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If I have a set of groups of individuals (e.g. people), and I want to calculate the typical group size (as observed by individuals), how do I do this?

Wikipedia refers to this as "mean crowding" or "Typical Group Size" but doesn't give a formula:

I believe the "typical group size" for a set of group sizes $\{a_1, a_2,..., a_n\}$ is simply:

$$\sqrt{\frac{1}{n}\sum_{i=1}^na_i^2 }$$

i.e. the root mean square or quadratic mean.

Is that correct? Is there a better way to characterise typical group size?

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up vote 2 down vote accepted

The precise definition (according to the Wikipedia link in your question) seems to be that the typical group size is the average size of the group as observed by the individuals of your population. Thus, if you have a population of $N$ individuals, split into $K$ groups with size $n_k$ (thus, $\sum_{k=1}^K n_k = N$), the typical group size would be $$ T = \frac{1}{N} \sum_{k=1}^K \left(\sum_{i=1}^{n_K} n_k\right) = \frac{1}{N}\sum_{k=1}^K n_k^2 $$ since the $n_k$ individuals in group $k$ all observe a group size of $n_k$.

It's entirely possible that it's for some reason more convenient to define the typical group size as $\sqrt{T}$ instead of plain $T$, but it then ceases to be the average group size as observed by the individuals.

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Thanks! My answer actually wasn't $\sqrt{T}$ because it had 1/K, not 1/N. – spookylukey May 17 '13 at 8:41
@spookylukey Hm, true. Then I don't see an obvious connection between the two formulas. Where did your formula come from? – fgp May 17 '13 at 9:13
my own head! more of a guess really, on the basis that I knew it must be related to the square of the items, but I didn't think it through. – spookylukey May 19 '13 at 5:52

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