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The Strike Zone as a Probability Density Function

The strike zone in baseball is supposed to be the area above the plate and between the batter's armpits and knees.  Seems simple enough, but  John Palmer suggests that it is more complicated than that.

Several days go, while watching some very erratic home-plate umpiring, I posted a short item to Facebook quoting myself, "The strike zone is a probability density function." I had originally made the statement during a play-by-play radio broadcast of a London Tigers baseball game over 20 years ago. The Tigers were a AA minor league team. The radio station manager asked me not to do that again and not to try to explain it on air.

This is an interesting way to think about the strike zone.  What John is implying is that the probability of a strike being called within the strike zone is determined by where it is in the strike zone.  In particular, John suggests that the closer the pitch is to the center of the zone, the more likely it will be called a strike.  He links to this post at FiveThirtyEightSports written by Etan Green that supports John's claim.  The following graph from the article illustrates John's belief about the strike zone.


Over at Facebook, King Banaian commented

(T)here is not just one strike zone. The dimensions are not just spatial but umpire specific. So even your three dimensional depiction of the PDF isn't quite right, and I question whether summing up along the umpire dimension gives us anything really meaningful about the strike zone."

The strike zone may also be situation-dependent.  For example, an umpire may slightly expand his strike zone if he realizes that he called an earlier ptch a ball when he should have called a strike.

Lastly, in watching my sons play baseball over the years, there is definitely a difference in strike zones between youth baseball and pro baseball.  In youth ball, the strike zone tends to move around more.  In the pros, the strike zone seems more consistent to my untrained eye.