s = √[(1/(n−1))Σ(xᵢ − x̄)²]). The five-number summary (minimum, Q1, median, Q3, maximum) is displayed as a boxplot, where the box spans the middle 50% of the data. Outliers by the 1.5×IQR rule fall below Q1 − 1.5×IQR or above Q3 + 1.5×IQR; on a boxplot, whiskers extend to the most extreme non-outlier values and outliers are plotted individually. Changing units (e.g., inches to centimeters) scales all statistics by the same factor.
| Statistic | Measures | Resistant to outliers? |
|---|---|---|
| Mean | Center | No |
| Median | Center | Yes |
| Standard deviation | Spread | No |
| IQR | Spread | Yes |
| Range | Spread | No |
When comparing two or more distributions of the same quantitative variable, address center, variability, shape, and unusual features for each group and make explicit comparative statements (e.g., 'Group A has a higher median than Group B'). Back-to-back stem-and-leaf plots, side-by-side dotplots, and parallel boxplots are common comparison displays. A z-score standardizes a value: z = (xᵢ − μ) / σ, where μ is the population mean and σ is the population standard deviation. A positive z-score means the value is above the mean; a negative z-score means it is below. Z-scores allow comparison of values from distributions with different means and standard deviations.
| Display | Best for | What to compare |
|---|---|---|
| Parallel boxplots | Comparing medians and spread across groups | Median, IQR, outliers |
| Back-to-back stemplots | Two small data sets | Shape, center, unusual values |
| Side-by-side dotplots | Small-to-moderate data sets | Shape, clusters, gaps |
| z-scores | Values from different distributions | Relative position in standard deviations |
An experiment imposes treatments on experimental units and measures a response variable; random assignment of treatments allows cause-and-effect conclusions. An observational study records variables without imposing treatments; even with a random sample, confounding variables prevent causal conclusions. A census collects data from every member of the population. Random sampling methods include simple random sample (SRS, every sample of size n equally likely), stratified sampling (divide into strata, randomly sample within each), cluster sampling (randomly select entire clusters), and systematic sampling (select every kth unit after a random start). Random selection from a population justifies generalizing results to that population.
| Study type | Imposes treatment? | Supports causation? | Supports generalization? |
|---|---|---|---|
| Experiment with random assignment | Yes | Yes | Only if units randomly selected |
| Observational study with random sample | No | No | Yes, to the sampled population |
| Observational study without random sample | No | No | Only to similar individuals |
Bias is a systematic error that causes a statistic to consistently over- or underestimate a parameter. Voluntary response bias occurs when only self-selected volunteers respond (typically those with strong opinions). Undercoverage bias occurs when part of the population has no chance of being selected. Nonresponse bias occurs when selected individuals do not respond and differ systematically from those who do. Response bias occurs when question wording, interviewer presence, or social desirability causes inaccurate answers. A well-designed experiment requires comparison of at least two treatment groups, random assignment of treatments, replication, and direct control of extraneous variables. Blinding (single or double) prevents knowledge of treatment assignment from influencing responses or assessments. A completely randomized design assigns all treatments at random across all units. A randomized block design groups similar units into blocks first, then randomly assigns treatments within each block. A matched pairs design is a special block design pairing each unit with itself or a very similar unit.
