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STATISTICAL MODELS FOR STUDENT PROJECTS WITH SPORTS THEMES

Robin H. Lock
Department of Mathematics, Computer Science and Statistics, St. Lawrence University,
United States of America
rlock@stlawu.edu

We describe several types of student project assignments that involve applications of statistical
models to address questions arising from sports data. Although we illustrate these ideas with
examples from specific sports, our goal is to provide sufficiently general guidelines to allow
instructors to adapt and extend the topics to different sports, teams, leagues or levels of play. Some
of the projects are accessible to students at the introductory levels while others are more
appropriate for a second course or even an undergraduate capstone/thesis. Topics include Bill
James’ so-called “Pythagorean law” for estimating team winning percentages, investigations of
home field advantage, logistic regressions on the chance of winning a match based on boxscore
statistics, the use of empirical Bayesian Stein estimators to project player performance over a full
season based on early season results, and methods for modeling outcomes in seeded tournaments.

INTRODUCTION
Some students are avid sports fans and/or active participants on athletic teams. Instructors
can find lots of questions that are of interest to sports enthusiasts and also serve to illustrate
important concepts about how we use techniques of statistics to address practical issues. Our goal
in this paper is to identify some common questions and templates of projects and activities that
appeal to students with interests in sports. In most of our examples (for this paper and in class) we
use data from professional sports that are popular in the United States: Major League Baseball
(MLB), the National Basketball Association (NBA), the National Football League (NFL) and the
National Hockey League (NHL) as well as various college/university level sports sponsored by
National Collegiate Athletic Association (NCAA). Obviously, these can be adjusted to sports,
teams and leagues that are more relevant to your own country and students.

HOMEFIELD ADVANTAGE
The concept of an advantage for the home team is well established among sports fans. But
how big an effect is it? This question provides lots of avenues for student investigations involving
inference for one or two means or proportions. For example, Figure 1 shows the difference
between points scored by the home and away teams for all 256 games from the NFL’s 2009 regular
season. One popular rule of thumb is that home field in (American) football is worth about an extra
field goal (3 points) for the home team. For this season the average margin was slightly less than
that, +2.21 points with a standard deviation of 16.48 points. A 95% confidence interval for the
mean size of the homefield advantage in the NFL would be between 0.2 and 4.2 points.

Figure 1. Homefield margins for n=256 NFL games in the 2009 regular season

Figure 1 also shows that the home team won 146 of the 256 games in the 2009 NFL
regular season. Treating this as a sample of all NFL games, we would estimate the proportion of
times the home team wins to be and a 95% confidence interval for the
In C. Reading (Ed.), Data and context in statistics education: Towards an evidence-based society. Proceedings of the
Eighth International Conference on Teaching Statistics (ICOTS8, July, 2010), Ljubljana, Slovenia. Voorburg, The
Netherlands: International Statistical Institute. www.stat.auckland.ac.nz/~iase/publications.php [© 2010 ISI/IASE]
ICOTS8 (2010) Invited Paper Lock

proportion of home winners would go from 51% to 63%. Note that the absence of any homefield
advantage would mean an average margin of zero and winning proportion of 50%, both of which
lie just outside of the respective confidence bounds. Thus one might also use the data to test (as a
mean or a proportion) whether a homefield advantage exists at all. For the 2009 NFL data the
respective p-values for these (one tail) tests would be 0.017 for mean margin and 0.012 for the
proportion of home wins - both relatively significant and consistent with each other.
Another way to explore possible advantages to playing at home is to compare team or
individual performance statistics. Table 1 shows shooting results for the 2009 MVP of the NBA,
LeBron James of the Cleveland Cavaliers, broken down between home and road games. Although
he had a higher proportion of field goals and free throws made at home, his three point shooting
was better on the road and none of these proportions is significantly different between home and
away (as shown by the p-values for a two sided test for each type of shot). One curious observation
is that he had quite a few more attempts in each of these categories in road games. So his average
number of field goals attempted per game at home is
significantly less (p-value<0.0001) than his mean attempts when playing away from Cleveland
.

Table 1. Home vs. road shooting proportions for LeBron James in 2008-2009

Field Goals 3 Point Shots Free Throws

Home Road Home Road Home Road
Made 354 435 49 83 257 337
Attempts 705 908 148 236 323 439
Attempts 0.502 0.479 0.331 0.352 0.796 0.768
P-value 0.36 0.68 0.36

The sort of analysis is the previous paragraph can be turned into a homework assignment,
small project or even an in-class activity (assuming an internet connection to find the data) for
sports-minded students. Have them pick a favorite team or player and statistic and test for a
difference in performance at home and on the road. Fortunately the data, including the home/road
splits, are often easily accessible on the web.

