In this article I have read that correlation is "invariant to differences in the mean and variability" (to be more precise, they talk about 1 - corr, which is however equivalent w.r.t. this property). I wonder why that is. Whereby I know that:
\begin{equation} \mbox{corr}(X,Y)=\frac{\mbox{cov}(X,Y)}{\mbox{var}(X)^{1/2}\cdot \mbox{var}(Y)^{1/2}} \end{equation} and \begin{equation} \mbox{cov}(aX+b,cY+d)=ac \cdot \mbox{cov}(X,Y) \end{equation}
b and d clearly signify changes in the mean values of two vectors (sampled from the respective random variables) - and correlation is visibly invariant toward such changes. However, I would like to understand in how far correlation is also invariant towards changes in variance of X and Y, but a and c seem to be just multiplicative factors and not the variances of these random variables...or am I wrong?