| Bias type | Cause | Direction of error |
|---|---|---|
| Voluntary response | Only self-selected volunteers respond | Typically overstates extreme opinions |
| Undercoverage | Part of population excluded from frame | Underrepresents excluded group |
| Nonresponse | Selected individuals do not respond | Depends on who declines |
| Response | Question wording or social pressure | Toward socially acceptable answers |
| Term | Definition |
|---|---|
| Parameter | A numerical summary of a population characteristic, such as the population mean μ or population standard deviation σ; usually unknown and estimated using a sample statistic. |
| Categorical Variable | A variable that takes category names or group labels as values; summarized with counts and proportions, not means or standard deviations. |
| Histogram | A graph for quantitative data that groups values into bins; bar height shows the frequency or relative frequency of values in each bin. |
| Skewness | The asymmetry of a distribution; right-skewed distributions have a longer right tail and the mean is typically greater than the median, while left-skewed distributions have a longer left tail and the mean is typically less than the median. |
| IQR | The interquartile range, calculated as Q3 minus Q1; measures the spread of the middle 50% of the data and is resistant to outliers. |
| 1.5×IQR rule | A value is a potential outlier if it falls below Q1 minus 1.5×IQR or above Q3 plus 1.5×IQR. |
| Box Plot | A graphical display of the five-number summary (minimum, Q1, median, Q3, maximum); the box spans the middle 50% of the data and whiskers extend to the most extreme non-outlier values. |
| Sensitivity to extreme values | The degree to which a statistic is affected by outliers; the mean and standard deviation are sensitive (nonresistant), while the median and IQR are not (resistant). |
| Z-Score | A standardized value calculated as z = (xᵢ minus μ) divided by σ; measures how many standard deviations a data value is above (positive) or below (negative) the mean. |
| Simple Random Sample | A sample of size n selected so that every possible sample of that size has an equal chance of being chosen; the basis for most random sampling methods. |
| Cluster Sampling | A sampling method that divides the population into groups (clusters), randomly selects some clusters, and includes all individuals in the selected clusters. |
| strata | Subgroups of a population that share a common characteristic; used in stratified random sampling to ensure each subgroup is represented in the sample. |
| Response Variable | The outcome measured on each experimental unit after a treatment is applied; the variable whose behavior the experiment is designed to study. |
| Experimental Unit | The smallest unit to which a treatment is assigned in an experiment; when experimental units are people, they are called subjects or participants. |
| Matched Pairs Design | An experimental design in which each unit receives both treatments in random order, or similar units are paired and each treatment is randomly assigned within each pair; reduces variability due to individual differences. |
Even when a random sample is used, an observational study cannot establish cause and effect because confounding variables are not controlled. Only random assignment of treatments in an experiment supports causal conclusions.
When a distribution is skewed or has outliers, the mean and standard deviation are pulled toward the extreme values and misrepresent the typical value. Use median and IQR instead, and explain why they are more appropriate.
Writing 'the distribution is skewed right with a median of 45' is incomplete. Always name the variable and units: 'the distribution of exam scores (in points) is skewed right with a median of 45 points.'
In stratified sampling, you randomly sample from every stratum. In cluster sampling, you randomly select entire clusters and include all members of those clusters. Mixing these up leads to wrong conclusions about which method reduces bias or variability.
The outlier fences are Q1 − 1.5×IQR (lower) and Q3 + 1.5×IQR (upper). A common error is subtracting 1.5×IQR from the median or adding it to the mean instead of using Q1 and Q3 as the anchors.
Free-response questions frequently present a graph or summary statistics and ask you to describe or compare distributions. A complete response addresses shape, center, variability, and unusual features for each group and uses explicit comparative language ('Group A has a larger median than Group B'). Omitting context or making vague comparisons are the most common reasons for losing points on these tasks.
Multiple-choice and free-response questions often describe a study and ask what conclusions are valid. The key reasoning chain is: random selection from a population justifies generalization to that population; random assignment of treatments justifies cause-and-effect conclusions; neither is possible without the corresponding random mechanism. Identifying confounding variables in observational studies is a related skill tested on the exam.
Exam questions ask you to calculate statistics such as the mean, median, IQR, standard deviation, or z-score, and then interpret them in context or explain which measure is more appropriate for a given distribution. Knowing that the median and IQR are resistant to outliers while the mean and standard deviation are not, and being able to explain why, is a recurring reasoning task across both multiple-choice and free-response sections.
Open the individual guides for Unit 1 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsWatch past review streams filtered to Unit 1 when you want a video walkthrough.
open videosOpen the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
open unit cheatsheetEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Stats Unit 1, Exploring One-Variable Data and Collecting Data, contains 13 topics covering investigative questions; populations, samples, variables, parameters, and statistics; categorical and quantitative displays; shape, center, variability, outliers, and z-scores; random sampling; sampling bias; and experimental design. These are the Fall 2026 CED topics used for the May 2027 exam.
Unit 1 accounts for 20–30% of the multiple-choice section. That range is a section weight, not a promise that every practice form will use the same percentage, so review the full unit rather than trying to predict one exact question count.
By the end of Unit 1, you should be able to construct and compare data displays, calculate and interpret summary statistics in context, choose a sampling method, and distinguish conclusions about a population from cause-and-effect conclusions. The exam rewards the method and the interpretation, so practice writing what a result means in the problem's context instead of stopping at a calculator output.
Common Unit 1 mistakes include describing a distribution without context, treating the mean as resistant to outliers, confusing random sampling with random assignment, or claiming causation from an observational study. Slow down long enough to identify the variables, population, and requested conclusion before calculating.
Start with the vocabulary and conditions, then mix short calculations with full-sentence interpretations. For each missed question, label whether the problem was choosing a method, doing the calculation, or interpreting the result. Finish with timed mixed practice so you have to recognize the method without a topic label.
The study-design ideas return whenever you verify inference conditions, while one-variable displays and summaries support later probability and regression work.