# Given a set of addresses, pick a geographically even subset for sampling

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:

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?

In order to be able to calculate estimates based on the typical design-based survey sampling paradigm (that is, using estimators based on the Horvitz-Thompson estimator), it's important to have a known probability of selection of each unit. For the estimate of error in the survey, it's important to have known joint probabilities of selection as well.

However, often if we want to spread out our sample relative to some measure, we ease up on the "known joint probabilities of selection" requirement and we perform Systematic Sampling. It's usually assumed fair to use the same variance formulas as in Simple Random Sampling, even though they aren't strictly correct.