Shapiro-Wilk Test For Normality

The Shapiro-Wilk test for normality is a very powerful test. This test is a regression-type test. It assesses how well the observed cumulative frequency distribution curve fits the expected cumulative frequency curve. The Shapiro-Wilk test is sensitive to both skewness and kurtosis. The Shapiro-Wilk and Anderson-Darling tests are comparable in power, with Shapiro-Wilk having a slight edge in many situations.


  1. Order the observations from low to high.

  2. Compute

    S² = (n-1)s²

    where s² is the sample variance.

  3. If n is even, k = n/2. If n is odd, k = (n – 1)/2. Then


    where a(n+i+1) for i = 1 to k, are found in tables.

  4. Compute the test statistic.

    W = b²/S²

  5. A p-value is generated to evaluate the significance of W.

The algorithms used by MVPstats are those of Patrick Royston, Applied Statistics, vol. 44 no. 4 (1995).

Accurracy is claimed for samples size from 3 to 5000. Sample size less than three will not produce a Shapiro-Wilk statistic.