# Pitch Type Linear Weights Explained

Yesterday David Appleman announced a new section at FanGraphs showing the Linear Weights Run Value for each pitcher and pitch type. He asked me to write a short explanation of how these values are calculated.

The run value of any event is the change in the expected number of runs scored over the rest of the inning from before and after the event happened. The expected number of runs scored is the average number scored from a given out and base-occupancy state. Let’s take Tuesday’s Oakland at Tampa Bay game as an example. At the top of the 1st inning with zero on and zero out the average team scores 0.55 runs. Orlando Cabrera hit a single off of Jamie Shields. Now with a runner on first and none out the average team scores 0.95 runs. So the run value of the single was 0.55-0.95=0.4.

You can do the same thing taking each pitch as as event rather than the outcome of each at-bat. To do this you need to know the run expectancy from each count, in other words, the average run value of all events from at-bats which pass through a given count. For example the run expectancy of a 3-0 count is 0.2, on average at-bats from that count are worth about half of a single.

Now we can run through Cabrera’s at-bat with Shields as an example of valuing each pitch in an at-bat.

0-0: Run Value 0.00

Pitch 1: Fastball for a ball

1-0: Run Value 0.03

So the value of that first pitch was 0.03 runs. On average the A’s will score 0.03 more runs than before Shields threw the pitch.

1-0: Run Value 0.03

Pitch 2: Fastball for a called strike

1-1: Run Value -0.02

The run value of that fastball was -0.02-0.03 =-0.05.

1-1: Run Value -0.02

Pitch 3: Fastball for a called strike

1-2: Run Value -0.08

The run value of that fastball was -0.06. You can see here that the run value of a strike (or any other event) is count-dependent.

1-2: Run Value -0.08

Pitch 4: Fastball for a ball

2-2: Run Value -0.04

The run value of that fastball was 0.04.

2-2: Run Value -0.04

Pitch 5: Change fouled off

2-2: Run Value -0.04

Since the count did not change the run expectancy did not, so the run value of this changeup was 0.

2-2: Run Value -0.04

Pitch 6: Fastball hit for a single

Runner on first no outs: Run Value 0.4

The run value of this fastball is the change in run expectancy, 0.4-(-0.04) = 0.44.

Shields threw five fastballs valued at 0.03, -0.05,-0.06,0.04 and 0.44. He threw one changeup that had a value of 0.00. These values are the change in run expectancy in the game, so a negative number is a good for the pitcher (fewer runs scored). On the player pages the numbers are flipped so a positive number indicates a good pitch, the number of runs saved by those pitches.

We hoped you liked reading Pitch Type Linear Weights Explained by Dave Allen!

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