The overall mean and variance of a Gaussian mixture of i components can be calculated from the means and variances of the individual components multiplied by their weights:
M = sumi(wi mi) and V = sumi(vi + mi2) - M2
where M, V = mean and variance of mixture
sumi(...) = sum over all i
wi, mi, vi = weight, mean and variance of ith component
sumi(wi) = 1
Is it then legitimate to use a single Guassian defined by M and V in place of the original mixture? I am only interested in descriptive stats here rather than inferential statistics. Are there any implications for uncertainty calculations?
It depends on whether M1,M2...Mn differ significantly and the quality of fit demanded by the application. One should always use the model with the minimum number of mixture components to achieve the required level of fit. If fitting one gaussian is no worse than fitting two after adjusting for degrees of freedom, then one makes sense.
I would say it depends on what you are modeling. what effects are being left out of the model by not using the mixture of distributions? If this is an important consideration the answer is no,.
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