To understand if Schmidt was a consistent homerun hitter, we will compare his homerun statistics with simulation results from a set of consistent Mike Schmidt clones who do hit homeruns with a constant probability of success during the years 1973 to 1987. These consistent clones will have careers that match Mike Schmidt's career in terms of the number of games played, the number of season at-bats for all years, and the overall rate of hitting a homerun.
The first step in setting up this simulation is to decide on the value for the constant probability of hitting a homerun.
During Schmidt's best 14 years, he hit 511 homeruns during 7413 at-bats for a overall rate of 511/7413 = .07. In the following, we use .07 for this constant probability.
Next, we have to define a schedule of games and at-bats for our consistent hitters to play in our simulation.
We want this schedule to mimic Schmidt's actual batting schedule during his career. Our basic data are the days on which Schmidt hit his homeruns. We also know from Table 1 the total number of games and number of at-bats for each of Schmidt's seasons.
Jay Bennett (personal communication) provided the printed preseason schedule of the Phillies for these 14 years. Other information about Schmidt's career is available from
Thorn and Palmer (1993) and Westcott (1995).
We note that Schmidt had only one significant injury during this 14 year period. This injury kept him from playing for a two-week period in May, 1982.
All baseball seasons generally start the first Monday in April and continue for 26 weeks or 182 days. The only short season occurred in 1981 where there was a player's strike and baseball play was canceled for 39 days.
To describe how we define a hitting schedule in our simulation, consider the year 1975. Schmidt hit 38 homeruns in 158 games that he played that season. We know the dates of the entire season of 162 Phillies games and the 34 days on which he hit his 38 homeruns. (There were four days on which Schmidt hit two homeruns this season.) We know Schmidt played 158 games, of which 158-34=124 were games on which no homeruns were hit. To find dates for these 124 no-homerun games, we randomly select 124 dates from the 162-34=128 available game days during the season. In a similar fashion, we randomly select no-homerun dates for all of Schmidt's seasons.
After we have the dates for all of Schmidt's games assigned, we have to allocate a fictional number of at-bats for each of his games. All that we know is the total number of at-bats for each season. We assume that Schmidt had 2, 3, 4, or 5 at-bats each game. To find the distribution of games of each type, we look at available at-bat records for a player similar to Schmidt. Von Hayes was a hitter for the Phillies in 1989. His batting record in 1989 was similar to Mike Schmidt in terms of the total number of at-bats (540), type of hitting (26 homeruns and 101 walks) and position in the batting order (middle). Looking at the boxscores for the 1989 season, Hayes had 1, 2, 3, 4, and 5 at-bats for the games he played with respective percentages 3, 7, 32, 47, and 12.
For Schmidt, we use a distribution of at-bats similar to that of Von Hayes. Using this distribution and the total number of at-bats for each season, we determine the number of games of each type. For 1975, for example, Schmidt had 562 total at-bats in 158 games. We assume that Schmidt had
13 games with 2 at-bats, 60 games with 3 at-bats, 69 games with 4 at-bats, and 16 games with 5 at-bats.
68 games with 3 at-bats. This distribution is similar to that of Von Hayes and gives the correct total number of at-bats (562) for this season. Finally, we assign these at-bat numbers randomly to the 158 games that Schmidt played in 1975. This procedure was repeated for all of Schmidt's seasons.
Now that we have created a season of game days and at-bats for each of Schmidt's 14 years, we can simulate homerun hitting for our ``consistent" hitter.
For every at-bat during each season, we simulate a coin toss where the probability of a homerun is .07.
By simulating a long sequence of these coin flips, we simulate a career of 13 seasons for our hypothetical hitter and keep track of the dates that each homerun was hit.
This entire process is repeated 2000 times to get homerun statistics for a large number of consistent hitters with similar careers as Schmidt.