Can Someone Please Explain Correlation Coefficient to me?

I know this is not going to help but I can't help you with the first one but the standard deviation is a real easy one to figure out and I dig ditches for a living.

yowza wrote:

I know this is not going to help but I can't help you with the first one but the standard deviation is a real easy one to figure out and I dig ditches for a living.

Good point. The world needs ditch-diggers too.

A correlation coefficient (r) is an index of the relatedness of two variables; e.g., how related are two populations or two sets of variables. r can vary between -1 and 1, but an r of 0 means no correlation.

The best way to understand the concept is to think of a simple x, y scattergram. A straight line of dots going up (from left to right, would indicate a correlation of 1, but if the line of dots declined as you go along the x axis, then you would have a correlation coefficient of -1. An r of 0 would be just a shotgun blast of data points on you x,y graph. If there is some relatedness then you'll be able to see a pattern. So if you see a linear trend in data points, you might have an r of 0.7--which indicates that as x goes up a unit of 1, y might go up .7, and but there is some variability along the way (the higher the r value, the less variablility as you approach true linearity).

Your r's of 0.34 and 0.36 indicate somewhat less relatedness (a weak correlation), and that there are likely other factors explaining the variablility between the populations of interest.

[disclaimer, I love stats, but never taught it, so maybe a math teacher could do better]

The correlation coefficient shows how precise the relationship is between two variables. If you plot a two-dimensional graph of coordinate pairs, which each point depicting the value of one variable for a given value of the other variable, you will have a bunch of points in the plane. If all the points lie on a straight line, the relationship between the variable is a precise linear one. A linear function rule could describe it perfectly. The "r" value for this relationship would be 1.00. If the points all lie nearly in a line, with a few barely off to one side here and there, there is still a strong correlation (in the linear sense) between the variables. Their relationship can almost, but not quite, be described by a linear rule. This might represent an "r" of 0.95. If the points are all over the place and there doesn't seem to be a reliable way of predicting the value of one variable based on any value of the other variable, the "r" is low. Neither 0.34 or 0.36 shows a particularly strong relationship between the variables in question. However, the set with 0.36 does indicate a tiny bit more of a dependent relationship.

Standard deviation is a measure of dispersal. In any group of values, which might represent heights, weights, mile times, etc., the mean, median and mode of the data are all ways to assess the central tendency of the group. That is, what are the most likely values you'll see. The standard devation is a univariable statistic which basically measures how spread out the data are from the mean. If you're measuring the wingspans of adult turkey vultures, there will be some variation in the values, but it's not likely that variation will be more than about a foot. Bird species tend to grow to within a few inches of certain size and only rarely are there exceptions (those are known in statistics as "outliers"). If the values you're gathering are the mile times of high seniors, it's a different story. You'll see times that are 3 minutes or more to either side of the mean. The standard deviation of that data set will a far larger value (especially relative to the value of the mean) than the s.d. of the wingspans of turkey vultures, which is probably only a few inches (with a mean of nearly 6 feet). So there is greater dispersal in the values of mile times. The standard deviation is just a "standard" way of measuring this "deviation."

So to speak.

thanks for the responses so far. i think i got a hold of the standard deviation. in fact, i am using cross country times and years of participation for this small classroom study, so i'm beginning to figure that out.

as far as correlation goes, that explanation was helpful too. the word picture of "shotgun" blast makes sense-- in this case, the XC times are literally scattered all over the place, with a small percentage having a correlation.

thanks for the help, this is running related!

grad student wrote:

the XC times are literally scattered all over the place

That's one reason such a graph is sometimes called a "scatter diagram." Ones with low correlation look like a paper target that was hit by a random blast of shot sprayed from a pellet gun.

To be more precise, the correlation coefficient measures the strength of a LINEAR (straight line) relationship. If two variables are related in a non-linear fashion (such as y=e^x), their correlation coefficient will be low.

A common rule of thumb is that .6 to .8 represents a strong correlation, and .8 or higher is very strong. In fact, if you get something like .99, that's pretty suspicious and it will make people think you fudged the data to get good results. Which isn't to say that you should fudge data in order to avoid that! If you happened to pick things that should be highly related, it might happen. Keep in mind that rule is talking about the absolute value of the correlation coefficient. So if you're given two coefficients, -.7 and +.7, and asked which shows a stronger relationship, the answer is neither.

Mr. Pearson wrote:

To be more precise, the correlation coefficient measures the strength of a LINEAR (straight line) relationship. If two variables are related in a non-linear fashion (such as y=e^x), their correlation coefficient will be low.

Yup. If the correlation between y and x is low, though, what you could do is see what the correlation coefficient is between y and ln(x), and between ln(y) and ln(x). (ln(x) is the natural logarithm of x.)

Then you could compare the three values, and see which is bigger. The bigger one is what you should use if you have to fit a model to the data.

a few clarifications:

correlation coefficient is r^2, not r. there's a big difference. r works only for linear situations, r^2 works for all relationships for which a regression or model can be done.

2nd r is NOT slope. the example of 0.7 has no bearing on the slope except that it would be positive as 0.7 is positive.

regression line slope is r*(sy/sx)

3rd, r^2 is always positive. it is a square and it's error reduced, that can't be negative.

finally, r^2 is a measure of this:

how much (what percent) of your squared error is eliminated by the regression equation as compared to using the average y-value as an estimate. this is not an easy concept to read and grasp, but there are some great diagrams out there such as in The Practice of Statistics (Yates Moore and Starnes).

ack, complete brain cramp. i shouldn't post when drinking. i was thinking coefficient of determination but writing about correlation coefficient. sorry.

the only comment of mine that stands is the one about slope. r does not indicate slope.

R^2 (and, by extension, R) is a function of both slope AND scatter around the regression line. If you have a scatterplot where the points are along a straight, perfectly horizontal line, it doesn't matter how closely the points fit that line, the correlation coefficient will be zero.

Your example of correlation coefficients of .34 and .36 does not really seem to show you anything except that the relationships appear to be weak. It might help for you to also look at the Root Mean Square Error (RMSE) and the slope of the line. But I'm not gonna get into RMSE because that can be confusing. If you post your entire computer output, perhaps we can help a little more.

you're a GRAD student and you don't know what correlation is ??

what school do you go to? undergrad? grad?

The correlation coefficient is the slope of the line generated by the changes in values between X and Y. These values don't necessarily have to fall ON that line, but within a finite area of limits based on the parameters of the experiment with the mean, or center of that dispersion being the linear representation of that grouping.

Appreciate your response BB . You 've lost on me on the HS senior mile being three minutes on either side . Unless you are including girls . Can you expound ? Thanks again .