Linear Transformation of Arithmetic Mean
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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:
- $\map T {\overline x} = \lambda \overline x + \gamma$
Proof 1
Follows from the definition of arithmetic mean and from Summation is Linear.
$\blacksquare$
Proof 2
This is a direct application of Expectation is Linear.
$\blacksquare$
Sources
- 2011: Charles Henry Brase and Corrinne Pellillo Brase: Understandable Statistics (10th ed.): $\S 5.1$