# Statistics and Probability Dictionary

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### Interquartile Range

The **interquartile range** (IQR) is a measure of variability,
based on dividing a data set into
quartiles.

Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

- Q1 is the "middle" value in the
*first*half of the rank-ordered data set. - Q2 is the median value in the set.
- Q3 is the "middle" value in the
*second*half of the rank-ordered data set.

The interquartile range is equal to Q3 minus Q1.

For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5. Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5. The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

## An Alternative Definition for IQR

In some texts, the interquartile range is defined differently. It is defined as the difference between the largest and smallest values in the middle 50% of a set of data.

To compute an interquartile range using this definition, first remove observations from the lower quartile. Then, remove observations from the upper quartile. Then, from the remaining observations, compute the difference between the largest and smallest values.

For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. After we remove observations from the lower and upper quartiles, we are left with: 4, 5, 5, 6. The interquartile range (IQR) would be 6 - 4 = 2.

See also: | AP Statistics Tutorial: Measures of Variability |