Spread of a two-stage assembly time

Question

A part is built in two independent stages. Stage one's time has variance $5$ and stage two's time has variance $11$ (both in seconds-squared). The total build time is

$T = S_1 + S_2.$

What is the standard deviation of the total build time?

Accepted Answer

The two stages run independently, so their variances add: \mathrm{Var}(T) = \mathrm{Var}(S1) + \mathrm{Var}(S2) = 5 + 11 = 16 	ext{ seconds-squared}. The reason we add variances and not standard deviations is that variance is the honest, additive measure of independent wobble; the standard deviation is a square root, and square roots do not add. So combine at the variance level, then convert back at the very end: \mathrm{SD}(T) = \sqrt{16} = 4 	ext{ seconds}. Takeaway. Even when the two parts have different spreads, independence means their variances still simply add. Combine on the varianc…

Spread of a two-stage assembly time

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