how do you interpret a correlation?

Answer 1

A correlation coefficient demonstrates the degree to which a change in one variable (the *dependent* variable) is associated with a change in another variable (the *independent*) variable.

The change can be 'increase in one is associates with an increase in the other' or 'increase in one is associated with a decrease in the other' - the latter case is known as an 'inverse correlation.' Therefore correlation coefficients run from -1.00 to + 1.00, with zero indicating no correlation at all.

The most crucial thing to bear in mind is that correlation does *not* necessarily imply cause. One may cause the other; an external cause may influence them both; or you could have stumbled on an accidental association which wouldn't stand up to wider investigation.

And if you're seriously interested in statistics and understand the terms that I'm about to use, the *square* of the correlation coefficient tells you how much of the change in the dependent variable is associated with the change in the independent variable. So if you have a correlation coefficient of 0.6, that means that 36% (i.e. 6 squared) of the variation in the dependent variable is accounted for by the change in the independent variable.

Also, the significance of the correlation depends upon the sample size. Beyond that, you'll have to look it up but I hope that helps. Most important is 'correlation does not mean cause.'

Answer 2

Correlation is defined as:
R = E[x*y] where x and y are two variables (either stochastic or deterministic). Correlation shows the similarity of the two variables. For instance if y behaves the same as x, i.e. y = k*x (with k constant) the R = k*E[x^2] which is the maximum for any variable x. If on the other hand x is independent of y (statistically) then E[x*y] = E[x]*E[y]. For E[y] = 0 => R = 0 meaning that x and y are not correlating at all. You can visualize it as any value of x, xi, multiplied by multitude of values of y (meaning y totally independent of x) giving sum(xi*y) = 0.
For deterministic variables the example would be

R(t,s) = E[sin(t)*sin(s)]
= E[1/2*(cos(t-s) - cos(t+s))]
1/2 for t-s = pi*k with k even
= 1/2*E[cos(t-s)] {-1/2 for t-s = pi*k with k odd
0 otherwise.

The interpretation is that sin correlates (is similar) to another sin if they are 0 or 2pi phase different (both are the same function), dis-correlate if they are 180 degrees out of phase, and do not correlate for any other phase difference.