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I am trying to show that my sample mean is approximately normalized. But, when I perform a shapiro wilk normality test, I obtain a p-value of 0.0002589. So, my sample mean doesn't tend to follow a normalized distribution.

It make a non sense for me because I made two plots which prove my sample mean is normalized.

enter image description here enter image description here

Someone could explain to me why I obtain a p value which doesn't fit with the two graphics please ?

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    $\begingroup$ I asked a question long time on repeated measures long time ago that included uing Sahpiro-Wilk test, and @Glen_b responded in comments on this test. Have a look at this: question $\endgroup$
    – NULL
    Aug 16 '17 at 18:17
  • $\begingroup$ Aslo, are you familiar with how to interpret q-q plots? What do you think about this plot that posted? $\endgroup$
    – NULL
    Aug 16 '17 at 18:19
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    $\begingroup$ Please explain how you produced these plots: what exactly do they purport to show? When you do, would you mind also explaining why you think the sample mean ought to be Normal? What is the basis for that? $\endgroup$
    – whuber
    Aug 16 '17 at 18:24
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I'm not sure I'm going to answer your specific problem exactly, but most often the issue with normality testing is the following.

The Shapiro-Wilk test does not test if your distribution is approximatively normal, but if the distribution is exactly normal.

It says no when the data contains enough information to say that the distribution is not normal. When there is enough information (a lot of data), it will always say "no" unless the distribution is exactly normal, which only happens in rare cases where an underlying "physical" process ensures the distribution is exactly normal. Thus, when you have enough data, you almost always have a very small p-value: normality rejected.

I think the answers on that thread explain the problem quite well: Is normality testing 'essentially useless'?

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