# Interquartile range

In descriptive statistics, the interquartile range (IQR), also called the midspread, middle fifty and middle of the #s, is a measure of statistical dispersion, being equal to the difference between the third and first quartiles. The interquartile range is a more stable statistic than the (total) range, and is often preferred to the latter statistic.

The interquartile range is the most commonly used interpercentile range. Since 25% of the data are less than or equal to the first quartile and 25% are greater than or equal to the third quartile, the IQR is expected to include about half of the data. The IQR has the same units as the data.

The IQR is used to build box plots, simple graphical representations of a probability distribution.

## Example

i x[i] Quartile
1 102
2 104
3 105 Q1
4 107
5 108
6 109 Q2 (median)
7 110
8 112
9 115 Q3
10 118
11 118

From this table, the length of the interquartile range is 115 - 105 = 10.

The median is the corresponding measure of central tendency.

## Interquartile range of distributions

The interquartile range of a continuous distribution can be calculated by integrating the probability density function (which yields the cumulative distribution function—any means of calculating the CDF will also work). The lower quartile, a, is the integral of the PDF from -∞ to a that equals 0.25, while the upper quartile, b, is the integral from b to ∞ that equals 0.25; in terms of the CDF, the values that yield 0.25 and 0.75 are the quartiles.

[insert equations here]

The interquartile range and median of some common distributions are shown below

Distribution Median IQR
Normal μ -1(0.75)≈ 1.349
Laplace μ $2b\ln(2)$ Cauchy μ $2\gamma$  