## The Value of Fouls

Stop me if you've heard this one before, but a strike is a strike, right?

Continuing my quest to be the world's greatest expert on the subject of strikes, let's turn our attention to the batter. This post investigates whether the type of strike makes a difference in the outcome of the plate appearance. In other words, is a batter more successful if they foul off strike one than if they swing and miss?

This is part one of a two part series. In this first post, we're going to examine the question retrospectively. That is, we'll look back at plate appearances that end at various counts and see if the outcome differs based on whether the strikes were called, swinging, or fouls. Part two will take a forward-looking approach - looking at each pitch as it happens and seeing whether things progress differently depending on the type of strike.

Let's look at a few different tables. The first one lists the wOBA for plate appearance that ended on a given count. The rest break that down further by looking at the type of the strikes (c = called, s = swinging, f = foul). I'm using wOBA as my measure of effectiveness since it's a nice rate stat that incorporate the run value of each plate appearance. All data is from 2000-2007 and ignores plate appearances by pitchers.

wOBA by Count
Count wOBA PA
0-1 .360 130686
1-1 .371 128341
2-1 .391 81998
3-1 .577 73001
0-2 .194 107667
1-2 .207 193890
2-2 .225 186221
3-2 .409 180828
wOBA by Count and Strike Type
Count Strike Types wOBA PA
0-1 c .356 86002
0-1 s .362 13560
0-1 f .369 30926
Count Strike Types wOBA PA
1-1 c .364 75263
1-1 s .371 15972
1-1 f .384 36402
Count Strike Types wOBA PA
2-1 c .382 46825
2-1 s .392 9345
2-1 f .407 24351
Count Strike Types wOBA PA
3-1 c .575 45504
3-1 s .587 6736
3-1 f .578 16333
Count Strike Types wOBA PA
0-2 2c .189 19682
0-2 1c 1s .171 20976
0-2 1c 1f .207 38140
0-2 2s .161 4488
0-2 1s 1f .194 11458
0-2 2f .216 12035
Count Strike Types wOBA PA
1-2 2c .206 32097
1-2 1c 1s .182 34219
1-2 1c 1f .222 67214
1-2 2s .163 8012
1-2 1s 1f .199 22558
1-2 2f .226 24088
Count Strike Types wOBA PA
2-2 2c .229 28302
2-2 1c 1s .206 29034
2-2 1c 1f .233 61289
2-2 2s .190 7202
2-2 1s 1f .220 21134
2-2 2f .243 23909
Count Strike Types wOBA PA
3-2 2c .408 18412
3-2 1c 1s .393 19384
3-2 1c 1f .418 40232
3-2 2s .377 4800
3-2 1s 1f .401 14182
3-2 2f .423 15416

A couple things to note right off the bat. I've ignored any plate appearances that ended with no strikes, since they're not really relevant to the question at hand. I've also ignored any foul balls beyond the first two strikes. I think those most likely do have an impact on the end result of the plate appearance, but I'm also assuming they're spread fairly evenly across all the different ways of getting to the two strike count. Finally, I'm assuming order of strikes doesn't matter. Whether the batter goes swinging strike followed by foul ball or vice versa doesn't matter.

So what do these results tell us? Well, the wOBA is a lot higher with 3 balls and a lot lower with 2 strikes. Makes perfect sense, since you can't walk without getting to 3 balls, and can't strike out without getting to 2 strikes. We'll come back to that later.

Looking at the raw results, it looks like there's some benefit to fouling off a pitch versus any other type of strike. Is it meaningful? Using the formula SQRT(wOBA * (1.1 - wOBA)/PA) (found here) for the standard deviation, looking at the 1 strikes counts, and hoping I'm doing this right (I'm using the smallest number of plate appearances as my sample size), we reach the 95% confidence level for 0-1, 1-1 and 2-1. 3-1 doesn't even really demonstrate the pattern in our sample, but I think that might be dampened by the number of walks at the 3-1 count.

So I feel pretty safe in saying that there's some advantage to fouling off the ball rather than swinging through it or taking the strike. Or, in the reverse, good hitters tend to make some sort of contact on strikes.

Teasing out some kind of meaning from the 2 strike counts is beyond my level of skill, but eyeballing it, it looks like there's something there too.

I'd like to filter out strikeouts and walks, since I was really most interested in whether the strike type indicated anything for balls put into play. So let's look at the same set of tables, showing wOBA on contact only.

wOBA on Contact by Count
Count wOBA PA
0-1 .353 128399
1-1 .366 126844
2-1 .388 81499
3-1 .424 35148
0-2 .335 59084
1-2 .344 111949
2-2 .357 114378
3-2 .392 84867
wOBA on Contact by Count and Strike Type
Count Strike Types wOBA PA
0-1 c .349 84607
0-1 s .356 13342
0-1 f .361 30254
Count Strike Types wOBA PA
1-1 c .360 74507
1-1 s .365 15746
1-1 f .378 35893
Count Strike Types wOBA PA
2-1 c .380 46569
2-1 s .389 9278
2-1 f .405 24186
Count Strike Types wOBA PA
3-1 c .419 21853
3-1 s .425 3031
3-1 f .438 8192
Count Strike Types wOBA PA
0-2 2c .313 11243
0-2 1c 1s .332 10228
0-2 1c 1f .334 22319
0-2 2s .350 1954
0-2 1s 1f .347 5891
0-2 2f .352 7008
Count Strike Types wOBA PA
1-2 2c .330 19303
1-2 1c 1s .348 17937
1-2 1c 1f .348 41175
1-2 2s .340 3679
1-2 1s 1f .349 12067
1-2 2f .359 14591
Count Strike Types wOBA PA
2-2 2c .351 18077
2-2 1c 1s .374 16386
2-2 1c 1f .351 39665
2-2 2s .364 3639
2-2 1s 1f .364 12165
2-2 2f .370 15299
Count Strike Types wOBA PA
3-2 2c .383 8874
3-2 1c 1s .408 8434
3-2 1c 1f .401 19845
3-2 2s .391 1871
3-2 1s 1f .400 6155
3-2 2f .409 7649

The results are pretty much the same. Still significant positives for foul balls at 0-1, 1-1 and 2-1, while not necessarily so for 3-1. And the 2 strike counts are still outside my realm of statistically competency, but appear to demonstrate a similar, if muted, effect.

What conclusions can we draw from these results? It appears that there's a correlation between fouling the ball off (as opposed to swinging and missing or taking a strike) and success in the at-bat. One possible explanation is that good hitters are more likely to foul off the ball. I'm not sure how true this is (see Adam Dunn), but it should be easily testable. Another possibility is that fouling off the ball indicates a near-miss and therefore suggests better contact when the ball is put into play.

The big assumption in looking at things this way is that there's no difference in how a plate appearance proceeds based on the type of strike. I don't think that's really the case. If a batter rips the first pitch into the seats just outside the foul pole, I'm betting the pitcher will respond differently than if the first pitch was swung on and missed by three feet. To check that assumption, part two will look at the same question as the at-bat unfolds, and will try to identify the effects of the various strike types on the whole at-bat, not just the ultimate outcome. I'll try to get the post out next week but there's a good chance it will slip a little bit.

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