Course Ratings and Slope, Difficulty, and Hole Stroke Index

[iacas]

I've been meaning to write this for awhile, and since the World Handicap System (WHS) is coming to most of the world this year, now's as good a time as any.

This article will assume that you're semi-familiar with the the concepts of course rating and slope, and really seeks to expel some basic myths and misconceptions.

Course Ratings Are the Primary Determinant of "Difficulty"

As you should know, when a golf course is rated (for "difficulty"), many, many, many numbers are generated for each hole. Most of these numbers are pretty objective: the width of the fairway, the length of the hole, elevation changes, the diameter of the green, the depth and size of green side bunkers, etc. A few are subjective, like "how difficult is it to escape if you hit your ball into those trees"?

Two numbers come out of this calculation: a scratch rating and a bogey rating.

The scratch rating is defined as the score a scratch golfer should shoot on rounds where he plays to his handicap index (of 0.0). The bogey rating is the same for a "bogey golfer," (who, oddly, isn't an 18.0 index, but closer to a 20.0, as they're about a 20 course handicap on a 113-slope course).

So, we have two numbers: the score for a scratch golfer and the score for a bogey golfer.

Some basic linear algebra and geometry are used as such:

• Two points define a line.
• We plot those two points on a graph, and draw a line connecting them.
• This line has a "slope" that tells us the "slope rating" of that set of tees.

Remember y = mx + b?

In this case, y is the course handicap, m is the slope (slope rating/113), x is the handicap index, and b is the course rating.

And that's just what we see in a graph:

This is a set of tees with a rating of 72.0 (note that I consider all tees to be par 72 for the simplicity's sake in this article) and a slope of 113.

y = mx + b
CH = 113/113 * (HI) + 0
CH = 1 * HI
CH = HI

This should make sense: on a 72.0 (par 72) course with a slope of 113, we have a basic line with a slope of 1. These golfers should shoot, on average for the eight rounds that count toward their handicap index, these scores:

HI	Score	To Par
0.0	72	0
3.0	75	+3
15.0	87	+15
+2.0	70	-2

We could keep the same slope and make the course rating 69.0, and the course would instantly be three shots easier for every golfer. And this leads into what seems to be the biggest misconception. Too many people look at one number - the slope - and use that to determine what the "difficulty" of the course is.

But that fails, because lines are defined by more than their slope: the y-intercept matters. Imagine a course with a rating of "100.0" and a slope of "102." Nobody in their right minds would say that course is "easier" than a 72.0/144-rated course.

Here's a chart of four fictitious golf courses:

The graph of the course handicaps (note that rounding creates some "bumps" in the lines when the slope is not the whole number 1 (113/113):

Showing the trend lines much more heavily:

 

What these graphs show you is that course "difficulty" is a function of both the "m" and the "b" - the course rating AND the slope.

Look at the 74.0/118 course (grey) and the 72.0/136 course (green). Despite a difference of 18 in their slope, for the majority of these golfers, the 74.0/118 course plays "more difficult" due to the higher starting point of 74.0. It's not until you get out to about a 14.0 index that you start to see the expected scores for the 72.0/136 course take over due to the slope.

Look at all of the lines, in fact: the yellow line (69.0/140) remains lower for most of the graph than even the two 72.0 courses (blue 113, green 136), and well below the grey course (74.0/118). But this is because the course rating varies by 3 and 5 (69 to 72 and 74), while the slope can only make up fractions of a stroke (140/113 =~ 1.24 course handicap strokes per 1.0 strokes handicap index).

So, the course rating is the primary determinant of a course's difficulty.

The slope tends to matter in only two situations:

• The difference in handicap indexes is LARGE.
• The course ratings are quite close together.

The first matters because the slope has more time to keep adding "tenths of a shot" to the course handicap. The second matters because it's easier to overcome a deficit of 0.2 or 0.3 than a deficit of 2.4 or 3.1.

But What about + Handicaps?

Next look at the 72.0/113 course in blue. This course crosses the 72.0/136 (green) course at 0.0, and for everyone with a handicap on the opposite side from + handicaps, the green course is "more difficult" than the blue course.

But on the left side of scratch, the green course is "easier" than the blue course? Why is that?

Because the slope is the relative difference in difficulty between a good player and a worse player.

The "0.0" seems to throw people off, but the fact that we have + handicaps tells you that's not the absolute lowest anyone can go. Think of it this way: if a 13.0 gets 6 shots from an 8.0 due to the slope, then a +5.0 should give up six shots to a 0.0 too (there are occasionally rounding things that change this a little, on either side). The slope is the same, and so a change in "x" (the handicap index) should result in the same difference in course handicap (the y axis) because the slope of the line is constant.

