Why are $x$ and $x^2$ correlated? I'm trying to understand why there tends to be correlation (as measured by the standard Pearson correlation coefficient) between $x$ and $x^2$ (for instance if $x$ is uniformly distributed). 
It's my understanding that the Pearson correlation coefficient only measures linear relationships. $x$ and $x^2$ are not linearly related.
 A: The Pearson correlation measures the amount of linear relationship -- it doesn't ignore variables that have a relationship that's not perfectly linear. If things increase and decrease together, some portion of their relationship is explainable as linear relationship (and some of it isn't).
For example, if $X$ is positive, then both $X$ and $X^2$ will increase or decrease together, and so be somewhat positively correlated. On the other hand if $X$ is negative, then $X^2$ will increase as $X$ decreases (becomes more negative). 
Here's a case where the population mean of $X$ is large compared to its spread, and so $X$ and $X^2$ have a high Pearson correlation:

In this case the population correlation is about 0.99867 and the sample correlation was about 0.99868.
If $X$ is both positive and negative then there are parts where $X^2$ increases as $X$ increases and parts where it decreases. This may result in an overall positive, negative or zero correlation (depending on the extent to which they cancel out).
