Ward (1963) provides a commonly used criterion for hierarchical clustering. It's based on the following definition (p. 237):
Given a set of ratings for 10 individuals, $\{2, 6, 5, 6, 2, 2, 2, 0, 0, 0\}$, a common practice is to use the mean value to represent all the scores rather than to consider individual scores. The "loss" in information that results from treating the 10 scores as one group with a mean of 2.5 can be indicated by a "value-reflecting" number, the error sum of squares (ESS).
The error sum of squares is given by the functional relation,
$$\text{ESS} = \sum_{i=1}^n x_i^2 - \frac{1}{n}\left( \sum_{i=1}^n x_i \right)^2$$
where $x_i$ is the score of the $i$th individual. The ESS for the example is […] 50.5.
If somebody asked me how to quantify the loss of information incurred by representing a vector with its mean, I'd say the SD or variance. Or if you wanted the sum of squares rather than the mean of squares, you'd multiply the variance by the sample size, and get $\sum_{i=1}^n (x_i - \bar{x})^2$. This is the sum of squared distances from the mean. So why would one use Ward's ESS instead of one of these quantities?
Ward, J. H. (1963). Hierarchical grouping to optimize an objective function. Journal of the American Statistical Association, 58(301), 236–244. doi:10.2307/2282967. Retrieved from https://web.archive.org/web/20050312103440/http://iv.slis.indiana.edu/sw/data/ward.pdf