Another way to think of it: add the course's par to the course handicap. If we called scratch golfers "72.0 golfers" on a 72.0-rated course, then you can see how a "67.0-rated golfer" should give up 5 shots to a "72.0 golfer" on a 72.0/113 course, and six shots to a 72.0/140 course.

67.0 * 140/113 = 83
72.0 * 140/113 = 89

Right?

The same math, essentially:

-5.0 * 140/113 = -6
0.0 * 140/113 = 0

Make sense?

Good. The final thing…

Stroke Indexes and "Hole Difficulty"

I'll try to keep this one short: the "stroke index" (handicap index) of the hole is NOT the "difficulty" of the hole.

It's a measure of where the higher handicapper is most likely to need a stroke against a lower handicapper.

Why are (or were, see the note below) par fives often the lower stroke index holes? Because:

• Better players tend to birdie or par them.
• Worse players tend to bogey them.

The increased distance gives a better player more chances to recover and more of an opportunity to show off their length. Consider the example of a one-foot putt versus a 50-foot putt: both the high and low handicapper are just going to tap in all the time on the one-footer, but the low handicap player is going to win a match of 50-foot putts against the worse player much more often.

To good players, par threes — often the high stroke index holes — are the "more difficult" holes relative to par. That's why they're traditionally the higher stroke index holes - the high handicapper isn't as likely to need a stroke against the better player.

Note: This stuff used to be calculated by courses literally turning in about 400 scorecards, which would all be entered hole-by-hole, and computed to determine which holes had the largest gaps between "better players" and "worse players." The holes would be ranked, the data massaged so that the first six holes didn't give out the strokes 1 to 5, and away we'd go.

The USGA and R&A have learned, however, that the actual location of strokes doesn't really matter all that much, so long as they're not clustered (like the example of stroke index holes 1-5 in the first six holes of the course). So, to make things simpler, they've come up with the idea of "triads" and are assigning stroke indexes via that method. You can read more about that method here: https://www.usga.org/content/usga/home-page/handicapping/roh/Content/rules/Appendix E Stroke Index Allocation.htm. In essence, it maintains the idea of the relative difficulty* while making things much simpler and not requiring all the "massaging" that was done before, and adequately spreads out the low- and high-stroke-index holes so that matches are not decided in the first six holes or before the last six holes are reached.

* To quasi-make up some numbers, a par 3 is likely to be 3.3 versus 3.5, for a total of 0.8 above par, while a par five is likely to be 4.9 versus 6.3, for a total of 1.2 above par.

[dagolfer18]

Very interesting! Here’s something I’ve been curious about for a long time now: why are the even-numbered stroke indexes usually on one nine and the odd-numbered indexes on the other?

[David in FL]

28 minutes ago, dagolfer18 said:

Very interesting! Here’s something I’ve been curious about for a long time now: why are the even-numbered stroke indexes usually on one nine and the odd-numbered indexes on the other?

The last paragraph, including the provided link, answers your question in detail...

[iacas]

3 hours ago, David in FL said:

The last paragraph, including the provided link, answers your question in detail...

Yes, @dagolfer18: https://www.usga.org/content/usga/home-page/handicapping/roh/Content/rules/Appendix E Stroke Index Allocation.htm.

[RFKFREAK]

On 4/11/2020 at 3:20 PM, iacas said:

I've been meaning to write this for awhile, and since the World Handicap System (WHS) is coming to most of the world this year, now's as good a time as any.

This article will assume that you're semi-familiar with the the concepts of course rating and slope, and really seeks to expel some basic myths and misconceptions.

Course Ratings Are the Primary Determinant of "Difficulty"

As you should know, when a golf course is rated (for "difficulty"), many, many, many numbers are generated for each hole. Most of these numbers are pretty objective: the width of the fairway, the length of the hole, elevation changes, the diameter of the green, the depth and size of green side bunkers, etc. A few are subjective, like "how difficult is it to escape if you hit your ball into those trees"?

Two numbers come out of this calculation: a scratch rating and a bogey rating.

The scratch rating is defined as the score a scratch golfer should shoot on rounds where he plays to his handicap index (of 0.0). The bogey rating is the same for a "bogey golfer," (who, oddly, isn't an 18.0 index, but closer to a 20.0, as they're about a 20 course handicap on a 113-slope course).

So, we have two numbers: the score for a scratch golfer and the score for a bogey golfer.

Some basic linear algebra and geometry are used as such:

• Two points define a line.
• We plot those two points on a graph, and draw a line connecting them.
• This line has a "slope" that tells us the "slope rating" of that set of tees.

Remember y = mx + b?

In this case, y is the course handicap, m is the slope (slope rating/113), x is the handicap index, and b is the course rating.

