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August 17, 2008
Books I Recommend and More on Z-Scores
First off, I'd like to point out a book that I carry around in my laptop case. I've found it tremendously helpful at working with statistics and refreshing some of my knowledge. It's an older book and I doubt it's even available on amazon anymore. You may have to Library.
Aigner, Dennis J. 1968. Principles of statistical decision making. Macmillan decision series. New York: Macmillan.
If you want to find it at the library, here's the Worldcat link. Also, as of today, there are four used copies on Amazon.com, for $2. Really, this book is a gem - it contains a concise summary of the foundation of Information economics and statistical decision theory. There's a bit of calculus and linear algebra but if you can get through this book, you'll be a pro.
Now, let's talk about z-scores. I've received a few emails asking questions about how it works. Here's the z-score equation (stolen from Wikipedia):
The Z value is X (the total points for a given player in a given game) minus the mean (all players at the same position against the same team) divided by the standard deviation (all players at the same position against the same team).
If your data is normally distributed, then all of your z-scores will fall in the range of -1 to +1, with -1 being below average and 1 being above average. The fantasy football data set, unfortunately, isn't normally distributed. However, it's pretty close and the vast majority of players fall in the -1 to +1 range. For example, here's the grid of Drew Brees' performance z-scores:
+----------+--------+------------+----------+--------+-------+ | name | id | pid | opponent | score | games | +----------+--------+------------+----------+--------+-------+ | D. Brees | 189811 | 00-0020531 | ARI | 0.202 | 2 | | D. Brees | 189812 | 00-0020531 | ATL | 0.631 | 5 | | D. Brees | 189813 | 00-0020531 | BAL | 0.494 | 2 | | D. Brees | 189814 | 00-0020531 | BUF | 1.099 | 2 | | D. Brees | 189815 | 00-0020531 | CAR | 0.134 | 5 | | D. Brees | 189816 | 00-0020531 | CHI | 0.338 | 2 | | D. Brees | 189817 | 00-0020531 | CIN | 0.823 | 2 | | D. Brees | 189818 | 00-0020531 | CLE | -0.338 | 3 | | D. Brees | 189819 | 00-0020531 | DAL | 1.768 | 2 | | D. Brees | 189820 | 00-0020531 | DEN | -0.600 | 7 | | D. Brees | 189822 | 00-0020531 | GB | 1.103 | 2 | | D. Brees | 189823 | 00-0020531 | HOU | 0.033 | 3 | | D. Brees | 189824 | 00-0020531 | IND | 0.277 | 3 | | D. Brees | 189825 | 00-0020531 | JAC | 2.003 | 3 | | D. Brees | 189826 | 00-0020531 | KC | 0.687 | 7 | | D. Brees | 189827 | 00-0020531 | MIA | -0.359 | 3 | | D. Brees | 189829 | 00-0020531 | NE | 0.553 | 2 | | D. Brees | 189830 | 00-0020531 | NO | 2.042 | 1 | | D. Brees | 189831 | 00-0020531 | NYG | 0.171 | 2 | | D. Brees | 189832 | 00-0020531 | NYJ | 0.010 | 3 | | D. Brees | 189833 | 00-0020531 | OAK | 0.435 | 8 | | D. Brees | 189834 | 00-0020531 | PHI | 0.731 | 3 | | D. Brees | 189835 | 00-0020531 | PIT | 0.323 | 3 | | D. Brees | 189837 | 00-0020531 | SEA | 1.533 | 2 | | D. Brees | 189838 | 00-0020531 | SF | 1.197 | 3 | | D. Brees | 189839 | 00-0020531 | STL | -0.210 | 2 | | D. Brees | 189840 | 00-0020531 | TB | 0.992 | 5 | | D. Brees | 189841 | 00-0020531 | TEN | -0.026 | 2 | | D. Brees | 189842 | 00-0020531 | WAS | -0.892 | 2 | +----------+--------+------------+----------+--------+-------+
You can see that our data isn't normally distributed (there are scores greater than 1) but most of the scores still fall in the -1,+1 range. What do these scores tell us? Historical performance indicates that he does better than other quarterbacks against quite a few teams (especially Jacksonville). I probably wouldn't start him in a game against Denver, though, his performance is less than the average (-.60) and he's played them 7 times.
So, what can we do with these scores? How can we make this useful? What I did is load the 2008 schedule into the database and build a grid of every game that every player is playing. From that, we can denormalize (go back to the X value). Here's what Drew Brees' predictions look like:
+----------+------+------------+-------+--------+------+ | name | id | pid | gid | points | team | +----------+------+------------+-------+--------+------+ | D. Brees | 5409 | 00-0020531 | 29534 | 8.83 | TB | | D. Brees | 5614 | 00-0020531 | 29551 | 10.85 | WAS | | D. Brees | 5782 | 00-0020531 | 29568 | 10.42 | DEN | | D. Brees | 5913 | 00-0020531 | 29580 | 11.72 | SF | | D. Brees | 6126 | 00-0020531 | 29602 | 11.74 | MIN | | D. Brees | 6182 | 00-0020531 | 29607 | 11.58 | OAK | | D. Brees | 6292 | 00-0020531 | 29618 | 10.14 | CAR | | D. Brees | 6482 | 00-0020531 | 29637 | 12.21 | SD | | D. Brees | 6707 | 00-0020531 | 29659 | 11.55 | ATL | | D. Brees | 6873 | 00-0020531 | 29679 | 12.01 | KC | | D. Brees | 7140 | 00-0020531 | 29703 | 11.46 | GB | | D. Brees | 7208 | 00-0020531 | 29713 | 8.83 | TB | | D. Brees | 7382 | 00-0020531 | 29726 | 11.55 | ATL | | D. Brees | 7465 | 00-0020531 | 29736 | 9.68 | CHI | | D. Brees | 7663 | 00-0020531 | 29755 | 12.75 | DET | | D. Brees | 7882 | 00-0020531 | 29776 | 10.14 | CAR | +----------+------+------------+-------+--------+------+
Couple interesting observations by looking at the predictions: first, even though Brees has no history against the Lions, he's still predicted to score at least 12.75 points because the average number of points against the Lions is higher than other teams. On the other hand, Brees performs well against Tampa Bay comparatively but is only predicted to score 8.83 points. However, a little bit of human intuition is needed in some of these situations - Tampa Bay used to have a great defense. Their defense this year isn't as great as in years past. This kind of interpretation is what makes Fantasy Football fun.
What's the end game here? Build the ultimate roster based on all players and their upcoming schedules. This is coming soon (before my draft day, August 31st but probably next week!)
Posted by haydenth at August 17, 2008 04:03 PM