Measuring quantization error for clustering - squared or not? 
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*When measuring the quantization error of a clustering, should the distance between samples/centroid be squared or not? I found both variants in the literature.

*Furthermore, is (squared) quantization error not the same as squared sum of errors?

 A: The point of the squared error is that it results out of the underlying assumption, that your data is distributed with a Gaussian random component. (Like noise on your measurements, e.g.)
The sum of squares error comes from the log probability. Say your points are distributed according to $p$, then you want to pick your parameters $\theta$ (for K-Means the centroids, e.g.) to maximize the probability of the data:
$$p(D) = \Pi_i p(x_i|\theta)$$
However, maximization of products is hard, so we maximize the log instead, which makes the product to a sum:
$$argmax_{\theta} \sum_i \log p(x_i|\theta)$$
Now, the density of the Gaussian distribution is someting like $\frac{1}{\sqrt{2\pi\sigma^2}}\,e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$. If you take the log of this, you are stuck with what you find in the exponent and some constant terms. You ignore the variance in most cases. And voilá, that's where the squared distance comes from.
Since the Gaussian punishes outliers very much (because of the square), sometimes it is more robust to assume a distribution that has heavier tails, like student's-t or Laplace.
If you only measure the absolute distance, this comes from a Laplace assumption.
Thus, it is a question that the statistician has to answer - it's more part of the model. Of course, you can use a model selection method (like cross validation) that does this part of the job.
