Shapiro-Wilk Test

The Shapiro-Wilk test is a test for normal distribution exhibiting high power, leading to good results even with a small number of observations. In contrast to other comparison tests the Shapiro-Wilk test is only applicable to check for normality.

The basis idea behind the Shapiro-Wilk test is to estimate the variance of the sample in two ways: (1) the regression line in the QQ-Plot allows to estimate the variance, and (2) the variance of the sample can also be regarded as an estimator of the population variance. Both estimated values should approximately equal in the case of a normal distribution and thus should result in a quotient of close to 1.0. If the quotient is significantly lower than 1.0 then the null hypothesis (of having a normal distribution) should be rejected.

In order to calculate the statistic W one has to perform the following procedure:

1. The sample of size n (x₁,x₂,...x_n) has to be sorted in increasing order, the resulting sorted sample will be designated by y₁,y₂,...y_n (y₁ < y₂ < ... < y_n).
2. Calculate the sum
   
3. a) if n is even, then b is calculated using mit k = n/2:
   ,
   b) if n is odd, b is calculated by using k=(n-1)/2, the median must not be included.
   The parameters a_n-i+1 depend on the sample size and have to be taken from a table published by Shapiro and Wilk
    
4. Calculate the test statistic W = b²/S²

critical thresholds Wα
n	α=0.01	α=0.05		n	α=0.01	α=0.05
5	0.686	0.762		16	0.844	0.887
6	0.713	0.788		17	0.851	0.892
7	0.730	0.803		18	0.858	0.897
8	0.749	0.818		19	0.863	0.901
9	0.764	0.829		20	0.868	0.905
10	0.781	0.842		25	0.888	0.918
11	0.792	0.850		30	0.900	0.927
12	0.805	0.859		35	0.910	0.934
13	0.814	0.866		40	0.919	0.940
14	0.825	0.874		50	0.930	0.947
15	0.835	0.881