The Shapiro-Wilk test, which is denoted by W, is a test of whether a data set has been drawn from an underlying normal distribution. The null and alternative hypotheses being tested are
\( H_0 \): The data have been drawn from a normal distribution
\( H_a \): The data are drawn from a non-normal distribution
The test is conducted as follows (from Gilbert 1987, pp. 158-160).
Compute: \( \large \quad d = \displaystyle\sum_{i=1}^n (x_i - \bar x)^2 \)
Order the \(n\) data from smallest to largest to obtain the sample order statistics: \( \quad x_{[1]} \leq x_{[2]} \leq \dotsb \leq x_{[n]} \)
Compute \(k\), where
\( k = \frac{n}{2} \) if n is even
\( k = \frac{n-1}{2} \) if n is odd
4. Find coefficients \( a_1 , a_2 , a_3 , \dotsb , a_k \) from Table A6 in Gilbert (1987, pages 259-260).
5. Compute: \( \large W = \frac{1}{d} \left[ \displaystyle\sum_{i=1}^k a_i ( x_{[n-i+1]} - x_i ) \right]^2 \)
6. Reject \( H_0 \) and accept \( H_a \) at the \( \alpha \) significance level if \(W\) is less than the appropriate percentile, \( W_{\alpha} \), of the W statistic given in Table A7 of Gilbert (1987, page 261).
Test provided courtesy of ProUCL.
Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring, Wiley, NY.
ProUCL. 2004. ProUCL Version 3.0 User Guide April 2004. Available for download from http://www.epa.gov/nerlesd1/tsc/tsc.htm