Why cant correlational studies show causality between two variables?

Answer 1

Correlation can only show that one variable increases linearly as another increases or decreases. It cannot show non-linear relationships. There can, therefore, be a perfect non-linear relationship and the correlation coefficient can be zero. For example y = x2 in the range (-a, a) for any positive number a,

Second, correlation cannot determine whether A causes B or B causes A. There is probably a good correlation between my age over the last 10 years and the number of white hairs on my head. However, I do not think that white hairs caused me to GROW older (I may look older, but that is another matter entirely).

Furthermore, when there are two correlated variable, there may not be any causal relationship between the two variables but there may be a third variable that causes both. There is a fairly good correlation between my age and the number of cars in the UK. My growing old did not increase the number of cars and the number of cars did not make me grow old. So there is no causal relation between them. Instead, both are correlated to time.