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Friday, April 01, 2011
How to figure out how much talent there is: example with NHL goalies
Someone supplied me with even strength shots faced and saves made for, well, I didn’t ask. It has 55 goalies, with an average of 2665 even strength shots faced, and Kipper leading at 4784. That’s probably about four seasons worth. Not important.
Step 1: Figure out how much one standard deviation is based on a binomial distribution. Vokoun faced 4249 shots, and the league average save percentage was .920, so one SD is sqrt (.92*.08/4249) = .0042.
Step 2: Figure out how much away you are from the mean. Vokoun’s save percentage was .931, and so was +.0108 from the mean.
Step 3: Figure out how many SD that is. .0108/.0042 = 2.59. That’s his z-score.
Step 4: Do it for all the goalies. (Thomas is 2.57, Luongo is 2.53… Holmqvist is -3.11, Raycroft is -2.34).
Step 5: Find the standard deviation of all the z-scores. In this case, for these 55 goalies, it’s 1.38.
Step 6: Rejoice if the number is substantially higher than 1.00. Happiness sets in at 1.10. You did good at 1.20. If you get 1.40, you’ve definitely found something.
Step 7: Figure out the average number of opportunities for each player. In this case, the average shots faced was 2665.
Step 8: Do this: 1 - 1/1.38^2 = 0.47. That’s your r or r-squared. (Longer story later. Just call it r for now.) That 1.38 was from Step 6.
Step 9: Do this: (1-r)/r * 2665. We get 2969. The 2665 is from Step 7.
That’s the key number. 2969. Let’s call it 3000. That’s how much you use to regress a goalie’s performance. You add 3000 shots of league average performance. So Vokoun’s 4249 shots at .931 save percentage gets added to 3000 shots at .920 save percentage for a best-estimate true talent level of .926. Holmqvist’s .900 with 1809 shots becomes .912. So, the observed difference between the two goalies (.031 saves per shot) becomes a true difference of .014.
A couple of important points:
1. This tells us how many talent there is FOR THE SAMPLE of goalies. Put in less goalies, the talent level will be tighter, and the regression will be higher. Put in more goalies, the talent level will be wider, and the regression will be smaller.
2. It’s best not to mix years without adjusting. As long as the seasons you are mixing up has a fairly static talent level and league averages, then don’t worry.
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