Truncating data reduces correlation? Here is an argument I came across: by limiting your sample by some range of one of the variables the (Pearson) correlation coefficient between the two variables is likely to be reduced.
I can't see the logic. I would think it could both increase or reduce, depending on whether the linear relationship is stronger or not for that interval at hand.
Any clues?
 A: There's a number of ways to look at it, but this is a pretty straightforward one:
Imagine for a moment we're looking at a regression problem. The squared correlation between the two variables ($r^2$) is $R^2$, the coefficient of determination, which is $1-\frac{s^2_\epsilon}{\text{Var}(y)}$. When you restrict the range of $x$, you also reduce the range of $y$, so $\text{Var}(y)$ goes down with it, while $s^2_\epsilon$ (the noise about the line) should hardly change, since it still has an expected value of $\sigma^2_\epsilon$. Here's an example of that:

Since the denominator of the fraction decreases while the numerator hardly changes, the fraction gets larger, so $R^2$ gets smaller, so $r^2(x,y)$ and hence $|r|$ will be smaller. So we really should expect that the size of the correlation decreases.
A: Thinking of a 2D plot of one variable plotted against the other, limiting the range for one variable means only looking at a vertical or horizontal "slice".
So my intuition is that the overall shape of the "cloud" of points will be more vertical or more horizontal, instead of "diagonal". A vertical or horizontal-looking cloud of points has zero correlation. So to me there is indeed an intuition that correlation is likely to decrease.
As a toy example, if your data points are (1,1), (1,20), and (20,20), you have 0.5 correlation, but if you limit the range of the first variable to [0,10] you are left with two points (1,1) and (1,20), and correlation =0. If you limit the second variable to [10,30] then you get two points aligned vertically, and again correlation =0. 
