Number of Pitches = Runs?

A couple of weeks ago I was listening to the Rockies radio broadcast team, Jeff and Jack, and Jack Corigan made the statement that basically said that as your (i.e. the pitcher) pitch count increases in the inning the more likely a run is going to be scored. From the sound of this it made me go hmmm. Two questions immediately popped up...1) is this true (logically you would think yes) and 2) what is on average the amount of pitches thrown in an inning that would allow a single run?

Well off to baseball reference where on their box score page they show the amount of pitches thrown in an inning along with the number of runs. By taking this data and plotting the number of pitches per inning, the runs scored for that number of pitches, and annotating in a running log how often this number of pitches occur in an inning, I was able to come up with the number of pitches thrown in 2008 by Rockies pitchers and what is the average number of pitches thrown to allow for one run in an inning (i.e., a 12 pitch inning occurred 81 times and a total of 12 runs were scored when 12 pitches were thrown in an inning or a rate of 0.12 runs). This is through Game 108 and I am missing about 7 runs (end of the season I will go back and tighten this up).

Graphically all this data looks like this:

This first plot shows the number of pitches versus the total runs. The size of the circle indicates how often this number of pitches were thrown. The next plot shows the distribution which indicates that 12 and 13 pitches were thrown in an inning approximately 161 times or accounting for about 17% of the total innings pitched..

The final plot shows the rate at which runs are scored. So for instance 26 pitches in an inning occurred 17 times and a total of 32 runs were scored. Thus giving a rate of 1.88 runs scored when 26 pitches are thrown. If you plot this rate you get this...

A relatively nice curve that has a Rsquare of about 0.79 (if you throw out two of the outliers you get upwards of 0.86). If you take the equation and solve for 1 you get approximately 20.6 pitches.

So there you have it...it does appear has you throw more pitches in an inning the likely of runs being scored increases. Follow on questions would be 1) what would the major league curve look like accounting for every pitch? and 2) could you establish pitcher effectiveness based on this (i.e., Cook has thrown 2,134 pitches this year, divided by 20.6 would indicate he should have given up 104 runs but presently he is at 64 allowed also he has averaged 13.9 pitches an inning which would equate to a rate of 0.47 runs per inning, he's pitched 154 innings so that equates to about 72 runs...)