Before we explain how we calculate these values, here is a brief overview of run expectancy. There are 288 possible base runner, out, count combinations: there are 8 possible base states (bases empty, man on first, etc.), 3 possible out states (0, 1, or 2 outs), and 12 possible counts (0-0, 0-1, etc.). Each one of these states has a unique run expectancy, or the number of runs a team is expected to score from that instance until the end of the inning. The run expectancy for a specific state is estimated by finding the average number of runs MLB teams scored â€“ from the occurrence of the state until the end of the inning â€“ in each instance of that state over the last 5 seasons. For example, a team with no outs, no baserunners, and a 0-0 count is expected to score 0.5 runs before the inning is over. Meanwhile, a team with no outs, the bases loaded, and a 3-0 count is expected to score 2.75 runs before the inning is over.

To calculate run expectancy effects, we simply find the difference between the team's run expectancy with the incorrect call and with the correct call. For example, imagine a bases loaded, 2 out, 3-2 count scenario. A strike call here ends the inning, while a ball call walks in a run and resets the count to 0-0. In one potential next state, the batting team is expected to score 0 runs (the inning would be over). In the other, the team would still be expected to score 0.74 runs. So, a missed call in this scenario would be worth 1.74 runs, or the difference in run expectancy between the two potential states plus the value of the run walked in. You can read an article one of our team members wrote for FanGraphs on this topic here.

This approach certainly has its limitations. As an example, many believe that if an umpire misses a called strike three, and then a batter hits a home run, that missed call should be worth more than if the batter had simply struck out on the next pitch. However, we continue to use this approach for at least two reasons: we can express umpire impact in terms of runs, the most fundamental unit of value in baseball, and we can assign an impact to any missed call, in any scenario.