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Fielding effectiveness can be measured at the team level by two numbers, which we call Fielding Efficiencies. We give the definitions elsewhere, but for convenience we will repeat them here.
The first measure we denominate FEbat, and it measures the team’s effectiveness at putting out batters who present an outs opportunity to the defense. It is simply the ratio of outs made by the defense on batters to the number of batters who put the ball into play for fielders. That measure is thus calculated like this. First, outs made by fielders on batters:
FObat = (AB - SO - H) + SF + SH
Then the available outs for fielders (BIP, Balls In Play) on batters:
BIP = BFP - SO - BB - HBP - CI - HR
And FEbat is just FObat/BIP.
The second measure, which we denominate FErun, is the percentage of runners who reached base safely (BR) but were subsequently thrown out by the defense. The BR can be expressed as:
BR = BFP - SO - (FEbat x BIP)
That is, it is all batters who came to the plate minus those who made out (which last is strikeouts and batters put out by the fielders).
The “actual fielder’s outs” on base runners (FOrun) can be reckoned as follows:
FOrun = BR - (R + LOB)
By the rules of scoring, every batter who reaches base must end up in one of three statistical categories: an Out, a Run, or a Left On Base. So, if we subtract Runs plus Left On Base from the total of men who initially reached base safely, we get the number put out on the bases; so, FErun, is simply FOrun/BR.
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If we want to see the effects of setting a team’s Fielding Efficiencies to all-MLB average levels, we need to first realize that such a chnage will affect the number of opponents who come to the plate (the BFP stat for pitching) in a game or a season. That is because the higher the opponents’ on-base percentage (that is, the worse the team’s FEbat), the more men they send to the plate in a given number of innings (or games)—and, of course, vice-versa). To start our reckoning, we must therefore see what the effects of changed fielding will be on BFPs.
Outs are made three ways: by strikeouts, by batters being put out, and by base runners being thrown out on the bases. The first of those is, of course, not dependent on fielding, but the other two are. To deal with them, we need to change numbers into rates.
The strikeout rate, which we write as SOrate, is simply SO/BFP. For the other two, it is more complicated.
If we look at the BIP value, as shown above, we see that none of its components depends on fielding. So we can reckon a BIPrate and know that it, like the SOrate, it is a fielding-independent given. We can thus use the all-MLB avereage FEbat value to calculate a new fielder’s-outs rate. Because FEbat is simply FObat/BIP—
FObat = FEbat x BIP
—thus the new FObat rate (which we write as FObat2rate) is just the MLB-average FEbat x BIPrate.
Next, we reckon the new rate at which runners are turned into outs. The new BR value is—
—where FEbatAvg is the all-MLB average FEbat value. Turning that into a rate by dividing by BFP, we get:
OB2rate = 1 - SOrate - (FEbatAvg x BIPrate)
Now, having that value, we can reckon the FOrun2rate as:
FOrun2rate = OB2rate x FErunAvg
So we now have all three components of the new rates at which the team gets outs: the SOrate, the FObat2rate, and the FOrun2rate. We add them to get the new overall outs rate for the defense:
OUTS2rate = SOrate + FObat2rate + FOrun2rate
Having that, we can now calculate the new BFP figure. (That requires assuming that Outs remain constant; since Outs are really innings played times 3, all we are assuming is that the team plays the same number of innings even with the changed fielding. That is probably true, though even if not, the diference will be very small.) Since OUTS = OUTSrate x BFP, we know that:
BFP2 = OUTS/OUTS2rate
So we take the (fixed-number) OUTS and divide it by the new OUTSrate we just found.
Finally, we can make an “adjustment factor” by taking BFP2/BFP—the ratio of the new somewhat increased or decreased BFPs that would result from MLB-average fielding to the actual BFPs. With that, we can adjust the team’s stats to allow for the changed number of BFPs.
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When we have adjusted every stat by the changed-PA ratio, our job is not over. Certain stats need further adjustments (as should be obvious by the fact that changed Fielding Efficiencies must mean changes in the Hits rate beyond those from changed PA opportunities. Also, the rate at which batters reach base on an error (ROE) must be assumed to change a bit as well (it is a small number, but we want the best accuracy we can get). For that one, we use the simplifying asumption that the rate varies in direct proportion to the improvement or degradation of FEbat by assuming MLB-average values. So ROE2rate = ROErate x (FEbatAvg/FEbat), and ROE2 is ROE2rate x PA2.
To get a corrected Hits figure, we start with this:
Hrate = (BR - BB - HBP - CI - ROE) / PA
Since none of those save BR and ROE are affected by fielding, we can write:
H2rate = BR2rate - BBrate - HBPrate - CIrate - ROE2rate
So H2, the new Hits total, is given by just H2 = H2rate x PA2. But that’s not quite all yet.
We have to assume that of the changed Hits total, the distribution of non-Home-Run hits remains unaffected: that is, the percentage of Hits that go for singles, doubles, or triples remains fixed. Granted, it is an assumption, but it seems a sound one. Better or worse fielding is not going to be restricted to singles. A fielder who muffs a ball, outfield or infield, may be giving up an extra-base hit—or one who successfully makes a difficult play may well be stopping an extra-base hit. So, that said, we get the new singles, doubles, and triples figures by taking the unadjusted ratios of hit type to all Hits (1B/H, 2B/H, 3B/H) and multiply the new Hits numbers by the corresponding ratio to get the new figure for that type of Hit.
(It shouldn’t need saying, but of course we round all the new-data figures to whole numbers before using them in any further operations.)
And that’s the whole gruesome business.
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