AP Statistics · Topic 1.8 · Unit 1

AP Stats 1.8: Boxplots and Five-Number Summary

By Jude Wallis · Published July 28, 2026

The five-number summary is the minimum, Q1, median, Q3, and maximum, and a boxplot graphs it. The box holds the middle 50% of the data. A mean above the median usually signals right skew; a mean below the median, left skew.

AP Statistics: Unit 1 (topics 1.8). Topic 1.8 (Graphical Representations of Summary Statistics for One Quantitative Variable) sits in Unit 1 of the redesigned AP Statistics course (effective Fall 2026, first exam May 2027). Unit 1 is the heaviest weighted unit at 20-30% of the multiple-choice section.

The five-number summary and the boxplot

Topic 1.8 turns summary statistics into a picture. The five-number summary is the minimum, first quartile $Q_1$ , median, third quartile $Q_3$ , and maximum. A boxplot graphs those five numbers: a box spans $Q_1$ to $Q_3$ with a line at the median, so the box holds the middle 50% of the data. Whiskers extend from the box to the minimum and maximum, and each whisker plus each half of the box marks off about 25% of the data, splitting the values into quarters.

When the data contain outliers, a modified boxplot draws the whiskers only to the most extreme values that are not outliers, and marks each outlier with an asterisk or other symbol. That keeps a single stray value from stretching a whisker across the whole plot.

Reading skew from a boxplot

A boxplot hides individual values but shows shape through the spacing of its five numbers. If the distribution is roughly symmetric, the mean and median are close together. If it is skewed right, the mean is usually larger than the median; if skewed left, the mean is usually smaller than the median.

On the plot itself, a longer right whisker or a median sitting toward the left of the box suggests right skew, and the reverse suggests left skew. See how to make a boxplot.

What a boxplot does and does not show

A boxplot is built entirely from the five-number summary, so it is compact and well suited to comparing groups, a use that returns in topic 1.9. The width of the box is the interquartile range, the distance between the quartiles, and it shows the spread of the middle half of the data at a glance.

What a boxplot leaves out is just as important. Because it plots only five numbers, it cannot reveal gaps, clusters, or a second peak, so two very different distributions can share the same boxplot. When those details matter, read a dotplot or histogram alongside it, and use the modified boxplot rules so outliers are shown rather than hidden inside a long whisker. Side-by-side boxplots on the same axis are one of the fastest ways to compare the center and spread of several groups at once, which is exactly what topic 1.9 asks you to do.

Five-number summary and a modified boxplot

Nine reaction times, in seconds, are 0.20, 0.22, 0.25, 0.26, 0.28, 0.30, 0.31, 0.34, 0.60. Give the five-number summary, check for outliers with the 1.5 IQR rule, and say where the whiskers end.

The data are ordered and $n = 9$ . Minimum $= 0.20$ , maximum $= 0.60$ .
Median: the middle (5th) value is 0.28.
Lower half (below the median): 0.20, 0.22, 0.25, 0.26. Its median is $Q_1 = (0.22 + 0.25)/2 = 0.235$ .
Upper half (above the median): 0.30, 0.31, 0.34, 0.60. Its median is $Q_3 = (0.31 + 0.34)/2 = 0.325$ .
IQR $= 0.325 - 0.235 = 0.090$ , so $1.5 \times \text{IQR} = 0.135$ .
Fences: upper $= 0.325 + 0.135 = 0.460$ ; lower $= 0.235 - 0.135 = 0.100$ .
Compare: $0.60 > 0.460$ , so 0.60 is an outlier. The minimum 0.20 is above 0.100, so no low outliers.

Five-number summary: min 0.20, $Q_1 = 0.235$ , median 0.28, $Q_3 = 0.325$ , max 0.60. The value 0.60 is an outlier. On a modified boxplot the right whisker ends at the largest non-outlier, 0.34, with 0.60 marked separately; the left whisker ends at 0.20.

Frequently asked questions

Can a boxplot show whether data are bimodal?

No. A boxplot only shows the five-number summary, so it hides peaks, gaps, and clusters. To see whether a distribution is bimodal, use a dotplot, stem-and-leaf plot, or histogram.