🚀 go-pugleaf

Normality Testing.

If you have a lot of samples, a straightforward and powerful test of normality
that does not require the use of tables is the method of moments. Compute
the moments of the distribution ( it is probably sufficient to compute the
first four moments but you can go higher if you so choose ). The normal
distribution has the property that :-

	if m == mean , v is the variance of the distribution then

	define mn = the nth moment ie : E[ (x-m)^n ]

	where E is the expectation, ^ means "to the power of "

	we have the following handy result :-

	mn = 0 if n is odd

	mn = (n-1)(n-3)...3.1.v^n  for n even.

Furthermore, this property is unique to the normal distribution.

Application :-

	compute the mean and variance of the sample distribution, then
compute 1/n*sum( x-m)^n ) where x is the random variable ( I am assuming
large n ; you need to adjust this to get an unbiased estimator for small 
n : use 1/Num Degrees of Freedom ).

Then see if the 3rd , 5th etc. moments are small or vanishing, and whether the
 4th, 6th ... follow the formula above. This is a pretty good test of
normality. It will differentiate between normal and lognormal samples
even for quite small sample sizes ( say, 30-50 observations ).

All my own humble opinion of course. 

Hope this helps, Phil.