shapiro_wilk_normality

The Shapiro-Wilk test for normality is thought by D’Agostino and Stevens (1986, p. 406) to be one of the best omnibus tests of normality. The function is based on the approximations and code given by Royston (1982a, b, c; and 1991). The minimum sample size is 3 and sample sizes as large as 5000 have been validated. In the Shapiro and Wilk test, W is given by

where x(i) is the i-th largest order statistic and

is the sample mean. Royston (1982 and 1991) gives approximations and tabled values that can be used to compute the coefficients ai, i = 1, …, n, and obtains the significance level of the W statistic.

Example

This example is taken from Conover (1980, pp. 195, 364). The data consists of 50 two-digit numbers taken from a telephone book. The W test fails to reject the null hypothesis of normality at the .05 level of significance.

#include <imsls.h>

#include <stdio.h>

int main()

{

int n_observations = 50;

float x[] = {23.0, 36.0, 54.0, 61.0, 73.0, 23.0,

37.0, 54.0, 61.0, 73.0, 24.0, 40.0,

56.0, 62.0, 74.0, 27.0, 42.0, 57.0,

63.0, 75.0, 29.0, 43.0, 57.0, 64.0,

77.0, 31.0, 43.0, 58.0, 65.0, 81.0,

32.0, 44.0, 58.0, 66.0, 87.0, 33.0,

45.0, 58.0, 68.0, 89.0, 33.0, 48.0,

58.0, 68.0, 93.0, 35.0, 48.0, 59.0,

70.0, 97.0};

float p_value, shapiro_wilk_w;

/* Shapiro-Wilk test */

p_value = imsls_f_shapiro_wilk_normality_test (n_observations, x,

IMSLS_SHAPIRO_WILK_W, &shapiro_wilk_w, 0);

printf ("p-value = %11.4f\n", p_value);

printf ("Shapiro Wilk W statistic = %11.4f\n",

shapiro_wilk_w);

}

Output

p-value = 0.3473

Shapiro Wilk W statistic = 0.9744