Ally -
Unless I am misreading you, why you would think all holes should have the same scoring distribution? I think most people believe the design of a hole affects the distribution of scoring on it one way or another.
David M. and I have argued before about what Sullie calls the "barbell" theory. The idea turns on what I think are a couple of uncontroversial notions. Golfers tend to play hard, penal holes more conservatively. They play down the middle of the fw, try to hit the green and are happy to get their par and move on.
Players tend to play strategic holes more aggressively. Low scores are possible if you are willing to take certain risks. Some players will manage those risks and be rewarded; others won't manage them and will pay a price.
Which is to say that we should expect differently designed holes to yield different scoring patterns. I'm not sure why that is a problematical idea.
The barbell theory says that if a hole is strategic, we should expect to see a wider spread of scores relative to scoring on other, less strategic holes on the course. Again, because strategic holes induce players to take more risks and play more aggressively. With the result that some will pull off those risks, others won't.
The 10th at Riviera is a pretty good example of that. Except for the 1st (which plays in effect as a par 4 for the pros) the 10th has the fewest number of par scores of any hole on the course. That ratio of non-pars to pars is .87. Which means there are almost as many non-par scores and par scores. That's the barbell. Of the non-par scores at the 10th there is a roughly 2:1 ratio of under par to over par scores.
(Ideally, that ratio of under to over par scores should be 1:1. That is the perfect barbell. But the scores at Riviera this week are for pros and they are very, very good golfers. I'd guess for single digit amateurs, the over and under scores on the 10th would be closer to 1:1.)
The only other hole at Riv. that comes close to having as few par scores is the 12th with an .80 ratio or pars to non-pars. The difference, however, is that almost all of the non-par scores at the 12th skew toward bogeys and doubles. The same is true of the 2nd, with a .75 ratio, but again the non par scores skew towards above par scores. (For single digit ams, I'd guess the 12th and 2nd would have fewer pars and far more above par scores, increasing that skew.)
No other hole at Riv can match the 10th's (a) number of non-par scores (with the exception of the 1st as noted) and (b) the mix of under and over par scores (roughly 2:1). Other par 4's that have relatively few par scores have non-par scores weighted heavily in one direction. (There are few par scores on the 1st because virtually everyone birdies it, for example).
You can take or leave the above as you wish. I think it is a useful way of thinking about how strategic a hole is. That is not a revelation about the 10th at Riviera. We already know that. I found, however, that the barbell theory can be helpful in confirming intuitions. For example, the 12th at TOC comes out quite well after applying the theory to scoring at recent Opens. I confirmed to me how really good the hole is.
In any event, a geeky, fun exercise.
Bob