Spread of a commute that is a bus leg plus a walk

Question

A commute has two parts that vary independently of each other: a bus leg with variance $16$ minutes-squared (standard deviation $4$ minutes) and a walking leg with variance $9$ minutes-squared (standard deviation $3$ minutes). The total commute time is

$T = B + W.$

What is the standard deviation of the total commute time?

Accepted Answer

The tempting mistake is to add the two standard deviations, 4 + 3 = 7. That is wrong. Spreads do not add; variances do, and only when the parts are independent. The rule for independent pieces is \mathrm{Var}(B + W) = \mathrm{Var}(B) + \mathrm{Var}(W). In words: when two random quantities have nothing to do with each other, the wobble in the total is found by adding their variances, not their standard deviations. Then take a square root at the end. Apply it. \mathrm{Var}(T) = 16 + 9 = 25 	ext{ minutes-squared}. \mathrm{SD}(T) = \sqrt{25} = 5 	ext{ minutes}. Notice the total spread of 5 is l…

Spread of a commute that is a bus leg plus a walk

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