Reason to normalize in euclidean distance measures in hierarchical clustering Apparently, in hierarchical clustering in which the distance measure is Euclidean distance, the data must be first normalized or standardized to prevent the covariate with the highest variance from driving the clustering. Why is this? Isn't this fact desirable?
 A: Anony-Mousse gave an excellent answer.  I would just add that the distance metric that makes sense would depend on the shape of the multivariate distributions.  For multivariate Gaussian, the Mahalanobis distance is the appropriate measure.
A: It depends on your data. And actually it has nothing to do with hierarchical clustering, but with the distance functions themselves.
The problem is when you have mixed attributes.
Say you have data on persons. Weight in grams and shoe size. Shoe sizes differ very little, while the differences in body mass (in grams) are much much larger. You can come up with dozens of examples. You just cannot compare 1 g and 1 shoe size difference.
In fact, in this example you compute something that would have the physical unit of $\sqrt{g\cdot\text{shoe-size}}$!
Usually in these cases, Euclidean distance just does not make sense. But it may still work, in many situations if you normalize your data. Even if it actually doesn't make sense, it is a good heuristic for situations where you do not have "proven correct" distance function, such as Euclidean distance in human-scale physical world.
