Interquartile range

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, 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 N(0,σ²) 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:

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

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

where CDF⁻¹ is the quantile function.

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

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

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

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

and the third quartile

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

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

See also

• Midhinge
• Interdecile range
• Robust measures of scale
• Outlier