Why the difference of two measurements is noisier, not quieter

Question

Two independent measurements X and Y are each taken with variance 9 (standard deviation 3). You report their difference, D = X - Y. What is the standard deviation of the difference D?

Accepted Answer

It feels like subtracting should cancel noise, but the opposite is true. Both readings carry their own independent wobble, and the difference inherits both wobbles at once. Flipping the sign of Y does not reduce how much Y jitters, so its full variance still contributes. The rule for a difference of independent quantities is the same as for a sum: the variances add. \mathrm{Var}(X - Y) = \mathrm{Var}(X) + \mathrm{Var}(Y). The minus sign disappears because variance squares the coefficient on Y, turning -1 into +1. Apply it. \mathrm{Var}(D) = 9 + 9 = 18, \qquad \mathrm{SD}(D) = \sqrt{18} \appro…

Why the difference of two measurements is noisier, not quieter

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