Yes, I more-or-less stole the above title from the 2004 Ken Ross book entitled A Mathematician at the Ballpark. Like that book, anyone familiar with middle-school (or junior high school) math, should have no problem with most of what we do here. However, I will try to go into baseball in more detail than the book did.
Paraphrasing slightly, I read somewhere the following facetious remark:
From a survey of 1000 random baseball fans
across the nation, 183% of them hate math.
If you are one of these 183%, then this series could be for you. Hopefully, even if you aren’t a baseball expert, but you would like to learn some baseball statistics, (now often called “sabermetrics”), these posts will help. I’m no expert myself, so we’ll learn together.
In this series of blog posts, each post will introduce a particular metric in baseball statistics as well as some of the math and baseball behind it. We illustrate all these notions using the Baltimore Orioles’ 2022 season.
This week we look at one of the most popular statistics you see on televised games: OPS or “On-base Plus Slugging,” which is short for on-base percentage plus slugging percentage. Don’t worry, we’ll explain all these terms as we go.
On-base percentage
First, On-Base Percentage or OBP is a more recent version of On-Base Average or OBA (the same as OBP but the SF term is omitted). We define
OBP = (H + BB + HBP)/(AB + BB + HBP + SF),
where
- H is the number of Hits (the times the batter reaches base because of a batted, fair ball without error by the defense),
- BB is the number of Base-on-Balls (or walks), where a batter receives four pitches that the umpire calls balls, and is in turn awarded first base,
- HBP, or Hit By Pitch, counts the times this hitter is touched by a pitch and awarded first base as a result, and
- SF is the number of Sacrifice Flies and AB the number of At-Bats, which are more complicated to carefully define.
The official scorer keeps track of all these numbers, and more, as the baseball game is played. We still have to define the expressions AB and SF.
First, SF, or Sacrifice Flies, counts the number of fly balls hit to the outfield for which both of the following are true:
- this fly is caught for an out, and a baserunner scores after the catch (so there must be at most one hit at the time),
- the fly is dropped, and a runner scores, if in the scorer’s judgment the runner could have scored after the catch had the fly ball been caught.
A sacrifice fly is only credited if a runner scores on the play. (By the way, this is a “recent” statistic, as they weren’t tabulated before 1954. Between 1876, when the major league baseball national league was born, and 1954 baseball analysts used the OBA instead.)
Second, AB, or At-Bats, counts those plate appearances that are not one of the following:
- A walk,
- being hit by a pitch,
- a bunt (or Sacrifice Hit, SH),
- a sacrifice fly,
- interference (the catcher hitting the bat with his glove, for example), or
- an obstruction (by the first baseman blocking the base path, for example).
Incidentally, the self-explanatory number Plate Appearances, or PA, can differ from AB by as much as 10%, mostly due to the number of walks that a batter can draw.
The main terms in the OBP expression are H and AB. So we naturally expect OBP to be approximately equal to the Batting Average, defined by
BA = H/AB,
For example, if we take the top 18 Orioles players and plot the BA vs the OBP, we get the following graph:
The line shown above is simply the line of best fit to visually indicate the correlation.
Example: As an example, let’s look at the Orioles’ All-Star center fielder, Curtis Mullins, who had 672 plate appearances and 608 at bats, for a difference of 672 − 608 = 64. He had 1 bunt, 5 sacrifice flies, he was hit by a pitch 9 times, and walked 47 times. These add up to 62, so (using the above definition of AB) the number of times he was awarded 1st base due to interference or obstruction was 64 − 62 = 2.
Mullins’ H = 157 hits break down into 105 singles, 32 doubles, 4 triples, and 16 home runs.
Second, let’s add to these his 126 strikeouts, for a total of 157+126+64 = 347.
The remaining 608 − 347 = 261 plate appearances were pitches hit by Mullins, but either caught on the fly but a fielder or the ball landed fair and Mullins was thrown out at a base.
These account for all of Mullins’ plate appearances. Mullins has a batting average of BA = 157/608 = 0.258 and an on-base percentage of OBP = 0.318.
Slugging percentage
The slugging percentage, SLG, (SLuGging) is the total bases achieved on hits divided by at-bats:
SLG = TB/AB.
Here, TB or Total Bases, is the weighted sum
TB = 1B + 2*2B + 3*3B + 4*HR,
where
- 1B is the number of “singles” (hits where the batter makes it to 1st Base),
- 2B is the number of doubles,
- 3B is the number of triples, and
- HR denotes the number of Home Runs.
