17.1.7.2 Algorithms (Normality Test)NormalityTestAlgorithm
ShapiroWilk normality test
Given a set of observations sorted into either ascending or descending order, the Shapiro Wilk W statistic is defined as:
where
is the sample mean and ai, for i=1, 2,...n are a set of mathematical weights, the values of which depend only on the sample size n.
The algorithm used by Origin is from the Applied Statistics Algorithm R94 described by Patrick Royston (1995). The function supports sample sizes of 3.
Degree of freedom (DF) is equal to the sample size.
KolmogorovSmirnov normality test
Origin calls a NAG function nag_1_sample_ks_test (g08cbc) , to compute the statistics. Please refer to related NAG document, for more details on the algorithm.
Lilliefors normality test
Lilliefors test is adapted from the KolmogorovSmirnov test, and the statistics is computed in the same way as that of KolmogorovSmirnov test. However, the pvalue is different because Lilliefors test does not care about the mean score and variance of the data while KolmogorovSmirnov test does. Dallal and Wilkinson (1986) Method is used for pvalue computation.
AndersonDarling Test
Given a set of observations sorted into either ascending order, the Anderson Darling statistic is defined as
where
is the cumulative distribution function of the distribution
D'AgostinoK Squared
 Skewness statistic
 Compute the Skewness from the data

 Compute
 The Skewness statistic can be computed with equation below

 Kurtosis Statistic
 Compute the Kurtosis from the data

 Compute the mean and variance of


 Compute the standardized moment of

 Compute

 The Kurtosis statistic can be computed by formula below

 D'Agostino's Chi2 Statistic
ChenShapiro Test
Given a set of observations sorted into either ascending order, the ChenShapiro statistic is defined as
where
and is the inverse of teh standard normal distribution
