British Prime Minister Benjamin Disraeli once remarked that

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1 GABREL COSTA, MCHAEL HUBER, & OHN SACCOMAN Cumulative Home Run Fequency and the Recent Home Run Explosion Bitish Pime Ministe Benjamin Disaeli once emaked that thee ae thee kinds of falsehoods: lies, damned lies, and statistics. While this may be debated on political and philosophicallevels, thee ae cicumstances whee statistics (and gaphs and chats) can shed some tuth and light. Pehaps this is one ofthose times. n this pape we define a measue which we call the cumulative home un atio (CHR. We conside evey playe in histoy who has slugged 500 home uns, and compute thei yea-to-yea total of home uns (HR) divided by thei at-bats (AB). Afte which, we compae this accumulated HRAB atio to the slugge's coesponding age. Fo example, fom Table 1 below, in 1914 Babe Ruth hit a HR in 10 AB. This gives the 19-yea-old Ruth a popotion of HR to AB equal to The next yea (1915), Ruth hit 4 HR in 92 AB; Ruth's cumulative home un atio ove his two yeas as a po becomes 9216 (4 HR in 102 AB) at age 20. Since he hit 3 HR in 136 AB in 1916, his CHR at the age of 21 is computed to be 9412 (? H R in 238 AB. The data in Table 1 continues Ruth's majo league caee and povides his CHR fo each season. Table 1. Babe Ruth Yea AB HR Age CHR Figue 1. Babe Ruth's Cumulative Home Run Ratio 17 \...,.;N'" Figue 1 shows the plot of Ruth's age (hoizontal axis) vesus his CHR (vetical axis) ove his caee. Ruth's CHR seems to level off in his mid to-ate 20s, and emains faily constant fo the next 10 yeas o so. Gaphically, we can see that Ruth's CHR levels off towad a limiting value of oughly.085 o 8.5 HR pe 100 AB fo his caee. n fact, this limiting value seems to "convege" to a "natual" o a "pedictable" numbe, since fom 1928 though the end of his caee in 19, his CH R hoveed nea this value. One paticula question which aises is the following: "Did most, if not all, 500+ home un hittes level off with espect to CHR aound the same age as Ruth?" n Figue 2, we pesent the CHR fo Hank Aaon, the all-time home un leade. We see hee that Aaon's CHR, like that of Babe Ruth's, has elatively little vaiation fom his mid-20!s onwad. We note that his final CHR value conveges nicely to appoximately.061, vey nea his CHR fom the age of 28 onwad. Figue 2. Hank Aaon's Cumulative Home Run Ratio 17 u

2 At the end of this aticle, we have povided the cumulative THE BASEBALL RESEARCH OURNAL home un atio chats-in oughly chonological ode, dating fom thei fist home un-fo all 20 membes of the 500 Home Run Club (see Appendix). Some inteesting obsevations about CHR can be made: Slugges like Mel Ott and Fank Robinson have CHRs which ae nealy constant, as eflected in thei chats by vitually flat lines. Most 500+ home un hittes attain a final CHR which could have been accuately estimated by the time they eached thei m id-to-late 20s. Multi-seasonal 50+ home un hittes like Ruth, Foxx, Mantle, 17 and Mays all level off by thei mid-20s. Thid baseman Eddie Mathews is the only membe of the 500 Home Run Club to have peaked in his ealy 20s and then have his CHR actually decline ove his caee. Fo the most pat, consistency in CHR is independent of lifetime batting aveage: Ted Williams at.344, Mel Ott at.304, o Reggie ackson at.262. Fo the most pat, consistency in CHR is independent of playes with fequent injuies (e.g., Mickey Mantle), inteupted caees (e.g., Ted Williams and Willie Mays), o longevity of caees (e.g., Willie McCovey). We have seen that as the slugges' caees pogessed, thei CHR eached a paticula value and emained thee. This was tue of Aaon, who had a slight incease in CHR in his ealy 30 1 s, but vey little "vaiability" theeafte; it was tue of Willie Mays; ditto fo Mike Schmidt. n fact, the CHR fo Hall of Fames such as immie Foxx, Mel Ott, Fank Robinson, and Reggie ackson have vitually the same gaphical tends as the Slugges who played befoe them. The bottom line appeas to be that the geat slugges have CHRs which level off (o each a limit) as they ente thei late 201S. While the obsevations above ae geneally tue, we find exceptions, paticulaly with playes who staed ove the past decade o so. The home un explosion ove the past 10 yeas has geneated many questions spanning quite a few aeas. Fo many, the names of Mak McGwie, Sammy Sosa, and Bay Bonds bing these questions and a host of emotions to the foegound. How can such playes hit home uns with an inceasing fequency so elatively late in thei caees? s the pitching that much wose? Ae othe factos involved? Figues 3,4, and 5 epesent the CHRs of the afoementioned McGwie; Sosa, and Bonds. The tends of these gaphs appea to be significantly diffeent than the CHRs of the othes. n fact, the following thee gaphs eflect unpaalleled pefomances, because of the inceasing-athe than the leveling-off-chrs. Figue 3. Mak McGwie's Cumulative Home Run Ratio jl. \.. - " Figue 4. Sammy Sosa's Cumulative Home Run Ratio 7' Figue S. Bay Bonds' Cumulative Home Run Ratio v """"" 30 'This patten also appeas with egads to Rafael Palmeio (see Appendix) FATHER GABREL B. COSTA and LEUTENANT COLONEL MCHAEL R. HUBER teach mathematics at the United States Militay Academy; PROFESSOR OHN T. SACCOMAN teaches mathematics and compute science at Seton Hall Univesity. They also teach couses on sabemetics. Thei all-time favoite playes ae Babe Ruth, Paul Blai and Willie Mays, espectively. All thee authos hold doctoates in mathematics, ae membes of SABR and have peviously published in the BR. 38

3 THE BASEBALL RESEARCH OURNAL Babe Ruth's CHR immie Foxx's CHR f 1 \.. l l7 pp l' f Mel Ott's CHR Mickey Mantle's CHR..." '" L """ 30 f7....:" l o 3D Mickey Mantle 39

4 THE BASEBALL RESEARCH OURNAL Ted Williams's CHR Willie Mays's CHR o... v LOo j Eddie Mathews's CHR Enie Banks's CHR " 1 a '\ \ f Hank Aaon's CHR Hamon Killebew's CHR a ".,. 30 Fank Robinson's CHR Willie McCovey's CHR \.....1> '" \. V'" o..c.'oo

5 THE BASEBALL RESEARCH OURNAL Reggie ackson's CHR Mike Schmidt's CHR a...:"" f 1_ 0, Eddie Muay's CHR Bay Bonds's CHR.bOOC. v Mak McGwie's CHR i\ \ _ Rafael Palmeio's CHR A 11 \... (), Ken Giffey.'s CHR Sammy Sosa's CHR.. -"" 30 30