# Cramer’s decomposition theorem - find normal distributions within a normal distribution.

I know that Cramer's decomposition theorem says that any normal distribution can be expressed as the sum of multiple normal distributions. I have been searching for a method to divide a data set that constitute a normal distribution to 5 datasets, each of which will be a normal distribution and the sum of which will constitute the 'big' normal distribution I have begun with. Can some of you wizards suggest me a procedure? If you know the SPSS procedure, that will be even sweeter. Thanks a ton, in advance.

-
What are the conditions on your data sets? – Raskolnikov Sep 6 '12 at 17:37
It's not a sum of normal distributions, but rather a sum of independent random variables that have normal distributions. The operation on distributions, as opposed to that on random variables, is convolution rather than addition. But I think I'd want to know what your data looks like before attempting an answer. – Michael Hardy Sep 6 '12 at 17:38
PS: I changed the spelling to Cramer, with only one "m". Purists write "Cramér", with an accent. – Michael Hardy Sep 6 '12 at 17:38
Variance components models, sometimes called random effects models, may be what you're looking for. Google those terms. – Michael Hardy Sep 6 '12 at 17:41
I should add that Cramer's theorem doesn't say that normal distributions can be decomposed in that way, but rather that they can be decomposed only in that way. I.e. if a normally distirbuted random variable is the sum of independent random variables, then all of the terms in the sum must themselves be normally distributed. – Michael Hardy Sep 6 '12 at 18:26