And that's just what we see in a graph:

This is a set of tees with a rating of 72.0 (note that I consider all tees to be par 72 for the simplicity's sake in this article) and a slope of 113.

y = mx + b
CH = 113/113 * (HI) + 0
CH = 1 * HI
CH = HI

This should make sense: on a 72.0 (par 72) course with a slope of 113, we have a basic line with a slope of 1. These golfers should shoot, on average for the eight rounds that count toward their handicap index, these scores:

HI	Score	To Par
0.0	72	0
3.0	75	+3
15.0	87	+15
+2.0	70	-2

We could keep the same slope and make the course rating 69.0, and the course would instantly be three shots easier for every golfer. And this leads into what seems to be the biggest misconception. Too many people look at one number - the slope - and use that to determine what the "difficulty" of the course is.

But that fails, because lines are defined by more than their slope: the y-intercept matters. Imagine a course with a rating of "100.0" and a slope of "102." Nobody in their right minds would say that course is "easier" than a 72.0/144-rated course.

Here's a chart of four fictitious golf courses:

The graph of the course handicaps (note that rounding creates some "bumps" in the lines when the slope is not the whole number 1 (113/113):

Showing the trend lines much more heavily:

 

What these graphs show you is that course "difficulty" is a function of both the "m" and the "b" - the course rating AND the slope.

Look at the 74.0/118 course (grey) and the 72.0/136 course (green). Despite a difference of 18 in their slope, for the majority of these golfers, the 74.0/118 course plays "more difficult" due to the higher starting point of 74.0. It's not until you get out to about a 14.0 index that you start to see the expected scores for the 72.0/136 course take over due to the slope.

Look at all of the lines, in fact: the yellow line (69.0/140) remains lower for most of the graph than even the two 72.0 courses (blue 113, green 136), and well below the grey course (74.0/118). But this is because the course rating varies by 3 and 5 (69 to 72 and 74), while the slope can only make up fractions of a stroke (140/113 =~ 1.24 course handicap strokes per 1.0 strokes handicap index).

So, the course rating is the primary determinant of a course's difficulty.

The slope tends to matter in only two situations:

• The difference in handicap indexes is LARGE.
• The course ratings are quite close together.

The first matters because the slope has more time to keep adding "tenths of a shot" to the course handicap. The second matters because it's easier to overcome a deficit of 0.2 or 0.3 than a deficit of 2.4 or 3.1.

But What about + Handicaps?

Next look at the 72.0/113 course in blue. This course crosses the 72.0/136 (green) course at 0.0, and for everyone with a handicap on the opposite side from + handicaps, the green course is "more difficult" than the blue course.

But on the left side of scratch, the green course is "easier" than the blue course? Why is that?

Because the slope is the relative difference in difficulty between a good player and a worse player.

The "0.0" seems to throw people off, but the fact that we have + handicaps tells you that's not the absolute lowest anyone can go. Think of it this way: if a 13.0 gets 6 shots from an 8.0 due to the slope, then a +5.0 should give up six shots to a 0.0 too (there are occasionally rounding things that change this a little, on either side). The slope is the same, and so a change in "x" (the handicap index) should result in the same difference in course handicap (the y axis) because the slope of the line is constant.

Another way to think of it: add the course's par to the course handicap. If we called scratch golfers "72.0 golfers" on a 72.0-rated course, then you can see how a "67.0-rated golfer" should give up 5 shots to a "72.0 golfer" on a 72.0/113 course, and six shots to a 72.0/140 course.

67.0 * 140/113 = 83
72.0 * 140/113 = 89

Right?

The same math, essentially:

-5.0 * 140/113 = -6
0.0 * 140/113 = 0

Make sense?

Good. The final thing…

Stroke Indexes and "Hole Difficulty"

I'll try to keep this one short: the "stroke index" (handicap index) of the hole is NOT the "difficulty" of the hole.

It's a measure of where the higher handicapper is most likely to need a stroke against a lower handicapper.

Why are (or were, see the note below) par fives often the lower stroke index holes? Because:

• Better players tend to birdie or par them.
• Worse players tend to bogey them.

The increased distance gives a better player more chances to recover and more of an opportunity to show off their length. Consider the example of a one-foot putt versus a 50-foot putt: both the high and low handicapper are just going to tap in all the time on the one-footer, but the low handicap player is going to win a match of 50-foot putts against the worse player much more often.

To good players, par threes — often the high stroke index holes — are the "more difficult" holes relative to par. That's why they're traditionally the higher stroke index holes - the high handicapper isn't as likely to need a stroke against the better player.

Note: This stuff used to be calculated by courses literally turning in about 400 scorecards, which would all be entered hole-by-hole, and computed to determine which holes had the largest gaps between "better players" and "worse players." The holes would be ranked, the data massaged so that the first six holes didn't give out the strokes 1 to 5, and away we'd go.

