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Just a note

When derived from real data, statistics only have a completely normal distribution when calculated from almost infinitely large samples. For N samples, each containing n observations, a statistic cannot be normal unless both N and n approach infinity - that is when the samples are at the limit of what is possible.

In this, highly theoretical situation, the statistic adopts what is known as its asymptotic behaviour, or limit distribution. Unfortunately, not all statistics are asymptotically normal - and some statistics approach their asymptotic behaviour very very gradually indeed.