(Statistics) Calculating Correlation Coefficient

**Timbuk2**

I'm a first year student doing Statistics as part of my degree at the moment.

We were told that to calculate the correlation coefficient (r), we compute it as such
[latex]r=\displaystyle\frac{SS_{xy}}{\sqrt{SS_{xx}}\sqrt{SS_{yy}}}[/latex]

Which is equivalent to
[latex]r=\displaystyle\frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2\sum(y-\bar{y})^2}}[/latex]

I understand it perfectly up until here, but we were told that it's easier to compute by hand if we write it in the following way.

[latex]r=\displaystyle\frac{\sum{xy}-\frac{\sum{x}\sum{y}}{n}}{\sqrt{(\sum{x^2}-\frac{(\sum{x})^2}{n})(\sum{y^2}-\frac{(\sum{y})^2}{n})}}[/latex]

And while this is considerably easier to compute by hand, I don't understand the equivalence.

How does [latex]\sum(x-\bar{x})(y-\bar{y})[/latex] equate to [latex]\sum{xy}-\frac{\sum{x}\sum{y}}{n}[/latex], for example?

Is it just an estimate (seeing as there is an n, the amount of pairs of data, included?) or is it an exact equivalence.

Thanks

Fremen

It's exact. You just need to go back to the definition of [latex]\bar{x}[/latex]

[latex]\sum(X-\bar{X})(Y -\bar{Y}) = \sum(XY) - n\bar{X}\bar{Y} [/latex]

(why is this true?)

then,
[latex]\bar{X}\bar{Y} = \left(\frac{\sum X}{n}\right) \left(\frac{\sum Y}{n}\right) [/latex],

so

[latex]n \bar{X}\bar{Y} = \frac{\sum X \sum Y}{n} [/latex].

In the language of expectations,

[latex] \mathbb{E}[(X -\mathbb{E}[X])(Y - \mathbb{E}[Y])] = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y][/latex].

This is true for all random variables.

Once you understand how the numerator works, just set X=Y and the whole thing will follow.

**Timbuk2**

Thank you that makes complete sense