GABRIEL COSTA, MICHAEL HUBER, & JOHN SACCOMAN Cumulative Home Run Frequency and the Recent Home Run Explosion ritish Prime Minister Benjamin Disraeli once remarked that Figure 1. Babe Ruth's Cumulative Home Run Ratio there are three kinds of falsehoods: lies, damned lies, and 0.10 statistics. While this may be debated on political and philo­ B 0.09 .,.;N'" ~- sophicallevels, there are circumstances where statistics (and 0.08 graphs and charts) can shed some truth and light. Perhaps this 0.07 r is one ofthose times. 0.06 I In this paper we define a measure which we call the cumu­ 0.05 I lative home run ratio (CHRJ. We consider every player in history 0.04 I who has slugged 500 home runs, and compute their year-to-year 0.03 I\.. / ~ total of home runs (HR) divided by their at-bats (AB). After which, 0.02 I we compare this accumulated HR/AB ratio to the slugger's corre­ 0.01 I sponding age. 0.00 J For example, from Table 1 below, in 1914 Babe Ruth hit a HR 17 20 25 30 35 40 42 in 10 AB. This gives the 19-year-old Ruth a proportion of HR to AB Figure 1 shows the plot of Ruth's age (horizontal axis) versus equal to 0.000000. The next year (1915), Ruth hit 4 HR in 92 his CHR (vertical axis) over his career. Ruth's CHR seems to level AB; Ruth's cumulative home run ratio over his two years as a pro off in his mid·to-Iate 20s, and remains fairly constant for the next becomes 0.039216 (4 HR in 102 AB) at age 20. Since he hit 3 10 years or so. Graphically, we can see that Ruth's CHR levels off HR in 136 AB in 1916, his CHR at the age of 21 is computed to be toward a limiting value of roughly .085 or 8.5 HR per 100 AB for 0.029412 (? HR in 238 AB J. The data in Table 1 continues Ruth's his career. In fact, this limiting value seems to "converge" to a major league career and provides his CHR for each season. "natural" or a "predictable" number, since from 1928 through the end of his career in 1935, his CH R hovered near this value. Table 1. Babe Ruth One particular question which arises is the following: "Did most, if not all, 500+ home run hitters level off with respect to Year AB HR Age CHR CHR around the same age as Ruth?" 1914 10 19 0.000000 In Figure 2, we present the CHR for Hank Aaron, the all-time 1915 92 4 20 0.039216 home run leader. We see here that Aaron's CHR, like that of Babe 1916 136 3 21 0.029412 1917 123 2 22 0.024931 Ruth's, has relatively little variation from his mid-20!s onward. We 1918 317 11 23 0.029499 note that his final CHR value converges nicely to approXimately 1919 432 29 24 0.044144 .061, very near his CHR from the age of 28 onward. 1920 458 54 25 0.065689 1921 540 59 26 0.076850 1922 406 35 27 0.078361 Figure 2. Hank Aaron's Cumulative Home Run Ratio 1923 522 41 28 0.078393 0.10 1924 529 46 29 0.079663 1925 359 25 30 0.078746 0.09 1926 495 47 31 0.080561 0.08 1927 540 60 32 0.083888 0.07 1928 536 54 33 0.085532 0.06 1929 499 46 34 0.086086 0.05 ~ 1930 518 49 35 0.086763 1931 534 46 36 0.086716 0.04 r 1932 457 41 37 0.086899 0.03 ur 1933 459 34 38 0.086159 0.02 1934 365 22 39 0.085025 0.01 1935 72 6 40 0.085010 0.00 17 20 25 30 35 40 42 37 THE BASEBALL RESEARCH JOURNAL At the end of this article, we have provided the cumulative Figure 3. Mark McGwire's Cumulative Home Run Ratio home run ratio charts-in roughly chronological order, dating 0.10 ~ from their first home run-for all 20 members of the 500 Home 0.09 jl ./' Run Club (see Appendix). 0.08 ~/ 0.07 /'\.. Some interesting observations about CHR can be made: -~ 0.06 / • Sluggers like Mel Ott and Frank Robinson have CHRs which 0.05 " are nearly constant, as reflected in their charts by virtually 0.04 flat lines. 