There is no name for the distribution of . The cutoff values for the statistics are calculated through Monte Carlo simulations.[2]
Interpretation
The
null-hypothesis of this test is that the population is normally distributed. Thus, if the
p value is less than the chosen
alpha level, then the null hypothesis is rejected and there is evidence that the data tested are not normally distributed. On the other hand, if the p value is greater than the chosen alpha level, then the null hypothesis (that the data came from a normally distributed population) can not be rejected (e.g., for an alpha level of .05, a data set with a p value of less than .05 rejects the null hypothesis that the data are from a normally distributed population – consequently, a data set with a p value more than the .05 alpha value fails to reject the null hypothesis that the data is from a normally distributed population).[4]
Like most
statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis (i.e., although there may be some
statistically significant effect, it may be too small to be of any practical significance); thus, additional investigation of the effect size is typically advisable, e.g., a
Q–Q plot in this case.[5]
Royston proposed an alternative method of calculating the coefficients vector by providing an algorithm for calculating values that extended the sample size from 50 to 2,000.[7] This technique is used in several software packages including
GraphPad Prism, Stata,[8][9] SPSS and SAS.[10] Rahman and Govidarajulu extended the sample size further up to 5,000.[11]
^Davis, C. S.; Stephens, M. A. (1978).
The covariance matrix of normal order statistics(PDF) (Technical report). Department of Statistics, Stanford University, Stanford, California. Technical Report No. 14. Retrieved 2022-06-17.
^Field, Andy (2009). Discovering statistics using SPSS (3rd ed.). Los Angeles [i.e. Thousand Oaks, Calif.]: SAGE Publications. p. 143.
ISBN978-1-84787-906-6.
^Royston, Patrick (September 1992). "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and Computing. 2 (3): 117–119.
doi:
10.1007/BF01891203.
S2CID122446146.
^Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia Tests". Stata Technical Bulletin, StataCorp LP. 1 (3).
^Rahman und Govidarajulu (1997). "A modification of the test of Shapiro and Wilk for normality". Journal of Applied Statistics. 24 (2): 219–236.
doi:
10.1080/02664769723828.