# Interpreting Shapiro-Wilk-Test, skewness and kurtosis in the light of a qqplot and histogram

Currently I'm trying to find out if my data with n=11 follows a normal distribution to decide how I process further. To find this out I use the Shapiro-Wilk-Test which gives me p < 0.05 and thus I reject the H1. Further, I took a look on the skewness and kurtosis of my distribution.

Shapiro-
Wilk-Test     Skewness                   Kurtosis
W     p     Statistic SE    Z      Statistic SE     Z
0.92  0.41  0.39      0.66  0.59   -0.99     1.27   -0.78


As -1.96 < Z < 1.96 I reject the H1 for skewness as well for kurtosis.

Then, I took a look on the qqplot, too. While there are some points that deviate from the line, most of them are pretty close to it. Together with the values of the Shapiro-Wilk-Test, skewness, kurtosis and the qqplot I assumed that my distribution is close to normal. However, taking a look on a histogram of my distribution reveals a distribution that is not even approximately normal.

My question is if using the Shapiro-Wilk-Test, skewness and kurtosis values such few data is useless and if I interpreted the qqplot wrong and should rather stick to the histogram?

• With only $n = 11$ obs, it is very difficult to say whether the population from which they were sampled is normal, uniform, exponential--or anything else. Shapiro-Wilk may reject normality, but the Q-Q plot looks fine. For such small sample sizes sample skewness and kurtosis could be almost any value (try moving one observation from below 50 into the gap between 110 and 130. Then recompute Shapiro-Wilk P-value, skewness and kurtosis. Almost no histogram of 11 obs will look normal, unless you contrive iit. // Also you haven't made it clear what "further processing "is yet to come.come. – BruceET Jul 15 '19 at 0:03
• When I tried digitizing your sample from the y-axis of your Q-Q plot and then running a Shapiro-Wilk test, I got P-value about 10%. I'm not saying you made a mistake, but my reconstructed sample can't be hugely wrong, so I suspect Shapiro-Wild doesn't say the actual sample his 'hugely' non-normal. In fact, your output seems to say it's 41% >> 5%. Maybe your question can benefit from some clarification. – BruceET Jul 15 '19 at 0:14
• Thanks for your comment. With further processing I mean which method for inference statistics I should use. I want to test for normality to decide if I have to use parametric or non-parametric tests. – Brain Damage Jul 15 '19 at 11:22
• Seems like I made a mistake when writing my post and I meant to say that Shapiro-Wilk gives me a p-value greater than 0.05. So, if I understand you right, kurtosis and skewness aren't reliable for such a small sample size? – Brain Damage Jul 15 '19 at 11:32
• With only 11 observations you may have trouble getting a significant result from a nonparametric test. Test of what? – BruceET Jul 15 '19 at 15:17