I remember the textbook definition of correlation coefficients from college: it's a measure of the covariance between two variables, telling you essentially how closely those variables move hand-in-hand.
What I've always wondered is this: is there an easy way of describing a correlation coefficient with real-life examples?
Let's say you have a dataset and see two binary variables with a correlation coefficient of 0.6. Can you say that changing ten of one of the variables will change six of the other variables?
My first reaction is to say no, you can't infer causation from correlation. But what if you really knew the relationship was causal? Then is this what a correlation coefficient of 0.6 means? Change ten of one, you change six of the other?
I get the feeling this is a misinterpretation of what the correlation coefficient is, but I'm not sure. If it's wrong, can anyone give a layperson's example of what the correlation coefficient means? To the extent that it's possible, I'm interested in what it means for what to expect when you start changing the data set after measuring the correlation.