Why a negative multiplier does not change the variance

Question

A quantity Y is built from X by Y = -3\,X + 7, and X has variance 16 (so its standard deviation is 4). What is the standard deviation of Y?

Accepted Answer

The interesting part here is the minus sign. Multiplying by -3 flips the values around and stretches them by a factor of 3. Flipping does not squeeze the data together at all; a value far above the average simply lands far below it. Spread is about distance from the center, and distance does not care about direction. The rule for Y = aX + b is \mathrm{Var}(Y) = a^2\,\mathrm{Var}(X), \qquad \mathrm{SD}(Y) = a\,\mathrm{SD}(X). The multiplier is squared (or you take its absolute value for the standard deviation), so the sign disappears. Apply it. With a = -3 and b = 7: \mathrm{SD}(Y) = -3 	imes…

Why a negative multiplier does not change the variance

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