Correlation of subsets vs correlation of full data set Suppose I have a dataset consisting of $3N$ elements ($X,Y$ pairs) and I split it evenly into three subsets. I know the Pearson correlations between $(X,Y)$ in each of these subsets, and suppose they are all equal to some value $k$, i.e. $r_1 = r_2 = r_3 = k > 0$. Is it possible for the correlation $r$ between $(X,Y)$ in the entire dataset to be, say $2k$ (or any number $> k$). 
I thought about this question for a while, I can easily see cases where the correlation $r$ will decrease or stay the same, however, I could not find a justification or a counter-example for it to become larger.
On the other hand I was trying to approach this problem through the linear regression point of view, and I can construct an example where three lines   corresponding to each of the subsets have smaller slope than the line fitted to the entire set, I understand that slope and correlation are connected, however, I could not get the desired answer for the correlation of a full dataset from this.
 A: Regardless of what $k$ might be, $r$ could have any value in the interval $(-1,1)$.  Statistically, this asserts that the correlations of disjoint subgroups of a population do not provide any information about the correlation of the population.
This is easy to see geometrically.  Take three clusters of points $(x_i^{(j)}, y_i^{(j)})$, $j=-1,0,1$, having any correlations whatsoever (and any counts, for that matter).  Visualize translate them by the vector $j(\lambda,\lambda)$ (first case) or $j(\lambda,-\lambda)$ (second case). Translation does not change the correlations within the clusters. As $\lambda$ grows in size, this moves the first and third clusters away from the third along a line of slope $1$ in the first case or slope $-1$ in the second case.  In the limit the clusters grow small compared to the separation $|\lambda|$, so the correlation must approach $1$ (in the first case) or $-1$ (in the second case).
Any intermediate correlation can be achieved with a modification of this approach: translate cluster $-1$ to any location and translate cluster $1$ to any other location.  Moving them around continuously changes the overall correlation $r$ continuously.  The previous construction shows $r$ can be arbitrarily close to $-1$ or $+1$. The Intermediate Value Theorem implies all intermediate values of $r$ will be attained as you continuously move the configuration from the first case to the second case.
Here, to illustrate, is a picture where each cluster consists of three uncorrelated points ($k=0$).  They have been positioned along the line $y=-x$ so they are separated by about five times the cluster diameters, making the overall correlation nearly $-1$:

The point clusters are distinguished by color.  The title displays $k$ and $r$, in that order.
