How can I compare distributions using only mean and standard deviation?

Question

I only have means and standard deviations of samples of two random variables. What technique can I use to determine how similar the distributions these describe are? Assume that the values are built from very similar samples. I'm looking for a mechanism to detect when the distributions deviate from one another by some threshold. Access to historical observations is limited.

Answer 1

There are a number of different tests that are generally used to compare samples to different distributions, such as Jarque-Bera, Anderson-Darling, and Kolmogorov–Smirnov (see this related question).

In your case, with just the standard deviation and mean, there isn't a whole lot to say. You need to assume a distribution (e.g. normal). You would be able to tell much more if you could at least get the skewness and kurtosis.

Answer 2

Be careful, remember that the mean and the standard deviation don't tell you the whole story: http://en.wikipedia.org/wiki/Anscombe%27s_quartet