# [S] Revised Question: Shapiro-Wilk test for several goups

Steven P. Millard (steven.p.millard@mci2000.com)
Wed, 28 Oct 1998 11:38:18 -0800

Hello,

Some people were confused about the question I posted yesterday, so I
will try to be more clear.

Suppose you have k groups with possibly different means and standard
deviations. Suppose you take a random sample from each of the groups
and you want to test whether all groups follow a normal distribution.
One possible way to test this is to create "z-scores" or "z-values"
within each group by taking the observations within the i'th group
(i=1,2,...,k) and subtracting the i'th group mean and dividing by the
i'th group standard deviation. You can then combine all of these
"z-values" from all of the k groups into one sample and apply the
Shapiro-Wilk goodness-of-fit test to this one sample. Has anyone seen
this proposed test in the published literature?

Sincerely,

--Steve M.

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Original Querry:

Hello,

Wilk and Shapiro wrote a paper in 1968 (Technometrics, Vol.10, No.4,
pp.825-839) suggesting a way to test simultaneously for normality for
several, say k, groups with possibly different population means and
standard deviations, based on random samples from each of the groups.
Wilk and Shapiro actually propose two similar procures, both based on
combining the k p-values from the k individual goodness-of-fit tests
for each group. (One procedure was proposed by Fisher in the context
of combining the results of several independent tests.)

Wilk and Shapiro's procedures are not really specific to the
Shapiro-Wilk test, since they involve simply combining p-values from
several independent tests.

Here's my question: I've seen it suggested that you simply compute the
z-scores for each group, combine the z-scores, and perform the
goodness-of-fit test on the z-scores. Has anyone seen this proposed
test in the published literature?