# Standard deviation of game results about predictions from a rating system

I'm very in to Sports analysis and am keen to look at finessing my analysis models that I have worked up (I don't have a great maths background, I've just done a little bit of reading).

Standard deviation of Game Results about prediction from a Ratings system (in this case least squares regression with a few tweaks) equals the stochastic element in a normal distribution function where:

x = the margin of victory that you are trying to get a % against (e.g. it could be a point spread e.g. % to cover) the mean = the predicted margin of victory derived in this case from my least squares regression to create power ratings for teams sigma = standard deviation of error on the difference between the predicted margin of victory and actual margin of victory based on past results.

Based on the above I would treat in Excel terms a win % for team A as:

1-NORMDIST(0.5,mean,sigma,true) - with the mean and sigma having values obviously.

Based on Stern "The Probability of Winning an American Football Game" American Statistician (August 1991) - the sigma value (stochastic element? - sorry this isn't my usual thing) is approximately 13.86. How would you go about refining this for other sports? - e.g. Australian Rules Football. I said this could be done for any sport (and I believe it can within reason unless the results or the sport doesn't lend itself to this type of analysis).

Looking at past AFL results from the end of the season point of view the standard deviations are as follows (have only started looking at these in the last two days):

2009: 31.34 (150+ results) 2010: 33.44 (150+ results 2011: 34.91 (150+ results) 2012: 25.62 (only approximately 70 results so far as it is mid season)

As you feed more data into the model for 2012 it will build up a more accurate picture but looking at past data I'm putting the sigma value for AFL at around 33.5 (above the 25.62 value). Is there anyway of defining a best fit value? Where I don't have a great background in this kind of thing, if someone could give me some gentle nudges in the right direction in terms of where I should be looking or what I could be applying I would be grateful.

I believe this can definitely be applied to other sports as well e.g.:

English Premiership:

2008/2009: 1.45 (Season with over 300+ results) 2009/2010: 1.52 2010/2011: 1.54 2011/2012: 1.64 (but when looking over the last 100 results around the 1.50 mark)

I've also looked at the percentages that this then gives for home wins, away wins and draws and they are virtually identical to the percentages for bookmakers odds (except they work to 105-106% instead of 100% so they have an edge which accounts for the small differences).

NBA:

2008/2009: 11.30 (Season with 1000+ results) 2009/2010: 11.56 2010/2011: 10.88 2011/2012: 11.36

It is common in sports to predict outcomes using logistic regression or if you want to predict a continuous response linear regression. In logistic regression the response Y has 2 values (win or lose in the case of a sporting event outcome) So the outcome is usually coded as 0 for loss and 1 for win. The model is y=a1 X1 + a2 X2 +a3 X3 +....+ e. The response y=log(p/1-p) where p is the unknown probability of success and X1, X2, X2,... are the vsrisble that are used to predict y. In linear regression you have the same form as with logistic regression except that y is the response rather than a function of it. e is the random component or stochastic element. It is there to express error (sometimes measurement error in determining y). In these models the error term is assumed to be additive. Also the Xi are assumed to be determined without error. Using the model and data for the Xis and the corresponding ys the regression parameters a1, a2,a3, ... are estimated (usually by least squares). Often the error term e is assumed to be normally distributed with me an 0. They are independent of the Xis and independent of each other. The variability of the error term is defined by the standard deviation sigma which is assumed to be constant (i. e. the same for each sample). For the normal distribution the mean and standard deviation uniquely determine the distribution. These models are general and the technique can be applied in any sport. The covariate and response variables can differ from sport to sport. It is important to keep the modeling assumptions in mind and if they don't seem to apply alternative approaches may need to be considered.

The American Statistical Association has a section on Statistics in Sports. Hal Stern whose article you cited is a member of that section. He has written substantially on sports. other members Jim Albert and Michael Schell have written books specifically on baseball. You might want to consider joining the association and the section. On the ASA website there is an eGroup for the section where questions like yours are discussed. I am a member but haven't actively done research in it. Many members of the section have though.

Other prominent members that I have not yet mentioned are Carl Morris and Scott Berry.

What is the sample size and do you have bins with fewer than 5 observations? I ask these questions because very small departures from normality can be detect in large samples. Also the chi square test is approximate and doesn't work well when there are bins with fewer than 5 observations. The median seems a lot larger than the mean which would indicate high skewness. But that doesn't show up in the skewness statistic.The standard deviation is large but the standard error is small which indicate a large sample size.

Since the ratio of the standard deviation tio the standard error should be the sqaure root of n, your sample size must be close to 600. I calculated 576 and looking at our data again I see count =567 is probably your sample size and that would make sense. I would like to see a histogram and a Q-Q plot. The conclusion is non-normal but it is not yet clear to me that the departure from normality is large enough to worry about.

• Thanks for the information - it's helpful to get a better understanding (we did this kind of thing at school but there was never a practical application so it was difficult to see where it was relevant). I would say what I am doing is essentially logistic but instead of coding 0 and 1, I am using actual games scores (which does affect the ratings as they are skewed by one-sided results). I am definitely keen to try and get a general approximation for e (if there is such a things as best fit for stochastic elements) and the given the past data you can get access to I think this achievable. Commented May 13, 2012 at 15:41
• The American Statistical Association has a section on Statistics in Sports. Hal Stern whose article you cited is a member of that section. He has written substantially on sports. other members Jim Albert and Michael Schell have written books specifically on baseball. You might want to consider joining the association and the section. On the ASA website there is an eGroup for the section where questions like yours are discussed. I am a member but haven't actively done research in it. Many members of the section have though. Commented May 13, 2012 at 16:02
• Other prominent members that I have not yet mentioneed are Carl Morris and Scott Berry. Commented May 13, 2012 at 16:02
• Thanks for the heads up. I'm keen to improve the models generally (side interest) and I'm just keen to refine the AFL one (hence why I worked up the last three years data to get a clearer idea of the e value as I figured the 2012 one was wrong or there wasn't enough data yet most likely). I know the model is generally solid though as it has gone 11/21 on general bet-able outcomes (so far this weekend - one game remaining) and 6 out of 7 on totals (which these models seem sharp on - 10% return off of my German Ice Hockey model - or the totals are badly handicapped). Commented May 13, 2012 at 16:15
• Q-Q stands for quantile vs quantile. It is like log paper for normal distributions. Things are scaled so that normally distributed data will fall close to a straight line. It provides an informal graphical check. I think redefining the bins will help with the validity of the chi square test. Commented May 18, 2012 at 19:13