On-base Plus Slugging
With all these definitions under own belt, finally we are ready to compute “on-base plus slugging”, that is the on-base percentage plus slugging percentage:
OPS = OBP + SLG.
Example: Again, let’s consider Curtis Mullins. He had 1B = 105 singles, 2B = 32 doubles, 3B = 4 triples, and HR = 16 home runs, so his TB = 105+64+12+64 = 245. Therefore, his SLG = 245/608 = 0.403, so his on-base plus slugging is OPS = OBP + SLG = 0.318 + 0.403 = 0.721.
This finishes our discussion of OPS. I hope this helps explain it better. For more, see the OPS page at the MLB site or the wikipedia page for OPS.
denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 5 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.
.
.)
denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 6 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.
having edge set: 
.
denote the vertex set. There are (up to isomorphism) exactly 2 4-regular connected graphs on 7 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.
. This is an Eulerian, Hamiltonian (by Ore’s Theorem), vertex transitive (but not edge transitive) graph.
. This is an Eulerian, Hamiltonian graph (by Ore’s Theorem) which is neither vertex transitive nor edge transitive.
denote the vertex set. There are (up to isomorphism) exactly six 4-regular connected graphs on 8 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.
. This is vertex transitive but not edge transitive.
and 
.
, 
, 
, 
.
. This is a strongly regular (with “trivial” parameters (8, 4, 0, 4)), vertex transitive, edge transitive graph.
and is generated by (5,6), (4,7), (3,4), (2,5), (1,2), (0,1)(2,3)(4,5)(6,7).
. This graph is not vertex transitive, nor edge transitive.
. This graph is not vertex transitive, nor edge transitive.
having edge set: 
. This graph is not vertex transitive, nor edge transitive.
denote the vertex set. There are (up to isomorphism) exactly 16 4-regular connected graphs on 9 vertices. Perhaps the most interesting of these is the strongly regular graph with parameters (9, 4, 1, 2) (also distance regular, as well as vertex- and edge-transitive). It has an automorphism group of cardinality 72, and is referred to as d4reg9-14 below.
or the complete graph 
. However, both d4reg9-3 and d4reg9-14 not only have harmonic morphisms to 
, they each may be regarded as a multicover of 
denote the vertex set. There are (up to isomorphism) exactly 59 4-regular connected graphs on 10 vertices. One of these actually has an automorphism group of cardinality 1. According to SageMath: Only three of these are vertex transitive, two (of those 3) are symmetric (i.e., arc transitive), and only one (of those 2) is distance regular.
.
.
.
.
. This graph is also called a star graph 
on 3+1=4 vertices, or the bipartite graph 
.
sends the red vertices in 
to the red vertex of 
is a harmonic morphism. Let 
be adjacent vertices of 
and 
“collapses” the edge (vertical) 
or (b) 
and the vertices 
and 
are adjacent in 
. In the particular case of this post (ie, the case of 
the green vertex is not adjacent to the blue or red vertex, none of the harmonic colored graphs below can have a green vertex adjacent to a blue or red vertex. In fact, any colored vertex can only be connected to a white vertex or a vertex of like color.
, plus the “obvious” ones obtained from that below and those induced by permutations of the vertices:
. 
can be described in a similar manner. Likewise for the higher 
graphs. Given a star graph 
, plus the “obvious” ones obtained from that above and those induced by permutations of the vertices with a non-zero color.
, one hovering over the other. Now, connect each vertex of the top copy to the corresponding vertex of the bottom copy. This is a cubic graph that can be visualized as a “thick” regular polygon. (The cube graph is the case 
.) I conjecture that there is no harmonic morphism from such a graph to 
for the complete bipartite (“utility”) graph 
. They are all obtained from either
, where 
. This graph has diameter 3, girth 3, and edge-connectivity 3. It’s automorphism group is size 4, generated by (5,9) and (1,8)(2,7)(3,6). The harmonic morphisms are all obtained from
.
, where 
.
.
.
.
. (This only differs by one edge from the one above.)
.
.
. Denote the 
-vector space of such functions by 
. We write an element of this space as 
, where the variables 
will be called coordinate variables. Let
. In this post, the cosets
). A function in 
which (a) is constant along some coordinate hyperplane 
, (b) whose restriction 
is constant along some coordinate hyperplane 
, (c) whose restriction 
is constant along some coordinate hyperplane 
, (d) and so on. This “nested” inductive definition might seem complicated, but to a computer it’s pretty simple and, to boot, it requires little memory to store. 
and 
then let 
denote the vector whose i-th coordinate is flipped (bitwise). The sensitivity of 
is
. Roughly speaking, it’s the number of single-bit changes in 
. The (maximum) sensitivity is the quantity
The block sensitivity is defined similarly, but you allow blocks of indices of coordinates to by flipped bitwise, as opposed to only one. It’s possible to 
, let 
be two independent, uniform random variables taken from 
, and let 
. The last banana game concerns the case 
. Here, I’m interested in investigating the question: What is 
? 
