Wiki Page, Math 13 - Perez Sandbox : Module 6 - Interquartile Range and Boxplots (1a of 3)

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

Describing Spread about the Median

Recall that when we describe the distribution of a quantitative variable, we describe the overall pattern (shape, center, and spread) in the data and deviations from the pattern (outliers).

Previously, we described the shape of a distribution using words like "left skew" or "bell shaped." We also learned more precise ways to describe the center of a distribution using the mean and the median, and learned how to choose between the two based on the shape of the distribution.

In the next two modules we focus on describing the spread of a distribution more precisely. First we will work on describing spread about the median. In the next module we will work on describing spread relative to the mean.

EXAMPLE Recognizing variability about the median

Consider the following two distributions of exam scores:

Both distributions have a median of 74.5. Which distribution has more variability?

The answer to this question depends on how we measure variability. Both distributions have the same range. The range is the distance spanned by the data. We calculate the range by subtracting the minimum value from the maximum value.

Range = Maximum value – minimum value

For both of these data sets, the range is 55 (here is how we calculated the range: 95 – 40 = 55). If we use the range to measure variability, we say the distributions have the same amount of variability.

But the variability in the distributions differ when we look at how the data is distributed about the median. Set B has a large portion of its data close to the median. This is not true for Set A. From this viewpoint, Set B has less variability about the median.

Quartiles as a Measure of Spread

Quartile marks divide the data set into four groups with equal counts. Each quartile contains 25% of the data in the distribution. The distance between quartile marks gives us a lot of information about the spread in the distribution.

EXAMPLE Quartile marks and the five-number-summary

We added dividers to show the quartile marks for the two sets of exam scores. Quartile marks divide the data set into four subgroups with the same number of individuals in each subgroup.

Notice: For a data set, there is an equal amount of data in each quartile.

Class A has 20 scores, so each quartile contains five scores (20 ÷ 4 = 5).

Class B has 32 scores, so each quartile contains eight scores (32 ÷ 4 = 8).

The quartiles together with the minimum and maximum scores give the five-number summary:

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

Notice: The second quartile mark (Q2) is the median.

Notice: Some quartiles exhibit more variability in the data even though each quartile contains the same amount of data. For example, look at Class B.

In Class B the first quartile mark is 71, so 25% of the data lies between 40 and 71. The eight scores in Q1 vary by 30 points. There is a lot of variability in this first quartile (Q1).

Compare this to the third quartile (Q3) for Class B: 25% of the scores in Class A are between 74.5 and 78.5. The 8 scores in Q3 vary by only 4 points. There is not much variability in Q3.

The Interquartile Range

How are quartiles used to measure variability about the median?

The interquartiles, Q1 and Q3, mark the middle half of the distribution. The interquartile range (IQR) is the distance between the first and third quartile marks. The IQR measures the span of the middle half of the distribution, so the IQR is a measurement of the variability about the median.

EXAMPLE The interquartile range

Here is the IQR for these two distributions:

Class A: IQR = Q3 – Q1 = 89 – 61 = 28

Class B: IQR = Q3 – Q1 = 78.5 – 71 = 7.5

As we observed earlier, Class B has less variability about its median. Its IQR is much smaller. The middle 50% of exam scores for Class B vary by only 7.5 points. The middle 50% of exam scores for Class A vary by 28 points.

A More Precise Description of Typical Values

The interquartiles, Q1 and Q3, mark the middle half of the distribution. Values between Q1 and Q3 are one way to describe an interval of typical values for the distribution, as opposed to the IQR which is the distance between the Q1 and Q3.

EXAMPLE Quartiles and typical values

We can also use the 1st and 3rd quartile marks to define an interval of typical values.

Class A: Typical scores for Class A are between 61 (Q1) and 89 (Q3).

Class B: Typical scores for Class B are between 71 (Q1) and 78.5 (Q3).

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