# Interquartile range

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Template:Ref improve 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

## Use

Unlike (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 (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.

## Examples

Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal Template:Maths Population

### 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
10 116
11 118

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:

${\displaystyle Q_{1}={\text{CDF}}^{-1}(0.25),}$
${\displaystyle Q_{3}={\text{CDF}}^{-1}(0.75),}$

where CDF−1 is the quantile function.

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

Distribution Median IQR
Normal μ 2 Φ−1(0.75) ≈ 1.349${\displaystyle \sigma \,}$
Laplace μ 2b ln(2)
Cauchy μ ${\displaystyle 2\gamma \,}$

### 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

${\displaystyle Q_{1}=(\sigma \,z_{1})+X}$

and the third quartile

${\displaystyle Q_{3}=(\sigma \,z_{3})+X}$

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