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

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

Boxplots

On the previous page, we learned about the five-number summary. At this point, you should know the following:

The five-number summary uses quartiles to identify the center and spread of a distribution.

The median (which is Q2) is a measure of center. We also view the median as a typical value that represents the distribution.

The values between Q1 and Q3 give a typical range of values.

The IQR is a way to measure the variability about the median.

Now we use the five-number summary to make a new type of graph, the boxplot.

Making a Boxplot

This movie shows the steps for making a boxplot of the distribution of ages for winners of the Oscar Award for Best Actress from 1970 to 2001. The five-number summary for this distribution is

Min: 21 Q1: 32 Median: 35 Q3: 41.5 Max: 80

Using the IQR definition of an outlier, there are three outliers: 61, 74, and 80.

When we make a boxplot, we start with a number line and scale it (in the movie the scale is 10), then we make the boxplot to this scale.

Here is a static version of this movie.

EXAMPLE Temperature of Pittsburgh vs. San Francisco

To compare the average high temperatures of Pittsburgh to those of San Francisco, we look at the following side-by-side boxplots and supplement the graph with the descriptive statistics of each of the two distributions.

(Note: boxplots can be drawn either vertically, as on the left, or horizontally, as on the right. The vertical and horizontal boxplots represent the exact same data.)

Monthly Average High Temperatures (Degrees Fahrenheit)

| Pittsburgh | San Francisco
Minimum | 33.7 | 56.3
Q1 | 41.2 | 60.2
Q2 (Median) | 61.4 | 62.7
Q3 | 77.8 | 65.4
Maximum | 82.6 | 68.7

When looking at the boxplots, the similarities and differences between the two distributions are striking. Both distributions have roughly the same center (medians are 61.4 for Pittsburgh and 62.7 for San Francisco). However, the temperatures in Pittsburgh have a much larger variability than the temperatures in San Francisco (Range: 49 vs. 12; IQR: 36.5 vs. 5).

From this we see that the weather in San Francisco is much more consistent than the weather in Pittsburgh throughout the year.

Because the temperatures in San Francisco vary so little during the year, knowing that the median temperature is around 63 is very informative. On the other hand, knowing that the median temperature in Pittsburgh is around 61 is practically useless, since temperatures vary so much during the year.

A statistician will refer to variability as "noise in the data" about the median or the mean.

When there is less noise in the data, the measure of center (in this case the median) is a good representation of the entire distribution. Is it important to always include information about spread, but when there is more noise in the data, it becomes much more important in our summary of the distribution.

EXAMPLE Comparing dotplots, histograms and boxplots

To deepen our learning of a new idea, we need to make connections to what we learned before. In this short movie we examine how boxplots relate to dotplots and histograms (4:34 minutes).

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Content by the Open Learning Initiative and licensed under CC BY.