If adding the average to a collection of numbers, will the average of the collection change?

Question

Consider the collection of integer numbers $\{a_1, a_2, \ldots, a_n\}$ having the average avg - also an integer number.

Will adding avg itself to the collection change the average of the collection i.e. will the average of $\{a_1, a_2, \ldots, a_n, \text{avg}\}$ also be avg ?

Simple examples seem to show this holds:

Ex1: $\{2, 2, 4, 4\}$, with avg = $3$. We have $\{2, 2, 4, 4, 3\}$ with average also being $3$.

Is this always true?

Thanks!

Answer 1

Yes it is. The average of a tuple $a = (a_1,...,a_n)$ is \begin{align*}\overline a = \frac{1}{n}\sum_{i=1}^n a_i\end{align*} So if $\overline a$ is the average of $a = (a_1,...a_n)$ and $b = (a_1,...,a_n,\overline a)$, then the average of $b$ is \begin{align*}\overline b &= \frac{1}{n+1}\sum_{i=1}^{n+1} b_i = \frac{\sum_{i=1}^na_i + \overline a}{n+1}\\ &= \frac{\sum_{i=1}^na_i + \frac{1}{n}\sum_{i=1}^n a_i}{n+1} \\ &= \frac{1}{n+1}\frac{n\sum_{i=1}^na_i + \sum_{i=1}^na_i}{n}\\&= \frac{1}{n+1}\frac{(n+1)\sum_{i=1}^na_i}{n} = \frac{\sum_{i=1}^na_i}{n} = \overline a\end{align*}