I have encountered this problem and I am struggling with the intuition.
I have a data sample $(X_i,Y_i)_{i=1}^n$ with sample correlation $\rho$. Let $\rho_1$ be the sample correlation for sample $(X_i,Y_i)_{i=1}^{n/2}$ and $\rho_2$ be the sample correlation for sample $(X_i,Y_i)_{i=n/2+1}^{n}$.
In my data, I have $\rho = 0.9$, $\rho_1 = 0.4$ and $\rho_2 = 0.4$. My understanding is that the Pearson correlation indicates the strength of a linear correlation between $X$ and $Y$. So if the correlation is not strong in $(X_i,Y_i)_{i=1}^{n/2}$ and $(X_i,Y_i)_{i=n/2+1}^{n}$, then how can the correlation be strong in the union of the data sets $(X_i,Y_i)_{i=1}^n$?
Looking at the formula, I guess it's possible since the means are different but I can't seem to grasp the intuition.