Hi Mike,

The Sn definition includes normalization factors, a sample size dependent scaling along with an additional standardized dispersion scaling.

My implementation in PCL (and by extension on the whole PixInsight platform) applies correction factors for finite sample sizes. I haven't mentioned them because they are actually irrelevant for images, since in this context we are working with very large sample sizes, where these corrections are immaterial. Specifically, the correction factor c for sample sizes larger than 9 items is:

For Sn:

For odd sample size n: c = n/(n - 0.9)

For even sample size n: c = 1

For Qn:

For odd sample size n: c = n/(n + 1.4)

For even sample size n: c = n/(n + 3.8)

For coherence with already existing statistical estimators in PCL, the normalization factors are not applied by default in my implementation. They are not used in the ImageIntegration tool either, since in this case we are comparing the same scale estimates between pairs of images, and hence the normalization factors cancel out. The factors that make these estimators converge to the standard deviation of a normal distribution are:

For Sn: k = 1.1926

For Qn: k = 2.2219