Normalizing a distribution of distributions.

I have a data set which consists of a bunch of means and standard deviations of normal distributions, and I want to normalize the data so that the means are normally distributed with mean 0 and standard deviation 1, and the standard deviations are appropriately converted. It's obvious how to modify the means (just compute the mean of means and use the usual Z-score), but how do I modify the standard deviations?

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The means are normally distributed between -1 and 1...: This is a bit of a contradiction in terms. :) –  cardinal Mar 31 '12 at 20:24
updated. My means are all between 0 and 10, and I want to get the means to be all between -1 and 1, but still normally distributed (with some reasonable scaling...). I think I can manage that if I get the data to be normally distributed with mean 0 and standard deviation 1. –  JeremyKun Mar 31 '12 at 20:25
Your means require a transformation. In this case it is $f(x) = \tfrac{1}{5}x - 1$. However, I would suggest that you consider treating your standard deviations as the roots of variances drawn from a $\chi^2$ distribution (though you will have to figure out the number of degrees of freedom). –  polarise Jan 16 at 23:42