# Interquartile range

In statistics, the interquartile range (IQR) is a number that indicates how spread out the data are, and tells us what the range is in the middle of a set of scores.

The interquartile range IQR is defined as:[1][2]

${\displaystyle \mathrm {IQR} =Q_{3}-Q_{1}}$

That is, it is calculated as the range of the middle half of the scores. The scores are divided into four equal parts, separated by the quartiles ${\displaystyle Q_{1},Q_{2}}$ and ${\displaystyle Q_{3}}$, after the scores have been arranged in ascending order (becoming bigger as one goes further). The second quartile ${\displaystyle Q_{2}}$ is also known as the median.[3]

The interquartile range is not sensitive to outliers (scores that are much higher or much lower than the other scores). In fact, it eliminates them.

## Example

Given the following 20 scores arranged from the smallest to the largest:

1, 2, 2, 2, 3, 4, 6, 8, 8, 8, 8, 8, 9, 11, 11, 14, 14, 15, 15, 29

We can put them into four different groups of five numbers each:

1, 2, 2, 2, 3 | 4, 6, 8, 8, 8 | 8, 8, 9, 11, 11 | 14, 14, 15, 15, 29

The groups are thus separated by:

${\displaystyle Q_{1}=3{.}5,\ Q_{2}=8,\ Q_{3}=12{.}5}$

Hence the interquartile range is:

${\displaystyle \mathrm {IQR} =Q_{3}-Q_{1}=12{.}5-3{.}5=9}$

If the observation 29 has accidentally been written down as 92 instead, then this number is an outlier. Notice that the interquartile range is not affected in that case.

## References

1. "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-10-13.
2. "InterQuartile Range (IQR)". sphweb.bumc.bu.edu. Retrieved 2020-10-13.
3. "Interquartile Range: Definition". stattrek.com. Retrieved 2020-10-13.