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

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

Using the IQR to Identify Outliers

A point is an outlier when it is substantially above Q3 or substantially below Q1. To make this statement more precise, we mark off a distance of one-and-a-half times the interquartile range (1.5 IQRs) above Q3 and below Q1. Any point outside of this range is considered an outlier.

We can write this idea as a formula:

A value is an outlier when

the value is greater than Q3 + 1.5 * IQR or

the value is less than Q1 – 1.5 * IQR

EXAMPLE Using IQR to identify outliers

For the data set in the dotplot below, Q1 = 15 and Q3 = 18, so the IQR = 18 – 15 = 3.

We need to determine one and a half times this distance, which is 1.5*IQR = 1.5 (3) = 4.5.

In the graph you can see this distance marked in green to the left of Q1 and in red to the right of Q3. Of course these distances are equal; both equal 1.5*IQR.

Now we need to determine the lower cut off for an outlier, which is a distance of 1.5*IQR below Q1:

Q1 - 1.5 * IQR = 15 – 1.5 * 3 = 15 – 4.5 = 10.5

This cutoff is shown as a green arrow on the dotplot.

The data point at 10 is considered an outlier because it is below 10.5.

Now we need to determine the upper cut off for an outlier, which is a distance of 1.5*IQR above Q3:

Q3 + 1.5 * IQR = 18 + 1.5 * 3 = 18 + 4.5 = 22.5

This cutoff is shown as a red arrow on the dotplot.

The data points at 24, 27, and 29 are considered outliers because they are above 22.5.

All of the points in purple are outliers by the 1.5*IQR definition of an outlier.

Let’s Summarize

The five-number summary of a distribution consists of the minimum, quartile 1 mark, median, quartile 3 mark, and maximum.

When we use the median as a measure of center, we use the IQR as the measure of spread.

The range is the distance between the smallest value and largest value in the data. It tells us how much the data varies overall. The IQR is also a distance. It is the distance between Q1 and Q3. It tells us how much the data in the middle half of the distribution varies.

The interval Q1 to Q3 describes an interval of typical or representative data values.

We can use the 1.5*IQR rule to identify outliers. Outliers are data values that are

more than 1.5 IQRs greater than Q3 (i.e., the value is greater than Q3 + 1.5 * IQR)

more than 1.5 IQRs less than Q1 (i.e., the value is less than Q1 − 1.5 * IQR)

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