# Talk:Shapiro–Wilk test

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## Some questions:

What is the criterion for W to be "too small"? What is the expected value for the order statistics? Is there a multi-variate generalization? PhysPhD 20:55, 16 May 2007 (UTC)

I think there should be some way of arriving at the a(i)'s... I've seen it like this: ai <- qnorm((i-0.375)/(n+0.25)) where qnorm is the inverse CDF. —Preceding unsigned comment added by 64.122.234.42 (talk) 21:35, 30 October 2007 (UTC)

I've found some table for critical values of criterion Wcrit in some old Russian book named "Основы математической статистики - Под ред.В.С.Иванова" which is roughly "Fundamentals of Mathematical Statistics - Edited by Ivanov V.S.". The table looks like this:

n alpha alpha n alpha alpha n alpha alpha
0.05 0.01 0.05 0.01 0.05 0.01
4 0.767 0.753 20 0.905 0.868 36 0.935 0.912
5 0.748 0.687 21 0.908 0.873 37 0.936 0.914
6 0.762 0.686 22 0.911 0.878 38 0.938 0.916
7 0.803 0.730 23 0.914 0.881 39 0.939 0.917
8 0.818 0.749 24 0.916 0.884 40 0.940 0.919
9 0.829 0.764 25 0.918 0.888 41 0.941 0.920
10 0.842 0.781 26 0.920 0.891 42 0.942 0.922
11 0.850 0.781 27 0.923 0.894 43 0.943 0.923
12 0.859 0.805 28 0.924 0.896 44 0.944 0.924
13 0.866 0.814 29 0.926 0.898 45 0.945 0.926
14 0.874 0.825 30 0.927 0.900 46 0.945 0.927
15 0.881 0.835 31 0.929 0.902 47 0.946 0.928
16 0.887 0.884 32 0.930 0.904 48 0.947 0.929
17 0.892 0.851 33 0.931 0.906 49 0.947 0.929
18 0.897 0.858 34 0.933 0.908 50 0.947 0.930
19 0.901 0.863 35 0.934 0.910

The null hypothesis is rejected if W < Wcrit. From this table we can deduce that Wcrit depends on so-called statistical significance level alpha (see article http://en.wikipedia.org/wiki/Statistically_significant), and on the actual number of experiments n. This test was specialized for small n (under 40-50), so if you have to test a larger sample, it's better to use other tests like Kolmogorov–Smirnov test (http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test) —Preceding unsigned comment added by 93.73.35.146 (talk) 07:47, 22 August 2010 (UTC)