Variance and Standard Deviation Formula Proof

The variance of a set of data is given by:

`sigma^2=frac{Sigma(x-bar x)^2}{n}`

But a simpler formula is:

`sigma^2=frac{Sigma x^2}{n} - bar x^2`

Proof:

`sigma^2=frac{Sigma(x-bar x)^2}{n}`

`=frac{Sigma(x^2-2x bar x +bar x^2)}{n}`

`=frac{Sigma x^2}{n}-frac{Sigma 2x bar x}{n}+frac{Sigma bar x^2}{n}`

Since `bar x` is a constant, we can factor it out of the sums like so:

`=frac{Sigma x^2}{n}-2 bar x frac{Sigma x}{n}+bar x^2 frac{Sigma 1}{n}`

We know that `frac{Sigma x}{n}` is `bar x` by definition, and `Sigma 1` is `n` since we are summing for a data ser of size `n`. Therefore we have:

`=frac{Sigma x^2}{n}-2 bar x^2+bar x^2 frac{n}{n}`

`=frac{Sigma x^2}{n}-2 bar x^2+bar x^2`

`=frac{Sigma x^2}{n}-bar x^2`

`QED`

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