1.8 Graphical Representations of Summary Statistics

Boxplots show the five-number summary

A boxplot displays the minimum, Q1, median, Q3, and maximum. The box spans the IQR (Q1 to Q3), with a line at the median. Whiskers extend to the smallest/largest values not flagged as outliers. Boxplots are great for comparing distributions across groups but hide gaps, clusters, and shape details visible in dotplots or histograms.

Identifying outliers with the 1.5×IQR rule

A value is a suspected outlier if it falls below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR). Modified boxplots plot these outliers as individual points, with whiskers drawn only to the most extreme non-outlier values. This rule gives a consistent, objective way to flag unusual observations.

Choosing and comparing graphs

Dotplots, stemplots, and histograms show individual data shape (clusters, gaps, shape); boxplots summarize center/spread and compare multiple groups side-by-side efficiently. When comparing distributions, always discuss shape, center, spread, and outliers in context, referencing units and using comparative language (e.g., "greater than," "less variable than").

Key terms

Five-number summary

Minimum, Q1, median, Q3, and maximum — the values used to construct a boxplot.

1.5×IQR rule

A value is a suspected outlier if it's below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).

Modified boxplot

A boxplot where outliers are plotted as separate points and whiskers extend only to the most extreme non-outlier values.

IQR (Interquartile Range)

Q3 − Q1; the range of the middle 50% of the data, represented by the box in a boxplot.

Boxplot vs. dotplot trade-off

Boxplots summarize center/spread and compare groups easily but hide shape details (gaps, clusters) that dotplots/histograms reveal.

Try an AP-style question

What does a boxplot display?

1. The mean and standard deviation
2. Every individual data value
3. The mode and range only
4. The five-number summary and any outliers