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?

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.
