If several samples are taken from a distribution, say Gaussian, each sample having size n1,n2,n3,... and the SD of the underlying distribution is estimated from each of the samples, how can those estimates be properly combined to make a better overall estimate of the population SD?
The specifics of the situation are that sets of measurements are taken with an instrument at different levels and thus can not be simply combined, but the uncertainty of the instrument appears not to be proportional to the magnitude of the measurement, so the SD at any level could probably be combined. I'm interested in the variability of single measurements, not their mean and the n's are sufficiently different to weigh the groups by them, I suppose. I don't know how to pool the variance estimates convincingly.
p.s. Not sure if this idea is valid, but I was thinking of simply taking the square root of the weighted mean of variances for each group - weighted by the inverse of the square root of sample/group size - if that makes any sense...
