Definitions

# Interquartile range

[in-ter-kwawr-tahyl, -til]
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.

Unlike the (total) range, the interquartile range is a robust statistic, having a breakdown point of 25%, and is thus often preferred to the total range.

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

For a symmetric distribution (so the median equals the midhinge, the average of the first and third quartiles), half the IQR equals the median absolute deviation (MAD).

The median is the corresponding measure of central tendency.

## Examples

Example 1

Data set in a table:

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 width of the interquartile range is 115 − 105 = 10.

Example 2

Data set in a plain-text box plot:

`                            +-----+-+`
`  o           *     |-------|     | |---|`
`                            +-----+-+`

`+---+---+---+---+---+---+---+---+---+---+---+---+   number line`
`0   1   2   3   4   5   6   7   8   9  10  11  12`

For this data set:

• lower (first) quartile ($Q1$, $x_\left\{.25\right\}$) = 7
• median (second quartile) ($Med$, $x_\left\{.5\right\}$) = 8.5
• upper (third) quartile ($Q3$, $x_\left\{.75\right\}$) = 9
• interquartile range, $mathrm\left\{IQR\right\} = Q3-Q1 = 2$

## 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 2 Φ−1(0.75) ≈ 1.349
Laplace μ 2b ln(2)
Cauchy μ $2gamma,$