Return to Answer

For example, you mentioned skewness and kurtosis - while those measures are certainly ways of identifying distributions that aren't Gaussian, and they can be combined into a single measure of deviation from Gaussian-ness* (and even form the basis of some common tests of normality), they're terrible at identifying distributions that have the same skewness and kurtosis as a normal but are distinctly non-normal.

A measure based on skewness and kurtosis is going to be terrible at identifying distributions such as these. Of course, if you're not worried about such possibilities, this may not matter - if you mainly want to pick up cases where the skewness and kurtosis deviate from those of the normal, a test based on those two measures is relatively powerful.

For example, you mentioned skewness and kurtosis - while those measures are certainly ways of identifying distributions that aren't Gaussian, and they can be combined into a single measure of deviation from Gaussian-ness* (and even form the basis of some common tests of normality), they're not great at identifying distributions that have the same skewness and kurtosis as a normal but are distinctly non-normal.

A measure based on skewness and kurtosis is going to be fairly poor at identifying distributions such as these. Of course, if you're not worried about such possibilities, this may not matter - if you mainly want to pick up cases where the skewness and kurtosis deviate from those of the normal, a test based on those two measures is relatively powerful.

It's bimodal, but has skewness 0, and kurtosis 3.00 (to two d.p. -- i.e. an excess kurtosis of 0), the same as the normal.

A measure based on skewness and kurtosis is going to be terrible at identifying distributions such as these. Of course, if you're not worried about such possibilities, this may not matter -f you mainly want to pick up cases where the skewness and kurtosis deviate from those of the normal, a test based on those two measures is relatively powerful.

The chi-square you mention probably refers to the chi-square goodness of fit test. It's generally a very weak test of goodness of fit for anything other than distributions over nominal categories. (Alternatively, it might be a reference to the asymptotic chi-square distribution of the Jarque-Bera type test. Be warned, the asymptotics there kick in very, very slowly indeed.)

It's bimodal, but has skewness 0, and kurtosis 3.00 (i.e. an excess kurtosis of 0), the same as the normal.

A measure based on skewness and kurtosis is going to be terrible at identifying distributions such as these. Of course, if you're not worried about such possibilities, this may not matter - if you mainly want to pick up cases where the skewness and kurtosis deviate from those of the normal, a test based on those two measures is relatively powerful.

The chi-square you mention probably refers to the chi-square goodness of fit test. It's generally a very weak test of goodness of fit for anything other than distributions over nominal categories. (Alternatively, it might be a reference to the asymptotic chi-square distribution of the Jarque-Bera type test. Be warned, the asymptotics there 'kick in' very slowly - you need a very large n for the distribution of the test statistic to be close to normal -- see Bowman and Shenton's paper -- which predates the one by Jarque and Bera. My own simulations from back in the 80s suggest that n=200 is not close to adequate - at least for the sort of adherence to significance levels I'd look for - but it's generally pretty reasonable by about n=500)

For additional useful points, see also here or here or here or here

For additional useful points, see also here or here or here or here