assume you are really asking the equivalent of the following question: what happens if you compute a correlation between two variables, at least one of which is "smoothed", for example, by some sort of "averaging". The answer, of course, is that you get a "large" correlation which doesn't mean anything.
The linear correlation coefficient is location and scale invariant. However, perform the following experiment. Take two sets of (independent) numbers and compute the correlation between their cumulative sums (proportional to the average). Examine the original and the (cumulative) scatterplots. You will find that a "correlation" has been induced by the "averaging". One way to explain this is to note that the smoothing is equivalent to a third variable (call it time) which is related to the original variables.
In any case, you should be careful about computing these things, or at least be careful about what you can infer.
