Fortran must die
I had always thought that given say a 100 cases the central limit theorem always worked. But today I read (in a book about multilevel analysis) that...
I actually did not think the Central Limit Theorem ever applied to raw data. I thought it applied to the distribution of the statistic.
The central limit theorem holds in practice..if the individual variances are small compared to the total variance. For example the heights of women in the United States follow an approximate normal distribution. The central limit theorem applies here because the height is affected by many small additive factors. In contrast, the distribution of heights of all adults in the United States is not so close to normality. The central limit theorem does not apply here because there is a single large factor -sex- that represents much of the total variation.