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I have a closest facility network analyses from the centroid of county to the closest medical facility. What's the best method to get the weighted averages of these drive times, using population as the weight. I am snapping all facilities and centroids to the road network, so now I am getting ridiculous averages like 6 hours. It is correct, but not feasible.

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I know little enough about statistics, but I'll take a stab at this because it sounds familiar. 'Best' is another story. I'm also a little unclear on exactly what you're trying to achieve.

Broad overview, so you have run your closest facility and generated a table with drive times. Next you would need to classify your populations (which happen to also be your incidents) to generate appropriate classes for a weighting scheme and determine what those weights would be. Once you have your classes, you would average the drive times for each class and then apply the weight, finally averaging the weighted averages for your single average drive time from any given centroid to its closest medical facility. Or perhaps the weight would better be applied to the drive times first based on class, then all averaged at once? Class definition is going to play a role too. Again, I'm not overly knowledgeable about statistical methods. But if you're using county centroids, that seems pretty generalized to start with.

More information about your goal might be helpful. This sounds like it might be also/better approached as a service area problem. I can also elaborate on any of the above if you like.

I assume this also relates to your other question: https://gis.stackexchange.com/questions/87303/is-there-a-method-to-add-attribute-weights-to-a-network-analysis

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