# Interpretation for sequence of slugging percentages

This is really more of a mathematical question with a baseball application, but it seems more appropriate to ask it here.

Because slugging percentage is defined as total bases divided by at bats, it does not appear to be meaningful to describe it in the sense of a probability. In contrast, a batting average is easily described as a probability and the complementary probability is also meaningful. Suppose that we're given batters A and B with batting averages of 0.300 and 0.280, resp., the product of the averages (0.300*0.280 = 0.084) describes the probability that both batters get a hit, assuming that at bats are independent. Similarly, the product of the complements (0.700*0.720 = 0.504) describes the probability that both are out. Lastly, the probability that exactly one of the batters gets a hit is given by 0.300*0.720 + 0.700*0.280 = 0.412.

But if I'm interested in slugging percentages, what would the interpretation be for a product of slugging percentages? Similarly, I assume that the complement of a slugging percentage would be 4-slg?

• Slugging percentage represents how likely a batter would hit a double, triple, or home run other than a single hit. Why does it have to be different from batting average? The higher the percentage, the more powerful a batter is. Don't you agree? – user10632 Oct 12 '16 at 18:50
• @Rathony I quibble with a few things. A speedy contact hitter could get a large slugging percentage as a result of turning singles into doubles and doubles into triples not because of power, but because of speed. Also, SLG, in combination with ISO or BA, can describe the likelihood of getting an extra base hit vs. a single, but not SLG alone. Again, my (mathematical/statistical) interest is in a concise interpretation for a product of slugging percentages. – Pistol Pete Oct 12 '16 at 19:07
• Batting average is incredibly in favor of a speedy contact hitter, too. The speedier you are, the higher the batting average will be. They are just statistics that baseball uses. Nothing more, nothing less. – user10632 Oct 12 '16 at 19:33
• @Rathony SLG does not represent how likely a batter would do any specific thing. (In fact, a batting average is also not a probability, but that's another issue.) SLG represents the number of bases reached by hit per at-bat, but that's not a probability of getting a base or of getting any particular kind of hit. XB% would be that (extra base-hit percentage). – Joe Oct 13 '16 at 14:34
• Personally I would find this question more appropriate on Cross Validated, as I think this is really a statistics question (how to interpret an outcome ratio versus a likelihood ratio), but I think it's at least technically on-topic here if you prefer to keep it here. But I think you may get superior answers on Cross Validated. – Joe Oct 13 '16 at 14:49

SLG is the expected value of total bases ("TB") of an at-bat. You would use it slightly differently from batting average, since it's measuring a different thing.

Let's see some examples.

``````Player A has a .300 BA, .450 SLG, 1.5 SLG/BA ratio
Player B has a .280 BA, .320 SLG, 1.1 SLG/BA ratio
``````

You can use batting average to measure, for example, the probability that one of two players gets a hit in their next at-bat. That would be `1-((1-.3)*(1-.28)) = 1-(.7*.72) = (.496)`. So these two players, on average, will get one hit between the two of them about half the time.

You cannot determine that with SLG, of course, as SLG does not measure proportion of at bats with hits. SLG measures total bases, though, so you can use that.

For example, in this case, you would say that the expected value of Player A's total bases after one at-bat is 0.450 (his SLG). So, in an average at-bat Player A records 0.45 bases. Player B, similarly, records 0.32 bases in an average at-bat. This all assumes that it is an offical at-bat, and not a walk or similar, of course.

If you want to combine the two, then, you would add the terms together. You would say, "The expected value of total bases after both Player A and Player B bat is 0.32+0.45 = 0.77", meaning after the two of them bat, on average 0.77 bases are recorded.

You could actually do this with batting averages, too, of course. 0.300 is also the expected value of the "hit" statistic after an at-bat. So after the two players both bat, the expected value of "hits" for the team for those two at-bats is 0.3+0.28 = 0.58.

This is fairly easy to see; you already calculated the joint probability of at least one hit (0.496), and the probability of both of them getting a hit is easy (0.3*0.28 = .084). Sum those two together and you get 0.58 again.

Another thing you can do with regards to conditional probability is combine batting average and slugging percentage. Batting average says "This is the probability of a hit", and Slugging percentage says "This is the expected number of bases from hits", right? So E(TB|H) is reasonable to calculate, and is simple (and this should be obvious): SLG/BA.

So 0.45 is SLG for player A, and 0.30 is BA for player A, so E(TB|H) = 1.5. Player B has a 1.1 SLG/BA ratio, so E(TB|H)=1.1.

What this means, though, is that you can use that figure to discuss the expectations between the two. If you want to know the expected number of bases given that one of them gets a hit, for example, you would say

``````E(TB|H) = Sum( (E(TB|H)*E(H) for each player) / (Joint probability of E(H))
= (E1(TB|H)*E1(H) + E2(TB|H)*E2(H)) / (1-(1-E1(H))*(1-E2(H)))
= (1.5*0.3 + 1.1*0.28)/(0.496)
= 1.55
``````

That is, when at least one of them gets a hit, the expected value of total bases is 1.55 from the two of them.

SLG works well with other statistics, to some extent, this way; you can derive things like isolated power (ISO), which tells you what proportion of their slugging percentage comes from XBH explicitly. You can't explicitly calculate the probability of an XBH from it, though, since the proportion of 2B|3B|HR can vary with identical SLG/AVGs. (Take one HR and one Single, and make them 1 2B and 1 3B, for example; identical SLG and AVG for those two results.)

The slugging "percentage" is "the mean number of bases a batter obtains per at bat". As such, multiplying two slugging percentages together doesn't have an intuitive "physical" meaning - it's a quantity with a dimension of "bases squared", which isn't something that really means anything; it's not like multiplying two batting averages which are just probabilities, and therefore their product is also a probability.