.
, we have
.
is defined as 
, but there’s a ![E(Z)=\sum_{k=0}^{n-1} P(Z>k)=\sum_{k=0}^{n-1}[1-P(Z\leq k)]](https://s0.wp.com/latex.php?latex=E%28Z%29%3D%5Csum_%7Bk%3D0%7D%5E%7Bn-1%7D+P%28Z%3Ek%29%3D%5Csum_%7Bk%3D0%7D%5E%7Bn-1%7D%5B1-P%28Z%5Cleq+k%29%5D&bg=ffffff&fg=323232&s=0&c=20201002)
.
of degree m+1 in the variable n that gives the value of the sum of the m-th powers of the first n consecutive positive integers:
.
but they do and Faulhaber’s results establish this existence.) These polynomials were discovered by (German) mathematician 
to 
. It was about 100 years later, in the early 1700s, that (Swiss) mathematician 
,
,
lower order terms.
be m independent, uniform random variables taken from 
. Again we ask: What is 
case generalizes to:
.
the white vertex is not adjacent to the blue or red vertex, none of the harmonic colored graphs below can have a white vertex adjacent to a blue or red vertex.
as the domain in this post. However, before we get to examples (obtained by using 
be a harmonic morphism from a graph 
vertices to the path graph having 
vertices. Let 
be the coloring map (identified with an n-tuple whose coordinates are in 
). Associated to f is a partition ![\Pi_f=[n_0,\dots,n_{k-1}]](https://s0.wp.com/latex.php?latex=%5CPi_f%3D%5Bn_0%2C%5Cdots%2Cn_%7Bk-1%7D%5D&bg=ffffff&fg=323232&s=0&c=20201002)
of n (here ![[...]](https://s0.wp.com/latex.php?latex=%5B...%5D&bg=ffffff&fg=323232&s=0&c=20201002)
is a multi-set, so repetition is allowed but the ordering is unimportant): 
, where 
is the number of times j occurs in f. We call this the partition invariant of the harmonic morphism. 
, 
, with associated
whose corresponding partitions agree, 
then we say 
and 
are partition equivalent.
examples!
, so we start with 
. We indicate a harmonic morphism by a vertex coloring. An example of a harmonic morphism can be described in the plot below as follows: 
sends the red vertices in 
, plus that induced by 
and all of its cyclic permutations (4+6=10). This set of 6 permutations is closed under the automorphism of 
and all of its cyclic permutations (4+7=11). This set of 7 permutations is not closed under the automorphism of 
and all 7 of its cyclic permutations (total = 7+11 = 18).
and all of its cyclic permutations (4+8=12). This set of 8 permutations is not closed under the automorphism of 
and all of its cyclic permutations (12+8=20). In addition, there is 
and all of its cyclic permutations (20+8 = 28). The latter set of 8 cyclic permutations of 
is closed under the transposition (0,3)(1,2) (total = 28).
and all of its cyclic permutations (4+9=13). This set of 9 permutations is not closed under the automorphism of 
and all 9 of its cyclic permutations (9+13 = 22). This set of 9 permutations is not closed under the automorphism of 
and all 9 of its cyclic permutations (9+22 = 31). This set of 9 permutations is not closed under the automorphism of 
and all 9 of its cyclic permutations (total = 9+31 = 40). 
is the graph whose SageMath command is 
then, instead of showing a graph, I’ll list the edges (of course, the vertices are 0,1,…,11) and the SageMath command for it.
.
.
and 
are graphs then a map 
) is a morphism provided
and 
then 
is an edge in 
and 
, where 
then 
.
is the dual map 
and 
, then 
, an edge 
is called horizontal if 
and is called vertical if 
. We say that a graph morphism 
is a graph homomorphism if 
. Thus, a graph morphism is a homomorphism if it has no vertical edges.
denote the star subgraph centered at the vertex v. A graph morphism 
is called harmonic if for all vertices 
, the quantity
and mapping to the edge 
.
sends the red vertices in 
to make 2018.
where p, q, r, s are primes.
where p, q, r, s are primes.
using the fewest number of distinct digits.
for integers 
Yet Another Mathblog 
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