# Normality test of data with large skewness and kurtosis

I have a data set and want to know whether it has a normal distribution. I use both Kuiper and K-S normality tests and the null hypothesis is rejected in both tests. This means that the data can have normal distribution. However, the skewness of data is about 2 and the kurtosis is about 10 suggesting the data cannot have a normal distribution. My question is which one should I trust for checking the normality? the normality tests or the skewness and kurtosis values?

• If you have rejected the null hypothesis, that means you have rejected the hypothesis that the data is Normally distributed. There is no disagreement between your two approaches. Commented Oct 7, 2021 at 19:53

Both of the tests you have run have normality as the null and non-normality as the alternative hypothesis. Therefore, when you reject the null hypotheses, you reject the notion of normality. This is consistent with your observation that the skewness and kurtosis indicate a lack of normality.

Additionally, you may be interested in the argument that such normality testing is essentially useless.

• Thank you for your comment and for the point you mentioned about null hypothesis. Yes, I made a mistake in interpretation. However, what if this happens: Null hypothesis is accepted, with the skewness of 0 and kurtosis of 1. Here null hypothesis is saying data can be normal but the kurtosis is far from kurtosis of normal data (that is equal to 3). How can we interpret this case?
– Aep
Commented Oct 7, 2021 at 20:57
• You do not accept the null hypothesis, so I will guess that you mean a large p-value. If I got a large p-value from the hypothesis test despite the kurtosis seeming non-normal, I would believe that I lacked enough data to make much of an assessment of the normality. Confidence intervals for kurtosis are funky, but I would bet that it would be rather wide and include $3$, as normality requires. Thus, while the observed kurtosis is smaller than expected, that is not inconsistent with normality, given the sample size.
– Dave
Commented Oct 7, 2021 at 21:01
• Thanks for your comment Dave. To make it clear for myself, do you mean the data is still normal considering the small value of kurtosis? Also, the size of my dataset is about 2000 and I think it is not small.
– Aep
Commented Oct 7, 2021 at 21:13
• No, my point is that, if the conditions you stated were true, I would conclude that I don't know enough to make a decision, the usual conclusion when we fail to reject the null hypothesis. // I would be flabbergasted if a KS test saw a distribution with empirical kurtosis of $1$ and failed to reject despite the power of $2000$ points. You're right that you have a fairly large data set.
– Dave
Commented Oct 7, 2021 at 21:23