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Journal of Statistics Education Volume 15, Number 1 (2007), jse.amstat.org/v15n1/datasets.huber.html
Copyright © 2007 by Michael Huber and Andrew Glen, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.
Key Words: Anderson-Darling Goodness-of-Fit Test; Exponential Distribution; Hitting for the Cycle; Memoryless Property; No-Hit Games; Poisson Process; Triple Plays.
From 1901 through the end of the 2004 season, there were 206 official no-hit games pitched in the American and National Leagues. According to the The Book of Baseball Records, there have been 13 “Near No-Hitters” in the Major Leagues from 1901 - 2004 (instances where the no-hitter had been broken up in extra innings), as well as 25 occurences of a pitcher not allowing a hit in an official game that was less than nine innings. Because these events do not meet the criteria set forth by Major League Baseball (MLB) as being a “No-Hit Game,” we did not include them in the data. A no-hit game (commonly known in baseball as a “no-hitter”) refers to a game in which one of the teams has prevented the other team from getting an official hit during the entire length of the game, which must be at least 9 innings by the current Major League Baseball definition. During this time span there were 225 batters who hit for the cycle – batters who had a single, a double, a triple, and a home run in the same game. In addition, there were 511 times from 1901 – 2004 in which a team turned a triple play in a game, which means that a team recorded three outs in an inning during a single at-bat. We only consider, by the way, regular season games in this article.
Obvious questions arise. First, how often do no-hitters, hitting for the cycle, and triple plays occur in a regular season? Also, can we model the number of each that we might expect to see in a season? Finally, do the chances of these events occurring change during different eras in baseball history?
for non-negative integers of x and some positive . For us the parameter will be a rate per unit time. Additionally, if X has a Poisson distribution with parameter , then both the expected value of X and the variance of X are equal to . If the annual number of no- hitters, cycles, and triple plays indeed follow Poisson processes, exponential distributions will model the distributions of times between consecutive occurences. A random variable T is said to have an exponential distribution if its probability density function is
where > 0. Additionally, if T has an exponential distribution with parameter , then the expected value of T equals 1/ and the variance of T is equal to 1/2. Note, then, that the mean and standard deviation are equal.
An important distinction of the exponential distribution is its “memoryless” property. It is well known that the only continuous distribution that models a memoryless process is the exponential distribution. The memoryless property implies that the time of the last occurrence of an event does not affect the time to the next occurrence of that event. Intuitively, we believe this to be at least approximately true of our three baseball events.
As mentioned earlier, from 1901 to 2004, there have been 206 no-hitters, 225 cycles, and 511 triple plays. These data can be readily found at a number of websites (see References). The most no-hitters pitched in a single season during this time period is eight, and the fewest is zero. There has been one season in which eight batters have hit for the cycle, and the fewest number of occurrences in a season is zero. There have been three seasons in which eleven triple plays have occurred, but only two seasons in which no triple play occurred.
Figure 1, Figure 2, and Figure 3 show the total number of occurrences for no-hitters, cycles, and triple plays per year, respectively, for 1901 – 2004. All three events have a high number of years when only one, two, or three events occured. Instances where more than three of each event occurred in a particular season were infrequent, especially in the no-hitter and cycles data sets (see Table 1). The mean number of no-hitters per year is = 1.98, or just about two no-hitters per season. The mean number of cycles over this period is = 2.16, and the mean for the number of triple plays over this period is = 4.91. Even though this last number is higher than for no-hitters or cycles, we will still consider triple plays to be rare events. To further indicate the rarity of these events, we note that from 1901 to 2004 there were 159,650 official games played. Consequently, roughly 0.13% of games were no-hitters, roughly 0.14% of games had a batter hit for the cycle, and roughly 0.32% of games had a triple play.
Figure 1. Number of No-Hit Games by Year (1901 – 2004).
Figure 2. Number of Cycles by Year (1901 – 2004).
Figure 3. Number of Triple Plays by Year (1901 – 2004).
|Count||No Hitters||Hit for Cycle||Triple Plays|
|0||18 (14.3)||18 (12.0)||2 (0.8)|
|1||30 (28.4)||27 (25.9)||6 (3.8)|
|2||21 (28.1)||16 (28.0)||12 (9.2)|
|3||21 (18.6)||21 (20.2)||12 (15.1)|
|4||6 (9.2)||13 (10.9)||17 (18.6)|
|5||3 (3.6)||5 (4.7)||18 (18.2)|
|6||3 (1.2)||3 (1.7)||9 (14.9)|
|7||2 (0.3)||0 (0.5)||11 (10.5)|
|8||0 (0.1)||1 (0.1)||5 (6.4)|
|9||0 (0.0)||0 (0.0)||7 (3.5)|
|10||0 (0.0)||0 (0.0)||2 (1.7)|
|11||0 (0.0)||0 (0.0)||3 (0.8)|
|12||0 (0.0)||0 (0.0)||0 (0.3)|
Using the probability mass function for the corresponding Poisson distribution, we can estimate the probability that there will be x rare events in a season (see Table 2). For example, the estimated probability that exactly six batters will hit for the cycle in a single season is
That is, there is an estimated 1.64% chance that exactly six batters will hit for the cycle in a season. Further estimated chances are listed in Table 2. Using Table 2 we see, for example, that there is a 97.68% chance that fewer than six batters will hit for the cycle in a single season (interestingly, in the 2004 Major League season, six batters did hit for the cycle). Also, multiplying these estimated chances by 104 – the number of seasons from 1901 through 2004 – we get the estimated counts displayed in Table 1.
