Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Expected value ▷ Linearity of the sample mean

Theorem: Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a sample from a random variable $X$ and let $\tilde{x} = \left\lbrace \tilde{x}_1, \ldots, \tilde{x}_n \right\rbrace$ be a linearly transformed version of this sample with

\[\label{eq:samp-lin} \tilde{x}_i = a x_i + b \; .\]

where $a$ and $b$ are constant real numbers. The sample mean behaves linearly under this transformation, i.e.

\[\label{eq:mean-samp-lin} \bar{\tilde{x}} = a \bar{x} + b \; .\]

Proof: The sample mean is defined as

\[\label{eq:mean-samp} \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \; .\]

Applying this definition to the linearly transformed sample, we obtain:

\[\label{eq:mean-samp-lin-qed} \begin{split} \bar{\tilde{x}} &\overset{\eqref{eq:mean-samp}}{=} \frac{1}{n} \sum_{i=1}^{n} \tilde{x}_i \\ &\overset{\eqref{eq:samp-lin}}{=} \frac{1}{n} \sum_{i=1}^{n} (a x_i + b) \\ &= \frac{1}{n} \sum_{i=1}^{n} a x_i + \frac{1}{n} \sum_{i=1}^{n} b \\ &= a \cdot \frac{1}{n} \sum_{i=1}^{n} x_i + \frac{1}{n} \cdot n b \\ &= a \bar{x} + b \; . \end{split}\]
∎
Sources:

Metadata: ID: P538 | shortcut: meansamp-lin | author: JoramSoch | date: 2026-05-28, 08:52.