Mean — Linear Transformation Rule

It is often useful to know how the mean of a data set changes when every observation is transformed in the same simple way — for example, converting every temperature from Celsius to Fahrenheit, every weight from kilograms to pounds, or (as in the step-deviation method of Section 2) shifting and scaling every class mark to make the arithmetic easier.

Why this is true. For a frequency distribution,

since $\sum f/\sum f = 1$.

This rule is exactly the algebra behind the step-deviation method: the step-deviation $u = (x-A)/h$ is itself a linear transformation of $x$ (with $a = 1/h$ and $b = -A/h$), so once $\bar{u}$ is found, the actual mean is recovered by reversing the transformation, $x = hu + A$, giving $\bar{x} = h\bar{u} + A$ — precisely the step-deviation formula of Section 2. Worked Example 8 demonstrates this connection explicitly. …