On Quora, someone asked about hypothesis tests for skewness and kurtosis. I wrote an answer but I’ll expand a bit here.
I can’t think of a reason to do this; I can’t think of a reason why you’d be interested in whether skewness or kurtosis is significantly different from 0 – you want to know if it is far from 0, which could happen even if it’s not significant and could fail to happen if it is significant.
Nevertheless, if you want to test this for some reason, you can. Most statistical packages will print out the standard error of the skewness and kurtosis, and, under the null, they are normally distributed, so if it’s more than 2 SE from 0, it’s sig.
If your package doesn’t output the SE of skewness and kurtosis then here you go:
Usually, when you are looking at skewness and kurtosis in this way, you are interested in normality; but there are broader tests for that, which may be more useful.