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Let $Q=\{q_1,\ldots,q_n\}$ ($n\in\mathbb N$, $n>1$) a collection of elements and $d:\,Q\times Q \longrightarrow \mathbb R^+$ a distance between a pair of elements (as a measure of similarity). Assume that all elements in $Q$ are different according to $d$. I define the dispersion in $Q$ as

$$\sigma^2 =\frac{1}{n-1} \displaystyle\sum_{i=1}^n\left(\mu_c-d(q_c,q_i)\right)^2,$$

where $q_c$ is a centroid in $Q$ and $\mu_c$ is the average of distances between the centroid and the other objects.

Then, I normalize this measure in the interval $[0,1]$ as the coefficient of variation:


My question is if this is well-defined.

I think that it is because $\mu_c$ is strictly positive (distances are positive, all elements are pair-wise different and there must be at least two elements in $Q$). My doubt is that in the Wikipedia they say that it should be computed only for data measured on a ratio scale, and I don't really understand what that means.

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up vote 1 down vote accepted

A "ratio scale" means that the scale you are measuring on has a natural and meaningful zero; i.e., it makes sense to multiply and divide. In the case of measuring distances, this is true.

A counterexample is something like temperature. Zero in Fahrenheit is different from zero in Celsius, and one is not "more correct" than the other. It doesn't make sense to say that 40 degrees Fahrenheit is "twice as warm" as 20 degrees Fahreinheit, even though 40=2*20. So this is not a ratio scale--it's what's known as an "interval scale"

In your case, distances are definitely measured on a ratio scale, so you're good to go.

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+1 Thanks. That was very helpful. – synack Jan 12 '13 at 21:38

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