|#||No Hitters||Hit for Cycle||Triple Plays|
Estimating the number of seasonal occurrences of any one of our rare phenomena by a single Poisson is at best questionable as, among other things, the number of games played during a season has varied over time. Everything else being equal, seasons with a greater number of games will tend to produce greater numbers of our rare events. As a refinement to the modeling process just presented, in Section 4 we look at separate models for each of several smaller, more homogeneous eras over the 1901 – 2004 time period.
We calculated the inter-arrival times between successive no-hitters for every season from 1901 through 2004. Using box scores found online at www.retrosheet.org, we verified that each game took place on the date given, then went to the previous day, counted all the wins and losses for all teams, and divided by two. The average inter-arrival time for a no-hitter is 772 games. Similarly, the average inter-arrival times for a cycle and triple play are 720 and 316 games, respectively.
At this point we discuss goodness-of-fit tests for the exponentiality of the data. In Figure 4, we present a comparison of the empirical distribution function (EDF) with the fitted exponential cumulative distribution function (CDF) for no-hit games, using the estimated lambda. Similar EDF versus fitted CDF graphs for cycles and triple plays may, of course, be produced.
Figure 4: EDF vs. Exponential CDF for No-Hit Games.
Graphically, no-hitter inter-arrival times seem to be well modeled by an exponential. We now look at formal statistical tests of an exponential model. We first consider Pearson’s chi-square statistic. It is easy to understand, but as we will indicate, it is problematic for testing exponentiality. To illustrate the use of the chi-square statistic, we consider the no-hitter data. A cell count of those inter-arrival times, as well as what we would expect to see for each cell, is shown in Table 3 below. Each line in the Table shows the interval for inter-arrival times in which 20 events occurred. For example, 20 no-hitters occurred with inter-arrival times (x) of 0 x < 56. Similar tables could be constructed for cycles and triple plays.
|IAT||Observed (obs)||Expected (exp)||(obs - exp)2/exp|
The sum of the last column in Table 3 is about 16.5. The corresponding p-value, on 10 degrees of freedom, is about 0.09 indicating the exponential null hypothesis for no-hitters cannot be rejected at the 0.05 significance level. The chi-square test is based on binned data. Selection of the bin size for data sets is sometimes more of an art than a science. Some choose to select bins of equal width on the horizontal axis. We have chosen to create equi-probable bin sizes instead, as this bin-size method has been shown to be unbiased and more accurate for approximating the null hypothesis (D’Agostino and Stephens, 1986, pg. 69). The basic idea behind the chi-square test is that the observed number of points in each bin should be similar to the expected counts. For goodness-of-fit in general, while the chi-square statistic is easy to understand and implement, it is problematic and suffers low statistical power. It reduces continuous data into discrete cells; thus, we lose the resolution of each inter-arrival time. It is well established that such loss of fidelity makes the chi-square statistic lack power. That is, compared to more powerful tests, this test will accept a false null hypothesis more often.
Consequently, as noted by D’Agosino and Stephens (1986), with continuous data it is much more appropriate to use EDF-based statistics that measure the squared distance between the EDF and the exponential model, thus relying on the actual value of each observation. We used the Anderson-Darling A2 statistic for testing the fit of exponential distributions (as outlined on pages 134 – 135 of D’Agostino and Stephens, 1986, a ‘case 2, rate parameter unknown’ test), and calculated adjusted test statistics of 0.81 for no-hitters (a p-value between 0.200 and 0.250), 1.03 for cycles (a p-value between 0.100 and 0.150), and a test statistic of 2.73 (a p-value less than 0.0025) for triple plays. For triple plays, there seems to be a significant departure from exponentiality. This will be further examined in the next section, where we will use the A2 statistic to examine exponentiality of our three rare events over eras within the 1901-2004 time frame.
|Dead Ball||1901 to 1919||43||22||107|
|Lively Ball||1920 to 1941||20||60||136|
|Integration||1942 to 1960||30||29||81|
|Expansion||1961 to 1976||56||31||66|
|Free Agency||1977 to 1993||37||46||81|
|Long Ball||1994 to 2004||20||37||40|
Each rare event was thus divided into these six eras and the count, mean inter-arrival times (denoted as Mean IA below), and lambda values were estimated. This information is shown in Table 5.
|No-Hitter||No-Hitter||Cycles||Cycles||Triple Play||Triple Play|
|Era||Seasons||Mean IA||Mean IA||Mean IA|
There are clear differences between the eras. For example, during the Dead Ball Era, the mean inter-arrival time of no-hitters was low, while the mean inter-arrival time of cycles per year was high. These two trends were reversed during the Lively Ball Era. The mean inter-arrival time of triple plays did not change significantly during these two eras. The Expansion Era saw an increase in no-hitters per year, while the recent (and current) Long Ball Era has seen a drop in the mean inter-arrival time of cycles. Triple play mean inter-arrival times are up in recent eras.
