AP Statistics - Unit 1 - Topic 1.8

Boxplots and Five-Number Summaries Graphical Representations of Summary Statistics

A boxplot turns the five-number summary into a picture. In AP Statistics, that picture helps you show center, spread, possible outliers, and clues about skew without listing every data value.

Skill 3.A Construct tabular and graphical representations of data and distributions.

Skill 4.A Describe and compare tabular and graphical representations of data, as well as summary statistics.

Lesson Overview

Use these boxes as your quick AP checklist before drawing or reading a boxplot.

Main Goal

Topic 1.8 is about displaying summary statistics for one quantitative variable, especially with a boxplot.

AP Focus

Be ready to connect a five-number summary to a boxplot and explain why a boxplot is the right graph for that summary.

Shape Clue

If the mean is larger than the median, the distribution is usually skewed right. If the mean is smaller, it is usually skewed left.

Big Limitation

A boxplot is great for summaries, but it does not show exact individual values, gaps, clusters, or how many peaks a distribution has.

Key Definitions

These terms are the language you need for boxplot questions.

Five-Number Summary

The five-number summary is the minimum, Q1, median, Q3, and maximum, written in order from smallest to largest.

Boxplot

A boxplot is a graph of the five-number summary. It is also called a box-and-whisker plot.

The Box

The box starts at Q1 and ends at Q3. It contains the middle 50% of the data.

Median Line

The line inside the box marks the median, the middle value of the ordered data set.

Whiskers

The whiskers extend from Q1 toward the small values and from Q3 toward the large values.

Modified Boxplot

If there are outliers, the whiskers stop at the most extreme data values that are not outliers, and outliers are plotted separately.

Formulas and AP Notation

These are the rules that control boxplot construction.

Five-number summary \[\text{min},\ Q_1,\ \text{median},\ Q_3,\ \text{max}\]

Always keep the order. A boxplot is built from these five positions.

Interquartile range \[\mathrm{IQR} = Q_3 − Q_1\]

The IQR measures the width of the middle 50% of the data.

Outlier fences \[\begin{aligned}\text{Lower fence} = Q_1 − 1.5(\mathrm{IQR})\\\text{Upper fence} = Q_3 + 1.5(\mathrm{IQR})\end{aligned}\]

Values below the lower fence or above the upper fence are flagged as possible outliers.

Mean and median clue \[\begin{aligned}\text{Right skew: mean} > \text{median}\\\text{Left skew: mean} < \text{median}\\\text{Symmetric: mean} ≈ \text{median}\end{aligned}\]

The mean gets pulled toward the longer tail more than the median does.

Read the Boxplot

This modified boxplot shows how each part of the display is placed.

Sample modified boxplot Outlier is plotted separately

min

median

largest non-outlier

outlier

AP Exam Skill Builder

The AP Exam often asks for construction, graph choice, and interpretation in context.

Build a Boxplot Like an AP Reader

Order the data if raw values are given.
Find the five-number summary: min, Q1, median, Q3, max.
Check for outliers with the 1.5 × IQR fences when needed.
Draw the box from Q1 to Q3 and draw a line at the median.
Draw whiskers to the smallest and largest non-outliers, then mark outliers separately.
Describe in context: center, spread, outliers, and likely skew.

Scoring Moves to Practice

Graph choice If a table gives min, Q1, median, Q3, and max, say a boxplot is appropriate.

Construction A correct boxplot needs a scale, a box from Q1 to Q3, a median line, whiskers, and separate outliers if present.

Shape from summaries Compare mean and median, then name likely skew with a reason.

Context Use the variable name and units. Do not just say "the data are spread out."

Worked AP-Style Examples

These model the calculation and the sentence you would write after it.

Example 1 - Construct a Boxplot

Find the five-number summary

Data: 18, 21, 24, 25, 26, 28, 30, 34, 36, 45.

Median: \((26 + 28) / 2 = 27\)

Q1: median of 18, 21, 24, 25, 26 is 24

Q3: median of 28, 30, 34, 36, 45 is 34

Summary: 18, 24, 27, 34, 45

AP sentence: Draw the box from 24 to 34, place the median line at 27, and extend whiskers to 18 and 45 because no outlier check flags either endpoint.

Example 2 - Modified Boxplot

Find the whiskers when there is an outlier

Data: 6, 8, 9, 10, 11, 12, 13, 14, 15, 17, 42.

Summary before fences: min 6, Q1 9, median 12, Q3 15, max 42

IQR: \(15 − 9 = 6\)

Fences: \(9 − 1.5(6) = 0\) and \(15 + 1.5(6) = 24\)

Conclusion: 42 is an outlier because it is above 24.

AP sentence: The right whisker should stop at 17, the largest non-outlier, and 42 should be marked separately.

Example 3 - Choose a Graph

A table gives min, Q1, median, Q3, and max

A prompt gives only this summary for daily screen time: minimum 1.2 hours, Q1 2.8 hours, median 3.4 hours, Q3 5.1 hours, maximum 8.6 hours.

Best graph: A boxplot, because a boxplot directly displays the five-number summary. A histogram would require the individual data values or frequency counts.

Example 4 - Mean, Median, and Skew

Use summary statistics to describe shape

A commute-time distribution has mean \(42\) minutes and median \(32\) minutes.

AP sentence: Because the mean is greater than the median, the distribution is likely skewed right; a few long commute times may be pulling the mean upward.

Common AP Mistakes

These are small errors that can cost points even when the idea is close.

Mistake: Drawing a histogram from five numbers

Fix: A five-number summary does not give frequencies, so use a boxplot instead.

Mistake: Letting a whisker reach an outlier

Fix: In a modified boxplot, whiskers stop at the most extreme values that are not outliers.

Mistake: Forgetting the median line

Fix: The line inside the box is required. Without it, the center is not shown clearly.

Mistake: Reversing skew rules

Fix: Right skew usually has mean greater than median. Left skew usually has mean less than median.

Mistake: Over-reading the boxplot

Fix: Boxplots show summaries, not exact peaks, gaps, clusters, or every individual value.

Mistake: No context

Fix: Say what the variable is and include units when they are given.

Flashcard Deck

Click a card to reveal the answer. Mark it for review or mastery as you go.

Card 1 of 15

Practice-only

Vocabulary

Click the card or press Show Answer when you are ready.

Multiple-Choice Practice

Answer one question at a time. You will get instant feedback and a review at the end.

Question 1 of 10

Final study tip: When you see min, Q1, median, Q3, and max, think "boxplot first." Then ask whether outliers change the whiskers and whether mean versus median gives a clue about skew.

FRQ-Style Practice

Prompt: A class collects data related to boxplots, five-number summaries, and outlier rules. Write a free-response answer that uses the correct vocabulary and statistical reasoning.

Identify the variable(s), population/sample, or study design feature requested.
Choose or describe the appropriate table, graph, summary statistic, sampling method, or design decision.
Write one contextual interpretation that uses statistical language rather than a vague everyday claim.

Scoring focus: Credit depends on precise vocabulary, context, and a justified choice or description.

Calculator and Technology Check

Output to read: Calculator or spreadsheet output gives \(n = 48\), mean \(= 16.2\), median \(= 15.4\), IQR \(= 4.8\), and one flagged high value.

How to interpret it: For boxplots, five-number summaries, and outlier rules, connect the output to the context: compare resistant and nonresistant summaries, mention units, and decide whether the flagged value changes the story.

Source note: Aligned to AP Statistics Course and Exam Description, Effective Fall 2026.