The USGA and R&A have learned, however, that the actual location of strokes doesn't really matter all that much, so long as they're not clustered (like the example of stroke index holes 1-5 in the first six holes of the course). So, to make things simpler, they've come up with the idea of "triads" and are assigning stroke indexes via that method. You can read more about that method here: https://www.usga.org/content/usga/home-page/handicapping/roh/Content/rules/Appendix E Stroke Index Allocation.htm. In essence, it maintains the idea of the relative difficulty* while making things much simpler and not requiring all the "massaging" that was done before, and adequately spreads out the low- and high-stroke-index holes so that matches are not decided in the first six holes or before the last six holes are reached.

* To quasi-make up some numbers, a par 3 is likely to be 3.3 versus 3.5, for a total of 0.8 above par, while a par five is likely to be 4.9 versus 6.3, for a total of 1.2 above par.

[Morgan Mullins]

Just curious, is there an average course rating?  Or a rating range that many/most courses fall within? Thanks.

[Carl3]

I went into my first season in WNY with a 5.8 HI and joined a club and saw my index climb to over 10 and it could have been higher if not for the soft cap that limits the upward rise. Some of it could have been the stress in moving, playing a little less and not playing with a regular group with the friendly, competitive rivalries. However, some of it I had wanted to attribute to perhaps the course having a lower slope and index than warranted. From the back tees the par 72 course plays 6,523 yards has a rating of 70.8 and a slope of 123. I read though the explanation that @iacas provided.

I did hear that perhaps for some courses, much of this evaluation could be done with pen and paper and that a course rater is not obligated to play the course. Elements of the course that I believe make this particular more difficult are some elevated greens, some of which are small and have some severe slopes. I also felt that the greens were kept at a super high stimp and some of the regular pin placements were severe. I had heard a rumor that a junior golfer had come in decades ago and had destroyed the course with a super low score and the greenskeep had vowed that this would never happen again. Now this is all speculation and perhaps rationalization of my poor performance, but I do have some interesting data that I would like to share and get some opinions on.

My district senior championship was played there last year and I show the top 24 finishers with their handicap index, their scores, differentials and then the average differential minus their handicap index. I have highlighted the finishers who were playing the event on their home course. As you can see these three players did super-well compared to their index and no one else seemed to come close to this performance. Another observation is that the winner and the 5th place finisher who are 5.0 and 4.8 index respectively blew away golfers with much, much better indexes. I don't think that this could be all due to home-course advantage. The 17th place finisher is a 10.6 and beat a scratch golfer and a 2.2 index. Part of this could be attributed to the course being more challenging than the slope and index would indicate. This would also tend to make the home course players have a higher HI than indicated. 

The winner had an amazing two rounds and I was wondering what the probability is for him shooting two rounds in a row below his index especially since it is his home course where he plays all of the time. 5th place shot super close to his index too.

Any observations and thoughts would be appreciated. The reason I bring this up is that I may have a chance to rejoin this coming season and am weighing the pros/cons.

Edited February 8, 2022 by Carl3

[iacas]

1 hour ago, Carl3 said:

I did hear that perhaps for some courses, much of this evaluation could be done with pen and paper and that a course rater is not obligated to play the course. Elements of the course that I believe make this particular more difficult are some elevated greens, some of which are small and have some severe slopes.

Course raters, even if they don't play the course, still TOUR the course and measure everything and take note of everything. They'll roll balls on the greens, stop and look at recovery shots from the trees, look at the depth of the bunkers or the elevation of the greens, visibility issues, etc.

So everything (and more) that you listed is noted/measured/included.

1 hour ago, Carl3 said:

My district senior championship was played there last year and I show the top 24 finishers with their handicap index, their scores, differentials and then the average differential minus their handicap index. I have highlighted the finishers who were playing the event on their home course. As you can see these three players did super-well compared to their index and no one else seemed to come close to this performance. Another observation is that the winner and the 5th place finisher who are 5.0 and 4.8 index respectively blew away golfers with much, much better indexes. I don't think that this could be all due to home-course advantage. The 17th place finisher is a 10.6 and beat a scratch golfer and a 2.2 index. Part of this could be attributed to the course being more challenging than the slope and index would indicate. This would also tend to make the home course players have a higher HI than indicated. 

The winner had an amazing two rounds and I was wondering what the probability is for him shooting two rounds in a row below his index especially since it is his home course where he plays all of the time. 5th place shot super close to his index too.

Any observations and thoughts would be appreciated. The reason I bring this up is that I may have a chance to rejoin this coming season and am weighing the pros/cons.

You can't just multiply them, but your first guy shot 1.1 and 3.9 better… which is 10:1 odds and 57:1 odds.