In statistical terms, what does it mean for a batter to have a consistent behavior in hitting homeruns? A hitter comes to bat many times during the season. One can think of each at-bat as a Bernoulli trial or a coin toss -- if he hits a homerun, it is a success; if he doesn't, it is a failure. (Here an at-bat is an official at-bat -- walks, hit-by-pitches, or sacrifices will be considered neither successes nor failures.) In this setting, a consistent hitter will be defined as one who has a constant probability p of hitting a homerun during a single at-bat. It is important to note that consistency does not mean that the hitter will hit homeruns in a constant pattern across a season. It does mean that the chance a player hits a homerun during a plate appearance will not change during the season.
The objective of this paper is to examine Schmidt's homerun hitting data for consistency. Specifically, is there evidence that Schmidt's homerun hitting ability, as measured by the Bernoulli probability p, changed over the course of his career? The concept of consistency is related to the notion of streakiness or dependence in a sequence of observations. We will see in Section 3 that the dates of Schmidt's homeruns over a season exhibit some clustering. If this pattern is significantly different from that of coin tossing with a constant p, there are two possible alternative models for this behavior. Perhaps the sequence of batting results is still independent, but Schmidt's ability to hit these homeruns is changing over the season. Alternatively, the clustering may be due to some dependence in the sequence of hitting results. If Schmidt is indeed streaky, then the probability that he hits a homerun will depend on his homerun performance in the preceding at-bats.
In this paper, we will not investigate Schmidt's hitting for streakiness for a couple of reasons. First, the available hitting data for Schmidt is not complete in that we do not have his hitting record for the entire sequence of at-bats during any season. Since we don't have data at this detailed at-bat level, we cannot search for streakiness among the sequence of at-bats. Second, it appears from the streakiness studies (see, for example, Tversky and Gilovich (1989) and Albright (1993)), that streakiness in sports, if it exists, is a very subtle characteristic and difficult to detect statistically. Thus we will analyze Schmidt's hitting data for consistency, where our alternative model is independent Bernoulli trials with changing values of the success probability p. In the following discussion, we will use expressions like ``hot", ``caught fire", and ``cooled down" to describe Schmidt's hitting, with the understanding that these expressions mean that Schmidt's homerun hitting ability may be changing over time.