The Anderson-Darling Goodness-of-Fit Test was applied to each era for each rare event, to examine whether or not each subset was indicative of exponential behavior (see Table 6). Note that there are significant departures from “exponentiality” in certain subsets of the data. (We use a significance level of = 0.05 in what follows.) The data for no-hitters for example, may be exponential in the totality of the data, but is apparently non-exponential in the ‘Lively Ball’ and ‘Free Agent’ eras. Interestingly in the case of triple plays, the process is non-exponential as a total process; however, all but one of the eras fails to reject exponentiality, albeit with different arrival rates. This appears to be evidence of a non-homogeneous Poisson process. Using a Potthoff-Whittinghill test for homogeneity of a Poisson process with parameter lambda unknown (Potthoff and Whittinghill 1966), we reject homogeneity with a very small p-value (<<0.001). Thus for triple plays, within eras we generally have no reason to reject exponential inter-arrivals, but between eras there are significantly different mean inter-arrival times.
|Stat Era||Years||A2||n||adj A2||adjusted p-value||Exponential?|
|all triple plays||1901 - 2004||2.73||511||2.73||< 0.0025||Reject|
|Dead Ball||1901 - 1919||0.48||107||0.48||> 0.25||Do Not Reject|
|Lively Ball||1920 - 1941||0.67||136||0.68||> 0.25||Do Not Reject|
|Integration||1942 - 1960||0.35||81||0.35||> 0.25||Do Not Reject|
|Expansion||1961 - 1976||1.05||66||1.06||= 0.01||Reject|
|Free Agent||1977 - 1993||0.51||81||0.51||> 0.25||Do Not Reject|
|Long Ball||1993 - 2004||0.72||40||0.72||> 0.25||Do Not Reject|
|all Cycles||1901 - 2004||1.03||225||1.03||0.10 < p < 0.15||Do Not Reject|
|Dead Ball||1901 - 1919||2.96||22||3.04||< 0.0025||Reject|
|Lively Ball||1920 - 1941||0.94||60||0.95||0.10 < p < 0.15||Do Not Reject|
|Integration||1942 - 1960||0.72||29||0.73||> 0.25||Do Not Reject|
|Expansion||1961 - 1976||0.72||31||0.73||> 0.25||Do Not Reject|
|Free Agent||1977 - 1993||1.01||46||1.02||0.10 < p < 0.15||Do Not Reject|
|Long Ball||1993 - 2004||0.90||37||0.91||0.15 < p < 0.20||Do Not Reject|
|all No-Hitters||1901 - 2004||0.81||206||0.81||0.20 < p < 0.25||Do not Reject|
|Dead Ball||1901 - 1919||0.96||43||0.97||0.10 < p < 0.15||Do Not Reject|
|Lively Ball||1920 - 1941||1.43||20||1.47||0.025 < p < 0.05||Reject|
|Integration||1942 - 1960||0.95||30||0.97||0.10 < p < 0.15||Do Not Reject|
|Expansion||1961 - 1976||0.63||56||0.64||> 0.25||Do Not Reject|
|Free Agent||1977 - 1993||1.92||37||1.95||0.01 < p < 0.025||Reject|
|Long Ball||1993 - 2004||1.08||20||1.11||0.05 < p < 0.10||Do Not Reject|
Devore, J. L. (1995), Probability and Statistics for Engineering and the Sciences, Fourth Edition, New York, NY: Duxbury Press.
Siwoff, S. (2004), The Book of Baseball Records, 2004 Edition, New York, NY: Seymour Siwoff, Elias Sports Bureau, Inc.
Potthoff, R. F. and Whittinghill, M. (1966), “Testing for Homogeneity: II. The Poisson Distribution”, Biometrika, 53(1/2), pp. 183-190.
Internet site for triple play data: tripleplays.sabr.org/tp_sum.htm accessed 1 Jul 2006.
Internet site for hitting for the cycle facts: www.Baseball-Almanac.com/feats/feats16d. shtml accessed 1 Oct 2004.
Internet site for no-hit game definition: mlb.mlb.com/NASApp/mlb/mlb/official_info/official_rules/foreword.jsp accessed 11 Feb 06.
Internet site for baseball eras: www.netshrine.com/era.html accessed 10 Feb 2006.
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Department of Mathematical Sciences
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