Linear Transformation of Arithmetic Mean

Theorem
Let $D = \set {x_0, x_1, x_2, \ldots, x_n}$ 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:
 * $\forall i \in \set {0, 1, \ldots, n}: \map T {x_i} = \lambda x_i + \gamma$

Let $T \sqbrk D$ be the image of $D$ under $T$.

Then the arithmetic mean of the data in $T \sqbrk D$ is given by:
 * $\map T {\overline x} = \lambda \overline x + \gamma$