0.03 • Most 500+ home run hitters attain a final CHR which could 0.02 have been accurately estimated by the time they reached 0.01 their m id-to-late 20s. 0.00 20 25 30 35 40 42 • Multi-seasonal 50+ home run hitters like Ruth, Foxx, Mantle, 17 and Mays all level off by their mid-20s. • Third baseman Eddie Mathews is the only member of the Figure 4. Sammy Sosa's Cumulative Home Run Ratio 500 Home Run Club to have peaked in his early 20s and 0.10 then have his CHR actually decline over his career. 0.09 • For the most part, consistency in CHR is independent of life­ 0.08 time batting average: Ted Williams at .344, Mel Ott at .304, 0.07 ... ~ or Reggie Jackson at .262. 0.06 • For the most part, consistency in CHR is independent of 0.05 7' players with frequent injuries (e.g., Mickey Mantle), inter­ 0.04 7 rupted careers (e.g., Ted Williams and Willie Mays), or lon­ 0.03 /' gevity of careers (e.g., Willie McCovey). 0.02 r- 0.01 We have seen that as the sluggers' careers progressed, their 0.00 17 20 25 30 35 40 42 CHR reached a particular value and remained there. This was true of Aaron, who had a slight increase in CHR in his early 301s, but very little "variability" thereafter; it was true of Willie Mays; Figure S. Barry Bonds' Cumulative Home Run Ratio· ditto for Mike Schmidt. In fact, the CHR for Hall of Famers such as 0.10 Jimmie Foxx, Mel Ott, Frank Robinson, and Reggie Jackson have 0.09 virtually the same graphical trends as the Sluggers who played 0.08 before them. The bottom line appears to be that the great slug­ 0.07 ~ .-/ gers have CHRs which level off (or reach a limit) as they enter 0.06 their late 201S. 0.05 ~ ~ While the observations above are generally true, we find 0.04 v """"" exceptions, particularly with players who starred over the past 0.03 decade or so. The home run explosion over the past 10 years has 0.02 generated many questions spanning quite a few areas. For many, 0.01 the names of Mark McGwire, Sammy Sosa, and Barry Bonds bring 0.00 these questions and a host of emotions to the foreground. How 17 20 25 30 35 40 42 can such players hit home runs with an increasing frequency so 'This pattern also appears with regards to Rafael Palmeiro (see Appendix). relatively late in their careers? Is the pitching that much worse? Are other factors involved? Figures 3,4, and 5 represent the CHRs of the aforementioned McGwire; Sosa, and Bonds. The trends of these graphs appear to be significantly different than the CHRs of the others. In fact, FATHER GABRIEL B. COSTA and LIEUTENANT COLONEL MICHAEL R. HUBER the following three graphs reflect unparalleled performances, teach mathematics at the United States Military Academy; PROFESSOR because of the increasing-rather than the leveling-off-CHRs. JOHN T. SACCOMAN teaches mathematics and computer science at Seton Hall University. They also teach courses on sabermetrics. Their all-time favorite players are Babe Ruth, Paul Blair and Willie Mays, respectively. All three authors hold doctorates in mathematics, are members of SABR and have previously published in the BRJ. 38 THE BASEBALL RESEARCH JOURNAL Babe Ruth's CHR Jimmie Foxx's CHR 0.10 0.10 0.09 0.09 ~ 0.08 ~ 0.08 0.07 r 0.07 pP ..... 0.06 f 0.06 0.05 1 0.05 ~ 0.04 J 0.04 ~ 0.03 I\. .I 0.03 I I~ 0.02 0.02 l' 0.01 J 0.01 f 0.00 J 0.00 J l7 20 25 30 35 40 42 l7 20 25 30 35 40 42 Mel Ott's CHR Mickey Mantle's CHR 0.10 0.09 0.08 0.07 ~ Jo ~ 0.06 ...." 0.05 """ ./ I'" ....:/" 0.04 f7 0.03 I 0.02 I 0.01 L 0.00 ~ 17 20 25 30 35 l7 20 25 3D 35 40 42 Mickey Mantle 39 THE BASEBALL RESEARCH JOURNAL Ted Williams's CHR Willie Mays's CHR 0.10 0.09 0.08 0.07 LOo-. 0.06 ~ 0.05 o....r I v j 0.04 0.... 0.03 0.02 0.01 0.00 17 20 25 30 35 17 20 25 30 35 40 42 Eddie Mathews's CHR Ernie Banks's CHR 0.10 0.09 0.08 0.07 ~ 0.06 I" 1 '\ ~ 0.05 a 0.04 \/ 0.03 If 0.02 0.01 0.00 17 20 25 30 35 17 20 25 30 35 40 42 Hank Aaron's CHR Harmon Killebrew's CHR 0.10 0.09 0.08 --.0..".,. 0.07 ~ 0.06 r 0.05 ~ I 0.04 r ~ I 0.03 ar 0.02 I 0.01 I 0.00 0.00 I 17 20 25 30 35 40 42 17 20 25 30 35 40 42 Frank Robinson's CHR Willie McCovey's CHR 0.10 0.10 0.09 0.09 0.08 0.08 0.07 0.07 .J'Oo.