Standard deviation of a length after converting inches to centimeters

Question

A batch of hand-cut boards has lengths in inches whose standard deviation is $0.5$ inch. You convert each length to centimeters using

$L = 2.54\,X,$

where $X$ is the length in inches.

What is the standard deviation of the lengths in centimeters?

Accepted Answer

Converting units here is pure stretching: every inch becomes 2.54 centimeters, with no shift added. Stretching the scale stretches the spread by the same factor. The rule for Y = aX (a multiplier, no shift) is \mathrm{SD}(Y) = a\,\mathrm{SD}(X), meaning the standard deviation just gets multiplied by the same conversion factor. Apply it. With a = 2.54, \mathrm{SD}(L) = 2.54 	imes 0.5 = 1.27 	ext{ centimeters}. As a check on the variance, \mathrm{Var}(L) = 2.54^2 	imes 0.25 = 6.4516 	imes 0.25 = 1.6129, and \sqrt{1.6129} = 1.27. Takeaway. A unit conversion is a pure multiplier, so the stand…

Standard deviation of a length after converting inches to centimeters

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