Why does a correlation not establish a casual relationship between two variables?

Answer 1

Correlation only establishes the fact that the two variables in question change together - either one increases as the other decreases or they both increase together. It is possible that changes in the first cause changes in the second, or that changes in the second cause changes in the first, or that there is some third variable that is causing changes in both.

For example, consider an infant and measure its height and vocabulary from age 2 to age 8. In normal circumstances these two variables (height and number of words he or she knows) increase together. But that does not mean that either of these factors causes the other. The obvious culprit here is time or age. Another possible, but less important factor may be nutrition. Whatever! The main point is greater height does not increase the child's vocabulary not does an increased vocabulary increase its height.