Standard Deviation For A Sum Of Normal Distributions

  It can be shown using the mathematical tools found in textbooks on Statistical Mechanics that the standard deviation for a sum of m normal distributions whose individual standard deviations are σ_j is,

So, we don't need to calculate the expected value integral E(x²) with the weighted sum of the individual normal distributions if the individual stds are already known. One can also show that the mean for the summed distribution is the weighted sum of the means for the normal distributions.