problem with handicap system

[Big Lex]

Ok, I read several of Knuth's papers on his website.

Thinking about it in terms of a normal distribution leads to the same conclusion that Knuth reaches in the paper attached to Will's message, but you guys are correct that the slope system is not based on specifically on assumptions of a normal distribution, SD, etc. You guys are correct on that score. But my posts are still valuable (to me anyway) in helping understand why the slope adjustment works correctly, as it is now.

The slope rating is derived from the slope of a regression line. The regression line is a line of best fit for a plot of scores on a course, with scores on the y-axis and course handicap on the x-axis. The higher the slope of this line, the greater the difference between the scores of players across all handicaps.

That's clear, right?

Regression lines are mathematical models of data which attempt to "fit" all of the data as best as possible. By definition, they extend infinitely in both directions...the course rating and bogey rating are simply two points on the line. If the regression equation/model is correct, points at any point on the line will be correct, not just those between the scratch and bogey points.

My understanding is that in developing the system, they modeled the idea mathematically, but also did an empiric study, actually plotting scores v. course hcp for several courses, and calculated regression lines. They did this for different courses, and then tested to see how well the equation predicted how people from different courses would do. The empiric study found that the slope equation worked. It allowed them to take someone's scores at course x and then predict a score on course y.

The second part of the project was figuring out how to rate courses to _predict_ the rating and slope, so that it wouldn't have to be empirically calculated at any given course.

This is the system we have now.

In any event, in reference to the original question, the normal distribution, SD, etc. is not what is specifically used in the HCP system, but the predictions generated by my "invented" explanation are precisely what is predicted by the slope system. The logic is the same.

On a course of high slope, the scores are more spread out. Just like there is a bigger difference on this course between scratch and bogey compared to a low slope course, there is also a bigger difference between pro and scratch. Since they are supposed to be farther apart, a difference of, say, 4 between a pro and a scratch is LESS impressive on a high slope course. If the course is really really hard and the scratch player is way over the pro (just like the bogey is way way over scratch), then the relative merit of a score below the scratch score is less exceptional than it is on an easier course.

I've wasted too much bandwidth maybe and I'm sorry, iacas.

One final point--on the elements of rating for scratch v. bogey--yes, there are different multipliers for the different categories, but they are the same elements. That's what I'm saying.

[DaveLeeNC]

snip

Just ran across this thread. I had a similar view (system didn't work when scores/indexes went below course ratings) until I came to an interesting realization. The slope actually measures how fast a course gets easier (bogey golf as the starting point) as the player's ability improves.

If you think of it that way, it all makes sense. Of course you can argue that we are outside the bounds of "sampling" (scratch and bogey), but that would apply to both "+ indexes" and to "indexes over 18". dave