A good measurement of normality should be like a confidence interval,
not a hypothesis test.
(1) It should measure how far the distribution is from normality, in a
way that is appropriate for what you want to do next.
This measure should be asymptotically stable as N -> infinity, like the
center of a standard CI.
(2) It should measure the uncertainty, so you can tell whether *all*
distributions consistent with the data are reasonably close to normal.
This should go to 0 as N -> infinity, like the width of a standard CI.
With such an inference method, if you look into the Mirror of Erised
and daydream about that truckload of data, you would see yourself
learning something useful, and you would not know ahead of time what it
was. So you have a motive to do the work.
With the tests, if you're honest with yourself, you say "If I had a
million data... I'd reject!" (Fans of the Canadian group "The Barenaked
Ladies" will know the tune to sing this to...) This
_Gedankenexperiment_ seems to make the collection of data pointless.
I presume that with certain sample sizes the 5% p-value corresponds
roughly to "normal enough for rock-n-roll" and that that is how these
things ever got to be used. (If I remember correctly, some manuals
recommend wildly nonstandard critical values for these tests, which is,
I suppose, some sort of kludge to get around this)
-Robert Dawson
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