Regression Slope vs. Correlation Coefficient

[vnysot]

Can someone clarify the difference between regression slope (i.e. b0 = COV(Y,X)/Var(X))
and correlation coefficient (i.e. r = COV(Y,X)/var(x)var)
Mathematically, I see the differences, but intuitively, I thought the correlation coefficient was the slope. Can someone please clarify why they’re different?
Many thanks

 

[cpk123]

correlation coefficient = cov(x,y) / (sigma x * sigma y) not var(x) * var as you have got it above.
and regression slope is the beta. (from SML days)
beta = cov(x,y)/var(x)
correlation coeff is not the slope, beta is the slope.
Hope I am right!

 

[vnysot]

Ok, got ya. I actually made a typo and indeed the denominator in the correlation calculation should be sigma x*sigma y.
Put differently then, how would you express these definitions in layman’s terms:
“slope” = as you move through y, you get a certain change in x
“correlation” = for a change in one value, you get a certain % change in another value
Sounds the same.. I’m trying to make clear these differences, as they are mathematically. Thanks

 

[cpk123]

both correlation and the regression slope are two different things
correlation is not the change of one value for a % change in another value. It is the “strength” of the linear relation between the two variables in a single variable regression. (and which is definitely not the slope).

 

[vnysot]

Great, thanks.

 

[busprof]

in a univariate regression, the two measures are related as follows
start with
beta = cov(x,y)/var(x) = cov(x,y)/(sigma x * sigma x)
then substitute = correlation(x,y)* (sigma x * sigma y) into the above equation
cancel out the sigma x in the numerator and denominator, and you get
beta = correlation (x,y) * (sigma y)/(sigma x)
so, the beta is just the correlation adjusted for the relative dispersions of x and y.
correlation coefficient = cov(x,y) / (sigma x * sigma y) not var(x) * var as you have got it above.
and regression slope is the beta. (from SML days)
beta = cov(x,y)/var(x)

 

[busprof]

oops - forget the mess at the end of my previous post - sloppy cut and pasting.