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 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.

or 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.IQR = Q₃ - Q₁

$d<nowiki>Insertnon-formattedtexthere</nowiki>$==Examples==

Boxplot (with an interquartile range) and a [[probability density ^{function]] (pdf) of a Normal N(0,1σ2) Population}

===Data set in a table==={| class="wikitable" |- ! 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 |}

From this table, the width of the hi interquartie range is 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 this data set:

• lower (first) quartile ($Q1$, $x_{.25}$) = 7
• median (second quartile) ($Median$, $x_{.5}$) = 8.5
• upper (third) quartile ($Q3$, $x_{.75}$) = 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:

$Q1={\text{CDF}}^{-1}(0.25)$

$Q3={\text{CDF}}^{-1}(0.75)$

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	$2\gamma \,$

See also

• Midhinge
• Interdecile range