How can I find the center of a cluster of data points?

Let's say I plotted the position of a helicopter every day for the past year and came up with the following map:

Any human looking at this would be able to tell me that this helicopter is based out of Chicago.

How can I find the same result in code?

I'm looking for something like this:

$geoCodeArray = array([GET=http://pastebin.com/grVsbgL9]);
function findHome($geoCodeArray) {
    // magic
    return $geoCode;
}

Ultimately generating something like this:

UPDATE: Sample Dataset

Here's a map with a sample dataset: http://batchgeo.com/map/c3676fe29985f00e1605cd4f86920179

Here's a pastebin of 150 geocodes: http://pastebin.com/grVsbgL9

The above contains 150 geocodes. The first 50 are in a few clusters close to Chicago. The remaining are scattered throughout the country, including some small clusters in New York, Los Angeles, and San Francisco.

I have about a million (seriously) datasets like this that I'll need to iterate through and identify the most likely "home". Your help is greatly appreciated.

UPDATE 2: Airplane switched to Helicopter

The airplane concept was drawing too much attention toward physical airports. The coordinates can be anywhere in the world, not just airports. Let's assume it's a super helicopter not bound by physics, fuel, or anything else. It can land where it wants. ;)