Thursday, April 19, 2007

Something Every Baseball Fan Should Know (About Statistics)

Update from Dave: An article in Baseball Prospectus (subscription) talked about the wisdom of Erick Aybar trying to steal 2nd in two games so far this season, and being thrown out to end both. Based on win expectancy, Dan Fox calculated the break-even point for trying to steal at 65%. Give Mike Scioscia (the Angels manager) some credit:

"With that combination--the time we had for the pitcher [coming to the plate] and throwing time [for the catcher]--I thought we had a better than 75 percent chance of making it. Erick got a decent jump. It took a perfect throw to get him, and they got it. If it was a 50-50 proposition, obviously we're not going to do it."

(Original Post 4/13/07 at 10:39 PM)

In route to losing their game against the Houston Astros tonight, the Phillies made a characteristically bone-headed base running mistake which drastically reduced their chances of winning. How can I say with confidence that it was "bone-headed"? Because the Run Expectation Matrix told me so!

The situation: It is the bottom of the 8th inning and the Phillies are losing 8-6. The Phillies have runners on 1st and 2nd, with 0 outs. Ryan Howard is up and hits a single into right field. Shane Victorino is rounding third base. A decision needs to be made: send the runner or hold him at 3rd base.

Likely Outcomes:
A: Victorino is held at 3rd base, resulting in bases loaded, with 0 outs.
B: Victorino tries to score and is thrown out, resulting in runners at 1st and 3rd with 1 out, the score still 8-6. (Since Ryan Howard is the runner at 1st and is slow as molasses so we will assume that he does not make it to 2nd as a result of the throw to the plate)
C: Victorino tries to score and is safe, resulting in runners at 1st and 3rd, with 0 outs, Phillies down 8-7. (Again, assume slow-poke Ryan Howard does not make it to 2nd)

Run Expectation Table: The table below, called the Run Expectation Table, shows the average number of runs scored in the league for any combination of runners and outs.

Bases with a RunnerNo OutsOne OutTwo Outs

2nd, 3rd1.971.450.59
1st, 3rd1.811.170.49
1st, 2nd1.580.960.45
1st, 2nd, 3rd2.371.650.82

(Side note #1: This table gives an amazing amount of information and is good for hours of entertainment. Stealing 3rd with 2 outs? Stealing 2nd with no outs? Walking the leadoff man? Sacrifice bunt? Side note #2: These are average numbers and specific circumstances can alter the actual run expectations, so be careful while using them to make generalized condemnations about specific in-game decisions.)

The Statistics: From the table, we would score 2.37 additional runs in the inning given case A, 1.17 runs for case B and 1.81 runs from case C. For case C, a run also scores on the play, so for the avereage total runs scored for each of the cases, we have A = 2.37, B = 1.17, C = 2.81. (MATH CENSORED) The result is that he must have a 73% chance of scoring or higher in order for it to result in more runs, on average. The announcer's intuition wasn't far off when he said "in this situation, you want to make sure that runner can score almost standing up".

That isn't really the whole story though because, although it's pretty counter-intuitive, we don't necessarily want to maximize the number of runs we will score. In the end, we want to maximize the chances of us winning the game. We are down by 2 runs, so we might want favor outcomes which score 2 or 3 runs over those that score 1 or 5+. We can accomplish this with something called the Win Expectation Matrix, which gives the probability of winning the game based on the game situation. Here "game situation" means the number of outs, bases occupied and inning. For our three cases, the probability of winning the game for each of the three cases above is A = 0.54, B = 0.31, and C = 0.63. (MATH CENSORED) We find that the runner needs a 72% chance of scoring safely in order for sending him to cause a rise in the chances or winning the game.

So what was the outcome? The Phillies sent Victorino, who was thrown out at the plate. The next batter grounded into a inning-ending double play, the Phillies lost the game and I'm home at 11pm on Friday writing about it. Such is life in Philadelphia.

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