# Golf handicap Q

Vikster

## Main Question or Discussion Point

OK, this is probably a bit too easy for you guys, but we're kind of stuck so help would be much appreciated.

My question is about finding an analytical solution to ”the expected golf handicap” of a virtual golfer with a given mean score and standard deviation within the USGA handicap system.

We are a group of people working on improving the Swedish handicap system and it would be very helpful to us if we could understand better how to analytically solve the expected value within the US system.

The principal problem can be formulated this way:

If you take a sample of two observations from a population with a given mean, M, and standard deviation, SD, and then discard the higher of the samples, what would be the expected value of the lower?

The golf handicap system is based on the mean hcp differential of the best 10 rounds of the 20 most current times 0.96, so the sample n would be higher, but the principal should remain the same.

This is as far as we’ve gotten:

In an infinitely large sample the solution where you discard the higher half of the observation to the hcp problem would have the solution: Hcp Index = (mean diff-0,675xSD)0,96. The mean differential is easily calculated based on the known mean score of the population, so it is not taken from the sample mean.

This solution is naturally based on the fact that as we discard the higher half of the scores in an infinitely large sample to calculate our mean diff we have only to calculate the mean of the lower “half” of the Gauss-clock of the standard normal distribution. The expected value in an infinitely large sample would be where alfa equals 0.25 which gives us an approximate Z of 0.675.

Our question is if there is a precise analytical solution to how we should modify the standard deviation to find a correct expected value of the mean of the 10 lowest in a sample of 20 from a population with a given mean and standard deviation. So the snag seams to be how to handle the standard deviation for a given sample size. The only thing we've come up with is using SE=SD/the square root of n, and that can not be correct since we don’t want to calculate the standard error for the mean of a sample in a case where we have the actual mean and standard deviation of the population already.

In more general terms I think we could probably manage if we got help solving the more principal problem which is finding the expected value of the lower of two samples taken from a population with a given mean and standard deviation.

Many thanks in advance from a sunny spring Stockholm.

Vikster
Wow, no one? Maybe it cannot be done (finding an analytical solution).

Vikster
Does any one know of a forum that might be able to help out with stats questions of this kind (I guess it’s not pure math)? Or is there something wrong with the question? Or are you guys just ignoring newbies? Anything would be helpful to get us back on track and I can tell quite a few has at least read my question.

Regards from winter land.

Vikster
Since no one here seems to know the answer to whether or not there is an analytical solution to the finding the expected value of the lower of two samples drawn from a normal distribution with a given standard deviation and mean, isn’t there anyone who can give me a tip on other forums of this kind where I might be able to find help?

I would REALLY be VERY grateful for any kind of help.

Cheers.

No-one plays golf probobly.

Isn't scoring in golf the other way around. You have to get the LOWEST score to win.

What's a handicap anyway?

Vikster
Originally posted by ObsessiveMathsFreak
No-one plays golf probobly.

Isn't scoring in golf the other way around. You have to get the LOWEST score to win.

What's a handicap anyway?
That's why you take away the higher 50% of the scores.

But by all means forget the golf, can you, or anyone else, help me with this:

In more general terms I think we could probably manage if we got help solving the more principal problem which is finding the expected value of the lower of two samples taken from a population with a given mean and standard deviation.
??

climbhi
Originally posted by ObsessiveMathsFreak
No-one plays golf probobly.
I play golf, its a great game. Except the problem is I hate and am terrible at statistics, sorry...

Vikster
Originally posted by climbhi
...the problem is I hate and am terrible at statistics, sorry...
No problem climbhi, I'm glad you took the time to reply. I was beginning to think maybe newbies were excommunicated by default.

Like I said, forget the golf, I'm just as happy if anyone can help out with the general problem of finding the expected value of the mean of the lower 50% of the observations in a sample of n observations drawn from a population with a normal distribution defined by a known mean and standard deviation.

yeah, vikster, sorry you re not getting many replies.

i avoid probability and game theory and such completely, i just don t like the stuff.

sorry.

Vikster
Originally posted by lethe
yeah, vikster, sorry you re not getting many replies.

i avoid probability and game theory and such completely, i just don t like the stuff.

sorry.
I don't blame you. If I was more into it, I would be able to answer this one myself I guess!