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# The Roto Grotto: rates versus opportunities

Z-Scores allow you to compare rate and counting statistics because they scale each category based on the average and standard deviation of that category. However, they do not address the issue of opportunity. As a reader pointed out last week, Felix Hernandez went one for three last season, but his .333 batting average was not more valuable in fantasy than the .319 batting average Ryan Braun produced over 598 at bats.

There are several reasons there is not an easy answer. First, the issue applies to counting stats as well as rate stats. Last season, Jose Reyes and Ben Revere each stole 40 bases, but Reyes needed 716 plate appearances while Revere needed only 553 plate appearances. If you were only concerned about winning steals, then Revere was clearly more valuable. You could replace him with a waiver-wire player for the 36 fewer games he played than Reyes and pick up a handful of extra steals.

The Reyes-Revere example illustrates the second complication. There is an additional underlying opportunity that owners have to consider, which is chances to start a player. Hypothetically, two players could have the same number of plate appearances and the same number of stolen bases but play in a different number of games. Over the course of the season, the difference in number of plate appearances for players batting in different spots in the order or batting in the same spot but on teams with different offensive levels can be significant, as Tristan Cockcroft of ESPN recently showed. In addition, players with clear platoon splits tend to pinch hit or be pulled in the late innings of games because of pitcher match-ups.

So, for every statistic, you need to account for opportunities to start a player, and with rate statistics, you need to account for differences in opportunities within each opportunity to start. Your league type will determine the number of opportunities you have to start a player, but I’ll consider leagues with daily lineups for this so I can use games played to approximate it.

First, I calculated each counting stat per games played. Returning to an earlier example, Jose Reyes stole 40 bases in 160 games in 2012, which is 0.25 steals per game. Ben Revere stole 40 bases in 124 games, which is 0.32 steals per game.

Next, I calculated the Z-Scores of counting stats per game. I followed the same method that I used to calculate the Z-Scores for season totals, but substituted the means and standard deviations of those per game statistics. Reyes and Revere had a zSB of 4.00 and 2.90, respectively.

For rate stats, the per game averages are the same as the season averages, but I can then scale those Z-Scores based on opportunities. Last season, Derek Jeter led baseball with 683 at bats. I can use that as my denominator. For example, Felix Hernandez had a raw Z-Score of about 2.45 because of his .333 batting average. However, if I scale that with his three at bats divided by the 683 maximum possible at bats, his scaled zAvg is only 0.01. In contrast, Ryan Braun has a scaled zAvg of 1.63 despite a lesser .319 average because of his 598 at bats.

Now, I finally have counting and rate stats apples to apples. A simple addition of the Z-Scores in each category provides an overall value, similar to a player rate. Here is the top-10 from 2012:

Player Season zHR zSB zRBI zRun zAvg zTotal
Mike Trout 2012 1.88 4.45 1.33 3.71 1.66 13.02
Ryan Braun 2012 2.64 2.06 2.11 2.13 1.63 10.57
Miguel Cabrera 2012 2.75 -0.51 2.92 1.96 1.95 9.07
Josh Hamilton 2012 3.01 -0.17 2.93 2.09 0.74 8.61
Andrew McCutchen 2012 1.60 1.04 1.41 1.99 1.80 7.85
Edwin Encarnacion 2012 2.82 0.42 2.12 1.54 0.63 7.52
Giancarlo Stanton 2012 3.16 -0.15 1.94 1.50 0.68 7.13
Jose Bautista 2012 3.05 -0.07 1.98 2.09 -0.14 6.92
Matt Kemp 2012 1.89 0.40 1.65 2.11 0.83 6.88

As you can see, different players reach the top in different ways. Four of the top-10 players are actually negative contributors in a category (or two).

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