Well no answers here yet, but I did find some information on the web. The best statistics I found were from here:
http://computerchess.org.uk/ccrl/404FRC/opening_report_by_eco.html
It has statistics for all 960 opening positions. "Regular" Chess (RNBQKBNR) has a white win rate of 55%. The biggest advantage for white was 63%. Quite a few of the positions had much smaller advantages for white.
The down side is that the statistics here are using chess engines, not human play, which could be significantly different, but it's still a good database of results.
I think it shows that Chess960 is not a lot worse than regular Chess when it comes to white advantage.
It occurred to me that Chess.com runs an awful lot of games of Chess960, and the results of all of those games are available.
One of the great criticisms of Chess960 is that white has a significant advantage in many opening positions. Does chess.com keep records that would confirm or refute that? Are there statistics that show a huge advantage in some positions? Has anyone pored through the statistics to determine trends, like whether or not the first turn advantage is more pronounced at low ratings than at higher ratings?
I have defended Chess960 as being at least as good as, and in some ways better than, Chess. However, if it turned out that the deck really was stacked against the black player, I would have to admit that the game was pretty much broken, and needed some emergency repairs to fix. The statistics from this site ought to be able to answer that question.