# Thread: Help with proof of Arithmetic mean

1. ## Help with proof of Arithmetic mean

$\displaystyle x = \frac{1}{n} \sum^{n}_{i = 1} x_{i}$

$\displaystyle x = \frac{1}{n} \sum^{n}_{i = 1} x_{i} - a + a$

$\displaystyle x = \frac{1}{n} \left( \sum^{n}_{i = 1} x_{i} - na \right) + a$ <- What the heck happened with that $\displaystyle na$ thing? The transformation from the previous step, to this one, and the next step?

$\displaystyle x = \frac{1}{n} \left[ \sum^{n}_{i = 1} x_{i} - a \right] + a$

$\displaystyle x = \frac{ \sum^{n}_{i = 1} (x_{i} - a) }{n} + a$

By the way on the x on the left hand side there should be a little bar on top, indicating average.

Thanks for any help.

2. Is this what you are missing?
$\displaystyle \begin{gathered} \sum\limits_{k = 1}^n {\left( {x_k - a} \right)} = \left( {\sum\limits_{k = 1}^n {x_k } } \right) - na \hfill \\ \frac{1} {n}\left[ {\left( {\sum\limits_{k = 1}^n {x_k } } \right) - na} \right] = \frac{1} {n}\left( {\sum\limits_{k = 1}^n {x_k } } \right) - a \hfill \\ \end{gathered}$

3. Originally Posted by Plato
Is this what you are missing?
$\displaystyle \begin{gathered} \sum\limits_{k = 1}^n {\left( {x_k - a} \right)} = \left( {\sum\limits_{k = 1}^n {x_k } } \right) - na \hfill \\ \frac{1} {n}\left[ {\left( {\sum\limits_{k = 1}^n {x_k } } \right) - na} \right] = \frac{1} {n}\left( {\sum\limits_{k = 1}^n {x_k } } \right) - a \hfill \\ \end{gathered}$
But didn't that $\displaystyle n$ of $\displaystyle na$ get factored out somehow?

4. Is this what you are missing?
$\displaystyle \frac{1} {n}\left[ {\left( {\sum\limits_{k = 1}^n {x_k } } \right) - na} \right] = \frac{1} {n}\left( {\sum\limits_{k = 1}^n {x_k } } \right) - \frac{1} {n}\left( {na} \right)$

5. Originally Posted by Plato
Is this what you are missing?
$\displaystyle \frac{1} {n}\left[ {\left( {\sum\limits_{k = 1}^n {x_k } } \right) - na} \right] = \frac{1} {n}\left( {\sum\limits_{k = 1}^n {x_k } } \right) - \frac{1} {n}\left( {na} \right)$
There we go, that works for me. Thanks Plato.