I have a database with more than 50000 observations. I have applied non-parametric tests to determine the normality of the data but in any case p<0.05, rejecting the null hypothesis of normality (in many cases, graphically, the histograms appear to follow a bimodal distribution).

However, to elaborate a table, I don't know if it would be better to determine mean and standard deviation or median and median absolute deviation, since following the Central Limit Theorem (CLT), when the sample is large enough, it can approximate a normal distribution.


Furthermore, for the determination of normality by means of statistical contrasts, since the Shapiro-Wilk test cannot be used due to the large number of observations, would it be better to use the Kolmogorov-Smirnov test or the Anderson-Darling test?

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    $\begingroup$ Although you describe your data (somewhat), your question lacks purpose: why are you testing the normality of data that appear obviously to be non-normal? It is also predicated on a fallacy: no matter how much data you have, they are not going to be Normal and the CLT says nothing about that anyway. Could you share with us what you really need to accomplish? $\endgroup$
    – whuber
    Apr 30 at 21:46
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    $\begingroup$ What tests for normality are in any sense non-parametric? $\endgroup$
    – Nick Cox
    May 1 at 8:37
  • $\begingroup$ You have made a common but nonetheless incorrect assumption about the central limit theorem. Please see my question from last year: stats.stackexchange.com/q/473455/247274. The central limit theorem has to do with a transformation of the original data, not them data themselves. // If you run a simulation that takes $10000$ bootstrap samples (size of $50000$, with replacement) from your data, calculate the sample mean each time, and plot a histogram of the means, my guess is that you will wind up with a normal-looking plot. That is (more or less) what the CLT says. $\endgroup$
    – Dave
    May 1 at 13:14

That histogram is clearly bimodal, so with 50000 observations that cannot be a fluke, and your data is not normal, so you do not need a test. See also Is normality testing 'essentially useless'?

But rather tell us why you think you need normality? What is your goal with your data?

  • 1
    $\begingroup$ Besides the important comments others have written above, forget about the central limit theorem protecting you in any meaningful way. It is a limit theorem, not a theorem that pertains to your sample size, especially in the presence of asymmetry or bimodality. I would summarize the data with a histogram having 200 bins, along with median and IQR. $\endgroup$ May 1 at 11:23

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