Jono
13th May 2005, 11:21 AM
Andy, which buttons ??
(h1-h2)/h1 x 100 ;)
Suffering from a bad case of insomnia, I typed this out last night … A bit of nit-picking, I know, but I believe the above formula favours the lower markers, especially those close to scratch handicap.
I know our resident math genius, Trung, has noted it already, but your equation, AndyP, for calculating improvement rating is somewhat faulty. What you are calculating is the handicap deduction as a percentage of the old handicap. So, if a 1 handicap got his handicap down to scratch, he'd get 100% improvement. Likewise, a 36 handicap who miraculously got his handicap down to scratch would get 100% improvement. The former only has to have a good round or two to get to scratch, but the latter would probably have to sell his soul to the devil in order to achieve a scratch handicap. Essentially, your rating system greatly favours the lower markers.
And what about those who have plus handicaps? Say you go from scratch to plus one? The calculator would say error because it can't divide by 0. Say you go from plus one to plus two? You would get a negative percentage.
There are two main problems using your equation to calculate improvement:
a) you are using a percentage … i.e. you are calculating a ratio. What you should be doing is multiplying the drop in handicap by a "difficulty multiplier". So you recognize that it is more difficult to drop shots as your handicap improves, but you don’t get ridiculous situations like the example above with the 1 and 36 handicap.
b) you are using the scratch handicap as the absolute zero. As you can see, once you reach scratch handicap, your equation breaks down. You need to set a lower mark as the absolute zero, recognizing that it is quite possible to drop one’s handicap into the plus range.
Here's my proposal for working out the "improvement" rating.
We need to make a couple of assumptions.
Assumption 1) What is the BEST possible handicap that one can have? I suppose if you are a machine, you can get birdie every hole and shoot 18 under par. What is the best handicap that a HUMAN can have on a course with ACR of 72? (i.e. average difficulty) Let's say plus 10. I think even Tiger would have difficulty averaging 10 under even on an easy course.
Assumption 2) What is the WORST possible handicap? Let's say 36.
Assumption 3) Using these two reference points (ie. 10 under par for best handicap and 36 over par for worst), we can work out a "difficulty multiplier". Let's say you start with 36 handicap. As you get closer and closer to the "perfect handicap" of plus 10, it gets harder and harder to drop a shot. And this increase in difficulty is not linear. The difficulty of lowering your handicap is INVERSELY proportional to how far your handicap is from the "perfect handicap". So, if your handicap is 36, lets say your difficulty multiplier is one. Let's say you HALVE your distance to the perfect handicap of plus 10. (36-(-10))/2 is 23. i.e. you are 23 away from plus 10 handicap which is a handicap of 13. The difficulty multiplier is now 2 (i.e. it is now twice as hard to drop a shot when you are 13 handicap than it was when you were 36). Lets say you halve your distance again ... (13-(-10))/2 is 11.5. i.e. your are 11.5 from plus 10 which is 1.5 handicap. The difficulty multiplier is now 4. i.e. it is twice as hard to drop a shot than it was at 13, and four times as hard as it was at 36.
Using this approach, we can come up with an equation to calculate the difficulty multiplier:
D (difficulty multiplier) is INVERSELY proportional to (your handicap - best possible handicap)
so D = k * 1/(your handicap - best possible handicap) where k is some constant.
For simplicity, we can let k = the number of strokes between the best possible handicap and the worst possible handicap. This way, we get D = 1 if your handicap is the worst possible handicap.
So D = (worst possible handicap-best possible handicap) / (your handicap - best possible handicap)
Let's use plus 10 as the best possible handicap and 36 as the worst.
I'll use H as "your handicap"
D = (36-(-10)/(H-(-10) = 46/(H+10)
So let's say your handicap is 36. Then your difficulty multiplier is 46/(36+10) which is 1.
Let's say your handicap is 13. Then your difficulty multiplier is 46/(13+10) which is 2.
Let's say your handicap is 5. Then your difficulty multiplier is 46/(5+10) which is approximately 3.
Let's say a 36 handicap, 13 handicap, and a 5 handicap all drop a shot in their handicap. The 36 handicap would multiply that shot by his D which is 1. So he would get an improvement rating of 1. The 13 handicap would multiply that shot by his D which is 2 and get an improvement rating of 2. 5 handicap would get an improvement rating of 3. So providing they all drop a shot, the 5 handicap would have the best improvement rating.
Now, theoretically, D should change for every shot you drop during the improvement period. Sort of like variable interest. You get interest on the interest. However, for simplicity's sake, let's use your AVERAGE handicap during the improvement period to calculate the difficulty multiplier.
So H = (h1+h2)/2
So if you started at 36 handicap and at the end of the period, you finished at 30. Then your average handicap would be 33. So H = 33.
Using this we can formulate an equation for the improvement rating.
I = improvement rating
I = drop in handicap * difficulty multiplier
I = (h1-h2) * D
= (h1-h2) * 46/(H+10)
= 46 * (h1-h2) / ((h1+h2)/2 + 10 )
Math geniuses like Trung, should be able to see instantly that as your handicap approaches the “perfect handicap” of plus 10 (i.e. H = -10), the difficulty multiplier D approaches infinity. When your handicap is the maximum possible (ie. 36) your difficulty multiplier is 1.
So the corollary of all this rambling is that the formula 46 * (h1-h2) / ((h1+h2)/2 + 10 ) gives us a much better estimation of your improvement in relation to other golfers, as opposed to the percentage formula (h1-h2)/h1 x 100. The latter favours the low markers too much.
