How a Celsius-to-Fahrenheit conversion changes the spread

Question

A thermometer's daily reading in Celsius, $C$ , has a standard deviation of $2$ degrees (so its variance is $4$ ). You convert every reading to Fahrenheit with the rule

$F = 1.8\,C + 32.$

What is the standard deviation of $F$ ?

Accepted Answer

Think about what the conversion does to the spread. Adding 32 just slides every reading up the scale by the same amount; sliding everything together does not change how spread out the readings are. Multiplying by 1.8 stretches the scale, and stretching does change the spread. The rule that captures both effects is: for Y = aX + b, \mathrm{Var}(Y) = a^2\,\mathrm{Var}(X). In words: the multiplier a gets squared and scales the variance, while the shift b drops out entirely. For the standard deviation this is even simpler, because the square root undoes the square: \mathrm{SD}(Y) = a\,\mathrm{SD}…

How a Celsius-to-Fahrenheit conversion changes the spread

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