career.

Let’s start with Ruth. From the table above, we see that the Babe had 8398 AB and 2062 BB for a total of 10,460 PA. Hence, if x is the number of additional AB, the following proportion preserves the AB to PA ratio:

To solve this equation for x , we merely “cross multiply” and isolate the unknown quantity to obtain x = 1638. So Ruth would get an additional 1638 AB. This implies that he would also receive an additional 402 BB, because 8398 AB + 1638 (additional AB) + 2062 BB + 402 (additional BB) = 12,500 PA.

So, if Ruth was just as good as he always was for these extra 1638 AB, then his projected HR total would be:

We note that the term

is nothing more than a prorating of the 714 statistic. But we are assuming that Ruth would be 5 percent less the hitter he was during the rest of his career. Therefore, the

term should be multiplied by 0.95, giving the true projected HR figure as:

Note that the left-hand side of this last equation has a “714” in both terms. If we factor out the 714 (and recall that the number “1” is always an understood coefficient of any term), we see that

In other words, if we multiply 714 by the coefficient 1.185, we get the projected cumulative HR total for Babe Ruth.

We call this the equivalence coefficient , because it gives us a “reasonable” estimate of the desired cumulative HR total, given our defined “equivalent” scenario.

With regard to Ted Williams, his additional AB compute to 2197, while his additional BB come to 576. Using the 5 percent better and 10 percent better assumptions (giving kickers of 1.05 and 1.10, respectively), we find that the equivalence coefficients for Williams are 1.299 and 1.314, respectively. So, a 5 percent better Williams would hit 521(1.299) = 677 HR, while a 10 percent better Williams would project to 521(1.314) = 685 HR.

We summarize the technique of computing the EC in Figure 5.1.

Figure 5.1 Computing the equivalence coefficient for batting

We see that the equivalence coefficient has enabled us to compile the entries in the following table, perhaps shedding some light to answer the questions we posed above:

• What would Williams’ totals be if he had not lost so much time?

• What if Ruth had not started out as a pitcher?

• Who was the greater hitter: Williams or Ruth?

Table 5.2 Williams versus Ruth using the equivalence coefficient

So, who was the greater hitter?

Some remarks are in order regarding this approach. First, the EC can be regarded as a mathematical model. As with most models, it can be tweaked. For example, we assumed that Ted Williams was “equally better” during the years he missed in both the 1940s and in the 1950s. We could have assumed that he was 10 percent better in the 1940s and 5 percent better in the 1950s. Clearly, this would have yielded different projections and made our model a bit more complicated (see the Hard Slider problem at the end of the chapter).

We also assumed that the proportions of AB to PA were constant . But if Williams was 10 percent better in the 1940s, perhaps he would have been even more selective regarding what pitches to hit, meaning that he might have had less than a total of 9903 AB, while drawing more than 2597 BB. How would this have affected his projected cumulative totals?

Also, this model could be enhanced by considering such entities as the hit-by-pitch (HBP) statistic, on-base-average (OBA) and both stolen bases (SB) and caught stealing (CS). In this way, more offensive categories would be included.

What about pitching? We mentioned legendary Dodger southpaw Sandy Koufax. Koufax recorded 2396 strikeouts (K) in 2324.3 innings pitched (IP) during his shortened career. What if, for the sake of argument, we assume that he had pitched an additional 800 innings? Can we use the EC approach with respect to pitching? Yes, we can.

To find Koufax’s strikeout EC, we basically duplicate the procedure we used with the

Let’s start with Ruth. From the table above, we see that the Babe had 8398 AB and 2062 BB for a total of 10,460 PA. Hence, if x is the number of additional AB, the following proportion preserves the AB to PA ratio:

To solve this equation for x , we merely “cross multiply” and isolate the unknown quantity to obtain x = 1638. So Ruth would get an additional 1638 AB. This implies that he would also receive an additional 402 BB, because 8398 AB + 1638 (additional AB) + 2062 BB + 402 (additional BB) = 12,500 PA.

So, if Ruth was just as good as he always was for these extra 1638 AB, then his projected HR total would be:

We note that the term

is nothing more than a prorating of the 714 statistic. But we are assuming that Ruth would be 5 percent less the hitter he was during the rest of his career. Therefore, the

term should be multiplied by 0.95, giving the true projected HR figure as:

Note that the left-hand side of this last equation has a “714” in both terms. If we factor out the 714 (and recall that the number “1” is always an understood coefficient of any term), we see that

In other words, if we multiply 714 by the coefficient 1.185, we get the projected cumulative HR total for Babe Ruth.

We call this the equivalence coefficient , because it gives us a “reasonable” estimate of the desired cumulative HR total, given our defined “equivalent” scenario.

With regard to Ted Williams, his additional AB compute to 2197, while his additional BB come to 576. Using the 5 percent better and 10 percent better assumptions (giving kickers of 1.05 and 1.10, respectively), we find that the equivalence coefficients for Williams are 1.299 and 1.314, respectively. So, a 5 percent better Williams would hit 521(1.299) = 677 HR, while a 10 percent better Williams would project to 521(1.314) = 685 HR.

We summarize the technique of computing the EC in Figure 5.1.

Figure 5.1 Computing the equivalence coefficient for batting

We see that the equivalence coefficient has enabled us to compile the entries in the following table, perhaps shedding some light to answer the questions we posed above:

• What would Williams’ totals be if he had not lost so much time?

• What if Ruth had not started out as a pitcher?

• Who was the greater hitter: Williams or Ruth?

Table 5.2 Williams versus Ruth using the equivalence coefficient

So, who was the greater hitter?

Some remarks are in order regarding this approach. First, the EC can be regarded as a mathematical model. As with most models, it can be tweaked. For example, we assumed that Ted Williams was “equally better” during the years he missed in both the 1940s and in the 1950s. We could have assumed that he was 10 percent better in the 1940s and 5 percent better in the 1950s. Clearly, this would have yielded different projections and made our model a bit more complicated (see the Hard Slider problem at the end of the chapter).

We also assumed that the proportions of AB to PA were constant . But if Williams was 10 percent better in the 1940s, perhaps he would have been even more selective regarding what pitches to hit, meaning that he might have had less than a total of 9903 AB, while drawing more than 2597 BB. How would this have affected his projected cumulative totals?

Also, this model could be enhanced by considering such entities as the hit-by-pitch (HBP) statistic, on-base-average (OBA) and both stolen bases (SB) and caught stealing (CS). In this way, more offensive categories would be included.

What about pitching? We mentioned legendary Dodger southpaw Sandy Koufax. Koufax recorded 2396 strikeouts (K) in 2324.3 innings pitched (IP) during his shortened career. What if, for the sake of argument, we assume that he had pitched an additional 800 innings? Can we use the EC approach with respect to pitching? Yes, we can.

To find Koufax’s strikeout EC, we basically duplicate the procedure we used with the

Alan Dean Foster

Alice; Hoffman

Allison M. Dickson

Victoria Peters

This Lullaby (v5)

Chris Taylor

Andrew Vrana

Maggie Robinson

Daniel Black