Editing Normal distribution (section)

=== Normality tests ===
{{Main|Normality tests}}

Normality tests assess the likelihood that the given data set {''x''<sub>1</sub>, ..., ''x<sub>n</sub>''} comes from a normal distribution. Typically the [[null hypothesis]] ''H''<sub>0</sub> is that the observations are distributed normally with unspecified mean ''μ'' and variance ''σ''<sup>2</sup>, versus the alternative ''H<sub>a</sub>'' that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:

'''Diagnostic plots''' are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
* [[Q–Q plot]], also known as [[normal probability plot]] or [[rankit]] plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it is a plot of point of the form (Φ<sup>−1</sup>(''p<sub>k</sub>''), ''x''<sub>(''k'')</sub>), where plotting points ''p<sub>k</sub>'' are equal to ''p<sub>k</sub>''&nbsp;=&nbsp;(''k''&nbsp;−&nbsp;''α'')/(''n''&nbsp;+&nbsp;1&nbsp;−&nbsp;2''α'') and ''α'' is an adjustment constant, which can be anything between 0 and&nbsp;1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
* [[P–P plot]] – similar to the Q–Q plot, but used much less frequently. This method consists of plotting the points (Φ(''z''<sub>(''k'')</sub>), ''p<sub>k</sub>''), where <math display=inline>\textstyle z_{(k)} = (x_{(k)}-\hat\mu)/\hat\sigma</math>. For normally distributed data this plot should lie on a 45° line between (0,&nbsp;0) and&nbsp;(1,&nbsp;1).
'''Goodness-of-fit tests''':

''Moment-based tests'':
* [[D'Agostino's K-squared test]]
* [[Jarque–Bera test]]
* [[Shapiro–Wilk test]]: This is based on the fact that the line in the Q–Q plot has the slope of ''σ''. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
''Tests based on the empirical distribution function'':
* [[Anderson–Darling test]]
* [[Lilliefors test]] (an adaptation of the [[Kolmogorov–Smirnov test]])