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 $\overline x$ be the arithmetic mean of the data in $D$.

Let $T: \R \to \R$ be a linear transformation such that:
 * $\forall i \in \left\{{0, 1, \ldots, n}\right\}: T \left({x_i}\right) = \lambda x_i + \gamma$

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:
 * $T \left({\overline x}\right) = \lambda \overline x + \gamma$