You found the choice that I made to prevent the math from being even more daunting. Glad you asked so that I can take it one level higher.

To everyone who is uncomfortable with a square root, go have fun! Live your life!! Read a novel instead, search my name, you’ll find one you like, but don’t proceed into the morass that follows.

Ahem.

The uncertainty in the number of passes, N-pass, is as you caught in the article: sqrt(N-pass) and the uncertainty in the sample is sqrt(N-total). That’s fine, up to arguing whether the statistics follow a Gaussian (it’s pretty close for anything over about 20, and besides, cocktail napkin calculations).

But the uncertainty on the ratio is trickier and I left that out because the correction is small. Doing it right, you have to make use “propagation of errors” and make an assumption that’s unwarranted (but good enough for cocktail napkins).

The uncertainty in the ratio f(a, b) = a/b, given the uncertainty in a, da, and the uncertainty in b, db, is:

d[f(a,b)] = sqrt[ (df/da)²(da)² + (df/db)²(db)²+
(df/da)*(df/db)*da*db*rho) ]

where rho = the correlation between a and b. We assume it’s negligible, which is highly debatable since, in our case the sample total is correlated to the pass/fail numbers.

When you do the algebra (and post it here for all to see ;) you can convince yourself that for cocktail napkin calculations, the total uncertainty in our case is

d(fraction that passes) ~ d(N-pass)/N-total = sqrt(N-pass)/N-total