Editing Interquartile range (section)

==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, ''Q''<sub>1</sub>, is a number such that integral of the PDF from -∞ to ''Q''<sub>1</sub> equals 0.25, while the upper quartile, ''Q''<sub>3</sub>, is such a number that the integral from -∞ to ''Q''<sub>3</sub> equals 0.75; in terms of the CDF, the quartiles can be defined as follows:

:<math>Q_1 = \text{CDF}^{-1}(0.25) ,</math>

:<math>Q_3 = \text{CDF}^{-1}(0.75) ,</math>

where CDF<sup>−1</sup> is the [[quantile function]].

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

{| class="wikitable"
|-
! Distribution
! Median
! IQR 
|-
| [[Normal distribution|Normal]]
| μ
| 2 Φ<sup>&minus;1</sup>(0.75)σ ≈ 1.349σ ≈ (27/20)σ

|-
| [[Laplace distribution|Laplace]]
| μ
| 2''b''&nbsp;ln(2) ≈ 1.386''b''
|-
| [[Cauchy distribution|Cauchy]]
| μ
|2γ
|}

===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 [[Normal distribution|normally distributed]], or Gaussian. If ''P'' is normally distributed, then the [[standard score]] of the first quartile, ''z''<sub>1</sub>, is −0.67, and the standard score of the third quartile, ''z''<sub>3</sub>, is +0.67. Given ''mean''&nbsp;=&nbsp;<math>\bar{P}</math> and ''standard&nbsp;deviation''&nbsp;=&nbsp;σ for ''P'', if ''P'' is normally distributed, the first quartile

:<math>Q_1 = (\sigma \, z_1) + \bar{P}</math>

and the third quartile

:<math>Q_3 = (\sigma \, z_3) + \bar{P}</math>

If the actual values of the first or third quartiles differ substantially{{Clarify|date=December 2012}} from the calculated values, ''P'' is not normally distributed. However, a normal distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and −0.67 and not be normally distributed (so the above test would produce a false positive). A better test of normality, such as [[Q–Q plot]] would be indicated here.