Question about Pearsons Correlation Coefficient

Topia

So, today I started my spring semester class. It's a fairly low level Psych course, and spent the whole day (3 hours!) discussing the scientific method and data analysis. Fun, right?

Anyway, we got to correlations, and the professor stated that "this graph, we can tell is more correlated because the line-of-best-fit is steeper than the previous". The graph itself WAS indeed more correlated, and more steep (a visual representation of the graph), however this struck me as funny, and wrong.

I went to her after class to talk to her and said "I don't think what you said is right, it would be very easy for me to manipulate these graphs' axis' so that the steepness of the less-correlated one is higher, and visa-versa." (keeping in mind, while drawing examples, that I left the Y-axis unchanged as they showed the same variable) She argued, and eventually I gave in and just left, and pondered to myself on the walk home.

I need someone to tell me if I'm right or wrong.

I started thinking that, maybe, she meant to say slope. A graph with more slope would be more correlated (a higher coefficient). Then I though of this: http://i.imgur.com/9J9Y2.png

That picture shows two graphs. Let us, for simplicities sake, imagine that they have the exact same axis, labelled and scaled. The one on the left would have a higher slope (steeper), than the one on the right, but does not the one on the right show a higher correlation?

Thanks in advance...

edit: if I'm unclear about something or you want me to go into more detail I will, I tried to make this short and quick.

Septus

I do believe that the closer the cluster of data around a best-fit line, the stronger the correlation. I'm trying to think of the implications of steepness, and I can't think of any, other than to show whether the correlation is positive or negative.

Erios

Assuming your axes are values and not, well correlation to something else, moving in lock step is a sign of correlation, the slope of the graph MAY be an indicator of the tendencies of that relationship. Enter "correlation" into wikipedia and you'll see what I mean; several 1 and -1 correlation graphs are given with different slopes.

Daenris

The prof is wrong, as is easily demonstrated in an example like the graphs you provide. The slope of the fit line on the first is clearly greater, however the second graph is clearly more highly correlated. Slope just tells you the direction of the observed correlation (via the sign of the slope).

Topia

Erios wrote: »

Assuming your axes are values and not, well correlation to something else, moving in lock step is a sign of correlation, the slope of the graph MAY be an indicator of the tendencies of that relationship. Enter "correlation" into wikipedia and you'll see what I mean; several 1 and -1 correlation graphs are given with different slopes.

From wiki:

Several sets of (x, y) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.

So I was right. Slope doesn't matter. I kinda wanna bring this up to her, but it's probably not worth my time, is it?

cncaudata

Your professor is wrong, but you are also a little confused.

http://en.wikipedia.org/wiki/Correlation_and_dependence

Slope has nothing to do with the correlation coefficient except that if the slope is zero, the correlation coefficient is undefined, and if the slope is positive or negative, the coefficient matches.

Topia

cncaudata wrote: »

Slope has nothing to do with the correlation coefficient except that if the slope is zero, the correlation coefficient is undefined, and if the slope is positive or negative, the coefficient matches.

No, I know those things, I guess I should have used the word "magnitude" in there somewhere. I just wasn't very specific. I have a hard time outright dismissing my professors (good job education system, right?) even when I am 99% sure I am right, so my mind was clusterfucked.