And the other corollary is that I have too much spare time on my hands … :wink: :lol:
(h1-h2)/h1 x 100 ;)
Suffering from a bad case of insomnia, I typed this out last night … A bit of nit-picking, I know, but I believe the above formula favours the lower markers, especially those close to scratch handicap.
I know our resident math genius, Trung, has noted it already, but your equation, AndyP, for calculating improvement rating is somewhat faulty. What you are calculating is the handicap deduction as a percentage of the old handicap. So, if a 1 handicap got his handicap down to scratch, he'd get 100% improvement. Likewise, a 36 handicap who miraculously got his handicap down to scratch would get 100% improvement. The former only has to have a good round or two to get to scratch, but the latter would probably have to sell his soul to the devil in order to achieve a scratch handicap. Essentially, your rating system greatly favours the lower markers.
And what about those who have plus handicaps? Say you go from scratch to plus one? The calculator would say error because it can't divide by 0. Say you go from plus one to plus two? You would get a negative percentage.
There are two main problems using your equation to calculate improvement:
a) you are using a percentage … i.e. you are calculating a ratio. What you should be doing is multiplying the drop in handicap by a "difficulty multiplier". So you recognize that it is more difficult to drop shots as your handicap improves, but you don’t get ridiculous situations like the example above with the 1 and 36 handicap.
b) you are using the scratch handicap as the absolute zero. As you can see, once you reach scratch handicap, your equation breaks down. You need to set a lower mark as the absolute zero, recognizing that it is quite possible to drop one’s handicap into the plus range.
Here's my proposal for working out the "improvement" rating.
We need to make a couple of assumptions.
Assumption 1) What is the BEST possible handicap that one can have? I suppose if you are a machine, you can get birdie every hole and shoot 18 under par. What is the best handicap that a HUMAN can have on a course with ACR of 72? (i.e. average difficulty) Let's say plus 10. I think even Tiger would have difficulty averaging 10 under even on an easy course.
Assumption 2) What is the WORST possible handicap? Let's say 36.
Assumption 3) Using these two reference points (ie. 10 under par for best handicap and 36 over par for worst), we can work out a "difficulty multiplier". Let's say you start with 36 handicap. As you get closer and closer to the "perfect handicap" of plus 10, it gets harder and harder to drop a shot. And this increase in difficulty is not linear. The difficulty of lowering your handicap is INVERSELY proportional to how far your handicap is from the "perfect handicap". So, if your handicap is 36, lets say your difficulty multiplier is one. Let's say you HALVE your distance to the perfect handicap of plus 10. (36-(-10))/2 is 23. i.e. you are 23 away from plus 10 handicap which is a handicap of 13. The difficulty multiplier is now 2 (i.e. it is now twice as hard to drop a shot when you are 13 handicap than it was when you were 36). Lets say you halve your distance again ... (13-(-10))/2 is 11.5. i.e. your are 11.5 from plus 10 which is 1.5 handicap. The difficulty multiplier is now 4. i.e. it is twice as hard to drop a shot than it was at 13, and four times as hard as it was at 36.
Using this approach, we can come up with an equation to calculate the difficulty multiplier:
D (difficulty multiplier) is INVERSELY proportional to (your handicap - best possible handicap)
so D = k * 1/(your handicap - best possible handicap) where k is some constant.
For simplicity, we can let k = the number of strokes between the best possible handicap and the worst possible handicap. This way, we get D = 1 if your handicap is the worst possible handicap.
So D = (worst possible handicap-best possible handicap) / (your handicap - best possible handicap)
Let's use plus 10 as the best possible handicap and 36 as the worst.
I'll use H as "your handicap"
D = (36-(-10)/(H-(-10) = 46/(H+10)
So let's say your handicap is 36. Then your difficulty multiplier is 46/(36+10) which is 1.
Let's say your handicap is 13. Then your difficulty multiplier is 46/(13+10) which is 2.
Let's say your handicap is 5. Then your difficulty multiplier is 46/(5+10) which is approximately 3.
Let's say a 36 handicap, 13 handicap, and a 5 handicap all drop a shot in their handicap. The 36 handicap would multiply that shot by his D which is 1. So he would get an improvement rating of 1. The 13 handicap would multiply that shot by his D which is 2 and get an improvement rating of 2. 5 handicap would get an improvement rating of 3. So providing they all drop a shot, the 5 handicap would have the best improvement rating.
Now, theoretically, D should change for every shot you drop during the improvement period. Sort of like variable interest. You get interest on the interest. However, for simplicity's sake, let's use your AVERAGE handicap during the improvement period to calculate the difficulty multiplier.
So H = (h1+h2)/2
So if you started at 36 handicap and at the end of the period, you finished at 30. Then your average handicap would be 33. So H = 33.
Using this we can formulate an equation for the improvement rating.
I = improvement rating
I = drop in handicap * difficulty multiplier
I = (h1-h2) * D
= (h1-h2) * 46/(H+10)
= 46 * (h1-h2) / ((h1+h2)/2 + 10 )
Math geniuses like Trung, should be able to see instantly that as your handicap approaches the “perfect handicap” of plus 10 (i.e. H = -10), the difficulty multiplier D approaches infinity. When your handicap is the maximum possible (ie. 36) your difficulty multiplier is 1.
So the corollary of all this rambling is that the formula 46 * (h1-h2) / ((h1+h2)/2 + 10 ) gives us a much better estimation of your improvement in relation to other golfers, as opposed to the percentage formula (h1-h2)/h1 x 100. The latter favours the low markers too much.
And the other corollary is that I have too much spare time on my hands … :wink: :lol: