Statistics: Appendix D: Proof for the formula of the arithmetic mean

donderdag 1 mei 2014

Appendix D: Proof for the formula of the arithmetic mean

The arithmetic mean is the balancing point of the distribution. This means that the ‘weight’ to the left of the mean, should be same as the weight to the right of the mean. In formula notation this gives:

$\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)=0$

Expand this sum into two:

$\sum_{i=1}^{n}\left(x_{i}\right)-\sum_{i=1}^{n}\left(\bar{x}\right)=0$

Move the second part to the right hand side:

$\sum_{i=1}^{n}\left(x_{i}\right)=\sum_{i=1}^{n}\left(\bar{x}\right)$

Simplify the right hand side:

$\sum_{i=1}^{n}\left(x_{i}\right)=n\bar{x}$

Divide both sides by n:

$\frac{\sum_{i=1}^{n}\left(x_{i}\right)}{n}=\bar{x}$

Q.E.D.

Statistics

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donderdag 1 mei 2014

Appendix D: Proof for the formula of the arithmetic mean

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