Linear Transformation of Arithmetic Mean

Theorem
Let $D = \left\{{x_0, x_1, x_2, \ldots, x_n}\right\}$ be a set of real data describing a quantitative variable.

Let $\bar{x}$ be the arithmetic mean of the data in $D$.

Let $T: \R \to \R$, $x_i \mapsto \lambda x_i + \gamma$ be a linear transformation.

Let $T\left({D}\right)$ be the image of $D$ under $T$.

Then the arithmetic mean of the data in $T\left({D}\right)$ is given by $T\left({\bar{x}}\right) = \lambda \bar{x} + \gamma$.