Wiki Page, Math 13 - Perez Sandbox : Module 6 - Interquartile Range and Boxplots (2b of 3)

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2026-01-17T21:15:51-08:00

In our experience boxplots are less intuitive than dotplots and histograms, so this page contains additional examples. Previously, we focused on making a boxplot. Now we focus on analyzing data using a boxplot.

EXAMPLE Analyzing data using a boxplot

In this short movie we discuss shape, center and spread by comparing and contrasting two boxplots. (5:38 minutes).

In the movie we compared two distributions using boxplots. We discussed how the boxplot contains information about center (median) and spread (range and interquartile range.) In the next example, we revisit these ideas again to deepen our understanding of the visual cues in a boxplot.

EXAMPLE Analyzing the spread in a distribution using a boxplot

Here is the five-number summary for two distributions of exam scores that we examined previously:

Class A: Min: 40 Q1: 61 Q2: 74.5 Q3: 89 Max: 95

Class B: Min: 40 Q1: 71 Q2: 74.5 Q3: 78.5 Max: 95

Below are the horizontal boxplots alongside horizontal dotplots.

Study this image to check your understanding of how the boxplots relate to the 5-number summaries.

This is a complicated image. Let's dissect it to examine some important ideas about boxplots. After you read each bullet, go back and look at the boxplots to check your visual understanding:

For these two distributions, the medians are the same (74.5), so the center of each box is the same. Just comparing medians is obviously not enough because the distributions are very different when it comes to spread.

Within a single distribution, a longer quartile has more variability in the data, not more data points. For example, in Class B the first quartile ranges from 40 to 71 (a span of 31 points); the second quartile ranges from 71 to 74.5 (a span of only 3.5 points). But both quartiles contain 8 scores.

The width of the box gives you information about the consistency in typical scores (the amount of noise relative to the median.) Class B has a short box in the boxplot. This indicates a small IQR, so the middle half of the data has little variability. In other words, typical scores in Class B are fairly consistent. We see this in the central clump of data in the dotplot.
Class A has a longer box in the boxplot. This indicates a large IQR, so the middle half of the data has a lot of variability. In other words, typical scores in Class A are not consistent. We see this in the dotplot; the data in the center of the distribution is very spread out.

Here are the boxplots drawn vertically.

StatCrunch can make horizontal or vertical boxplots, so you can choose whichever orientation you like best.

Recognizing the Shape of a Distribution Using a Boxplot

With a dotplot we can more easily discern the shape of a distribution. Dotplots are particularly useful for small data sets. Histograms and boxplots are better for summarizing large data sets. When a data set is large, all three graphs are useful in describing the shape of a distribution. When a data set is small, a boxplot is not as useful.

EXAMPLE Boxplot for a skewed distribution

Below are two graphs of the distribution of ages for men who belong to a gym.

The dotplot shows a distribution skewed to the right. There are fewer older men who belong to the gym.

The boxplot also shows the right skew. The lower half of the distribution is less spread out (blue arrow) than the upper half (red arrow). The greater spread in the upper half of the data is due to the large spread in the 4th quartile (green arrow.)

EXAMPLE Boxplot of a (somewhat) bell-shaped distribution

Below are two graphs of the distribution of shoulder girth for men who belong to a gym.

The histogram shows a distribution that is somewhat symmetric with a central clump (sort of bell-shaped.) Many men have shoulder measurements in the central part of the distribution and measurements taper off in a fairly symmetric way on both ends.

The boxplot also suggests a symmetric distribution with a central clump. The interquartiles are shorter (red arrows) than than the outer quartiles (green arrows.) This tells us that the middle half of the data is clumped more tightly together. The interquartiles are also roughly the same length, which suggests symmetry in the middle of the distribution. The outer quartiles are longer and roughly the same length, which suggests tails tapering off in both directions in a symmetric way.

Note: Some statistical packages offer two options: a boxplot and a modified boxplot. In a modified boxplot, outliers are marked with an asterisk (*) or a dot. We drew modified boxplots in our examples in our movie.

For a boxplot that is not modified, the tails extend to the minimum and maximum values. In this type of boxplot, we cannot see outliers. In the examples we used boxplots that are not modified.

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