# Interquartile range

Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal N(0,σ2) Population

In descriptive statistics, the interquartile range (IQR), also called the midspread or middle fifty, is a measure of statistical dispersion, being equal to the difference between the upper and lower quartiles,[1][2] IQR = Q3 −  Q1. In other words, the IQR is the 1st quartile subtracted from the 3rd quartile; these quartiles can be clearly seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed mid-range, and is the most significant basic robust measure of scale.

## Use

Unlike (total) range, the interquartile range has 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 (where 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.

Filtering of outliers (see below).

## Examples

### Data set in a table

ix[i]Quartile
1102
2104
3105Q1
4107
5108
6109Q2
(median)
7110
8112
9115Q3
10116
11118

For the data in this table the interquartile range is IQR = 115 − 105 = 10.

### 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 the data set in this box plot:

• lower (first) quartile Q1 = 7
• median (second quartile) Q2 = 8.5
• upper (third) quartile Q3 = 9
• interquartile range, IQR = 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 other means of calculating the CDF will also work). The lower quartile, Q1, is a number such that integral of the PDF from -∞ to Q1 equals 0.25, while the upper quartile, Q3, is such a number that the integral from -∞ to Q3 equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

$Q_1 = \text{CDF}^{-1}(0.25) ,$
$Q_3 = \text{CDF}^{-1}(0.75) ,$

where CDF−1 is the quantile function.

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

DistributionMedianIQR
Normalμ2 Φ−1(0.75)σ ≈ 1.349σ ≈ (27/20)σ
Laplaceμ2b ln(2) ≈ 1.386b
Cauchyμ

### Interquartile range test for normality of distribution

The IQR, mean, and standard deviation of a population P can be used in a simple test of whether or not P is normally distributed, or Gaussian. If P is normally distributed, then the standard score of the first quartile, z1, is -0.67, and the standard score of the third quartile, z3, is +0.67. Given mean = X and standard deviation = σ for P, if P is normally distributed, the first quartile

$Q_1 = (\sigma \, z_1) + X$

and the third quartile

$Q_3 = (\sigma \, z_3) + X$

If the actual values of the first or third quartiles differ substantially[clarification needed] from the calculated values, P is not normally distributed.

## Interquartile range and outliers

Figure 3. Box-and-whisker plot with four close and one far away extreme values, defined as outliers above Q3 + 1.5(IQR) and Q3 + 3(IQR), respectively.

The interquartile range is often used to find outliers in data. Outliers are observations that fall below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR). In a boxplot, the highest and lowest occurring value within this limit are drawn as bar of the whiskers, and the outliers as individual points.