The three measures that vary with n (Extreme Spread, Diagonal, and Figure of Merit) are range statistics that lack convenient functional forms. However both the mean and standard deviation of their expected value, as well as quantiles, scale directly with σ, so it is sufficient to calculate those statistics once for σ = 1 and multiply the resulting values by the desired σ. Media:Sigma1RangeStatistics.xls contains quantiles and moments for n up to 100.
What extreme spread should I expect for 5-shot groups from my rifle? The extreme spread median from the table for 5 shots is 3.0 σ. I've determined my rifle has precision σ = ½MOA. If I take five shots at 100 yards we would expect half my groups to be less than 3.0/2 = 1.5MOA \(\approx\) 1.6".
Multiplying the rest of the distribution data for that row by my 0.5MOA we can also say that the extreme spread of my 5-shot groups should exhibit the following distributions:
- 50% between (1.2, 1.8)MOA
- 80% between (1.0, 2.1)MOA
- 95% between (0.8, 2.4)MOA
Over many tests I have found my rifle produces 5-shot groups with an average extreme spread of 1MOA. What extreme spread should I expect if I instead start shooting 10-shot groups? The table shows that the ratio of expected extreme spreads on 10-shot groups is 1.24 times the value on 5-shot groups. So my a priori expectation would be for 10-shot groups to average 1.24MOA.