Help with a Correlation coefficient proof

This is the problem as I received it, I hope that someone can explain how to do it, as I am at a loss

Correlation Coefficient Definition*

r=(∑((x_{i}-x̄)/Sx))∗((y_{i}-ȳ)/Sy))/n-1

Computational Formula for r**

r=((n∑x_{i}y_{i}-(∑x_{i})(∑y_{i}))/(√(n∑x_{i}^{2}-(∑x_{i})^{2})∗(√(n∑y_{i}^{2}-(∑y_{i})^{2}))

1) How do you get from Definition to Computational? Show that * is the same as **

2)* and ** are for samples, show that you get ** for population correlations coefficient.

I would really appreciate any help on this as I am truly lost on how to do this. Thanks

Re: Help with a Correlation coefficient proof

Hey marks1630.

Can you expand the Sx and Sy terms out, collect terms and then simplify? Most of this is just substitution of definitions to show that you get an equality.

Re: Help with a Correlation coefficient proof

Thanks chiro, I kind of get that but I just don't know how to go about substituting or where to start.

Re: Help with a Correlation coefficient proof

What are the definitions of Sx and Sy? First get these in explicit form, plug these in, expand everything and collect terms.

There are proofs of this result that you can find but if you want to do it yourself, this is pretty much how to do it.

Also for these kinds of proofs, doing things like Sigma (xi) = n*x_bar is pretty much how to get to the answer. It's the same kinds of tricks like when you complete the square to solve various integrals and simplify algebraic expressions.

If you post the part where you get stuck, then we can read it and make comments on that specific problem.

Re: Help with a Correlation coefficient proof

Sx and Sy are just the Standard deviation of x and y, at least that is my understanding of it.

Re: Help with a Correlation coefficient proof

Plug in the actual formula for Sx and Sy in terms of the observations and the mean of both samples, and then expand that whole thing out and simplify.

If it's just the standard error, then this will be 1/(n-1) * sum (x_i - x_bar) for Sx and a similar one for Sy: when you plug these in, you will get a massive equation involving only the x_i's, the y_i's and the means of the two.

The rest is just algebra and expanding everything out, making substitutions (like the kinds I mentioned above) and then simplifying.

You know what the end result should look like, so that will give you hints as to how your next algebraic transformation should be.