Psychology Wiki

Shapiro-Wilk test

Redirected from Shapiro-Wilk

34,203pages on
this wiki
Add New Page
Talk0 Share

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory

In statistics, the Shapiro-Wilk test tests the null hypothesis that a sample x1, ..., xn came from a normally distributed population. The test statistic is

W = {\left(\sum_{i=1}^n a_i x_{(i)}\right)^2 \over \sum_{i=1}^n (x_i-\overline{x})^2}


  • x(i) (with parentheses enclosing the subscript index i) is the ith order statistic, i.e., the ith-smallest number in the sample;
  • \overline{x}=(x_1+\cdots+x_n)/n\, is the sample mean;
  • the constants ai are given by
(a_1,\dots,a_n) = {m^\top V^{-1} \over m^\top V^{-1}V^{-1}m}
m = (m_1,\dots,m_n)^\top\,
and m1, ..., mn are the expected values of the order statistics of an iid sample from the standard normal distribution, and V is the covariance matrix of those order statistics.

The test rejects the null hypothesis if W is too small.


  • Shapiro, S. S. and Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)", Biometrika, 52, 3 and 4, pages 591-611.
This page uses Creative Commons Licensed content from Wikipedia (view authors).

Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random Wiki