In Swiss tournaments, is it common to play weaker players when playing White? I was looking at the statistics for my tournament games in SCID, and found something surprising. When I played opponents rated under 1000, 13 out of 15 times I've had White. Conversely, when I played opponents rated 2000 or higher, 10 out of 15 times I've had Black.

I'd accept 10 out of 15 as being coincidence. But 13 out of 15 seems unlikely to happen purely by chance; the odds are less than 1%.

Is there a reason why a person would tend to play stronger players as Black and weaker players as White? Or is it just a coincidence?
Your 1% chance calculation is wrong, because "observations" are not independent from each other within a single tournament. When you play one round as White, you are very likely to play the following one as Black. Also, after a win, you are very likely to face a stronger player in the next round. So your tournament could go something like this.

WHITE, weak opponent. WIN

BLACK, strong opponent. LOSE

WHITE, weak opponent. WIN

... (you get the idea)

This means that if, in 3 consecutive tournaments, you get to play a weak opponent as White (let's say that, by ranking, you are about 100th in a pool of 200 players), you can easily get pairings such as those you are showing (with a 1 in 8 chance, and also a 1 in 8 chance of getting the exact opposite)