## The distance between the centroid of $P$ points and the centroid of a subset of the points

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Imagine I have an $(p_1, ..., p_N) \in P$ points, on a two-dimensional plane, patterned in a rectangular or hexagonal lattice arrangement in a circle of radius $R_c$, with a spacing between the points of $r_s$.

Let $C_P$ be the centroid of the $P$ points. If I randomly select some subset of $k$ points from $P$, and I compute the centroid of these $k$ points, $C_k$, what is the probability that the distance between $C_p$ and $C_k$ is $\leq D$?

Update: I have specified that the $P$ points in the circle are in a rectangular or hexagonal lattice arrangement (whichever is easiest to analyze).

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The optimal circle packings in a disk are complicated and not known in general, and I doubt you really care about that level of complexity. The exact probability for small $D$ will depend on the exact optimal configuration. If you instead let the $P$ points be IID uniformly random, then you can say something. You can also compare the centroid of $k$ points with the center of the disk. One tool for this is the Central Limit Theorem, although you may want to use an effective version such as the Berry-Esseen Theorem or a large deviations result to get a bound for a particular $k.$ – Douglas Zare Aug 27 at 13:28
@Douglas Zare I have updated the question to specify that the $P$ points should be patterned in a rectangular or hexagonal lattice arrangement (whichever is easiest to analyze).  Also, I do care about circle packing, it's really interesting, but it was inappropriate for me to confound this question with a very difficult geometry (et. al.?) problem. – CKura Aug 27 at 14:44

## 1 Answer

I might not be understanding the question, but the centroid is just the mean of the sample, so for mildly large samples from a mildly large set it will be (bivariate) normally distributed, and all the statistics are easy to compute.

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