Say I am modeling test scores. As a given, an individual student’s test score has a standard deviation of σ=5. If the distribution of all students’ test scores has an stdev of σ=12, I can model this using a normal distribution with σ=10 then add the σ=5 variation to each test score, as σ=sqrt (102 + 52) = 12, which matches the measured distribution. My question is: if the measured distribution is not exactly normal, how do I create the σ=10 distribution from the measured σ=12 distribution? Can this be done? More importantly, how can I justify that the σ=10 distribution reflects the individual students?….. For a shortened version: I have 5 test scores, 33,44,50,55,66, (σ=12). How do I adjust these to get σ=10? If I simply move the two outer scores in by three points, I get σ=10, but these two outlier students are no longer accurately modeled.