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There are a certain number of houses spread randomly across a city. We have addresses and GPS coordinates for each house.

How can we select a geographically representative, fixed-size sample out of the houses in the selection?

To illustrate with an example, say we have about 100 houses on a map like the one below:

enter image description here

Each house has been identified as a likely supporter in a political campaign, and ideally we visit all of the houses and solicit support. However, it isn't possible to visit all of the houses; we can visit 25 of the houses. By spreading the 25 visits as evenly across the map as possible, we can leverage the fact that neighbors talk to each other.

In other words, how do we select a subset of the GPS coordinates so the entire set of coordinates is as well represented as possible?

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I don't think there is a 100% crisp mathematically one-and-only-one correct way to do this. Here are 2 things that occurred to me:

(1) divide your city into regions, with an equal number of houses per region. Randomly sample regions by weighting them by area, then pick houses uniformly within a region.

(2) pick houses uniformly, but reject a house with probability proportional to the reciprocal of the density of nearby houses. (As estimated, say, by the distance to the nearest other house.)

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You shouldn't have to do anything special to select your random sample in this case. You already have your "frame". Just assign random values to the list with rand(), sort them, and take the top 25. Svetlana Cheusheva talks you through it here: https://www.ablebits.com/office-addins-blog/2018/01/31/excel-random-selection-random-sample/

But 25 samples won't be enough to give you a very representative sample. There are formulas, like Cochran's, that address the quantity question. I talk about it in this post, where I have a similar problem, but no "frame" to rely on: How to apply Cochran's formula to geographic sampling

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