MODELING WINNING PERCENTAGE

Offense or defense?
One of the age-old debates among fans of many sports is relative merits of a prolific
offense vs. a stingy defense. While obviously a team would like to excel in both areas, does one
aspect of the game have a stronger influence on winning than the other? Figure 2 shows the
relationships between goals scored (per game) and goals allowed (per game) and the standings
points earned by 30 teams in the National Hockey League (NHL) during the 2008-2009 regular
season. We use standings points rather than straight winning percentage since ties are common in
ice hockey and the NHL uses an overtime and then sudden death shootout to award either one or
two points in tie games. Although the correlations and percentages of variability explained by each
predictor on its own are similar (r2=0.58 for goals scored, r2=0.51 for goals allowed) we might give
a slight nod in favor of offense as the slightly stronger predictor of team success for these data.
Of course, most sports fans will agree that performing well on both offense and defense is
the best way to gain a high winning percentage. If we use a multiple regression model for the 2008-
2009 NHL data we obtain the following fitted model

and R2=91.7%, indicating that a significantly greater proportion of the variability in standings
points can be explained by accounting for both offensive and defensive performances of NHL
teams in the same model. But multiple regression might often be beyond the scope of an
introductory statistics course. No problem – one can easily capture a similar effect creating a new
predictor Diff as the difference between offensive and defensive scoring rates for each team and
use a simple linear regression model. For the 2008-9 NHL data the prediction equation becomes

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and the R2 value remains at 91.7%. Might the ratio (GoalsFor/GoalsAgainst) be a better predictor?
It would be relatively easy for your students to check this or other possibilities.

Figure 2. Predicting NHL standings points based on goals scored or allowed

“Pythagorean” Theorems
Here’s another example of predicting winning percentage based on scoring/defending
abilities. Bill James (1981) introduced a formula in his Baseball Abstract for estimating the
winning percentage of a team based on its runs scored (RS) and runs allowed (RA).

Given the use of squares and sums of squares this was dubbed the “Pythagorean” theorem
of baseball. Later analysis showed that a better exponent for baseball was closer to 1.83 and other
sports might have quite different exponents depending on the nature of scores in the sport. If we
assume a general formula of the form

the question of interest is how do we estimate a value for the exponent,
? Although this is a non-
linear relationship, it is not difficult for a student with a spreadsheet to start with data on winning
percentage along with points scored and allowed (either average per game or total for a season),
then do a least squares analysis by “brute force”. They can compute a column of “predicted”
winning percentages using the formula above for a specific value of
stored in some cell, put the
difference from the actual winning percentages in another column and add a formula to compute
the sum of squared errors. Once this is all set up in a spreadsheet, it doesn’t take long to do a “trial-
and-error” search to find an optimal exponent. For example, using data for the 2008-2009 regular
season for the 30 teams in the NBA, an exponent of minimizes the sum of squared
errors between estimated and actual winning percentages at 0.0134. For the 2008-2009 NHL data
of the earlier example .

WHO SHOULD WIN?

Binary Logistic Regression

A project for a second statistics courses asks students to find their own data to use for
fitting a binary logistic regression model. Once again, sports are a popular choice and a standard
template for a project is to use information from a sample of boxscores from a favorite sport to
model the probability of winning a contest (avoiding the obvious predictors of the amount each
team scores!). For example, how is the probability of winning a baseball game related to the
number of hits a team gets in the game? Figure 3 shows a fitted logistic regression curve for the
probability of a win based on the number of hits using boxscores from each of the 2009 World

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Series Champion New York Yankees regular season games. The actual game results (0=loss,
1=win) are shown with jittering on the plot. The equation of the estimated logistic curve is

which produces an odds ratio of . Thus we see roughly a 24% improvement in the
odds of winning a game for every extra hit the Yankees get.

Figure 3. Logistic curve for Yankee wins based on hits

Seeded Tournaments
Many elimination tournaments (e.g. Grand Slam Tennis, World Matchplay Golf, many NCAA
Championships) are organized with a seeding system where the entrants (players or teams) are
ranked by perceived ability and a draw pits the top seeds against the lowest seeds in the early
rounds. Assuming the seeding is an accurate reflection of relative ability we should be able to
estimate a probability of either team/player winning a particular match, based on the relative seeds.
For example, the NCAA’s Men’s Basketball Tournament (dubbed “March Madness”)
generates much interest among fans (including many college students) who attempt to forecast the
bracket, i.e. predict the results for all 64 teams in the tournament. The teams are separated into four
regions with teams being seeded 1 to 16 within each region. The number #1 (top) seed plays the
#16 (lowest) seed in the first game of each region. In the 24 years since this format was instituted
the #16 seed has yet to win a game (although some have come very close to an upset). Thus the
NCAA tournament might appear to be fairly predictable, but most fans will tell you that the
predictability wanes as other seeds are compared. So suppose that Team A seeded #i plays Team B
seeded #j–how might we estimate the probability that Team A wins based only on the seeds?
Here’s one easy method

but we might prefer to use results from past tournament play to obtain more accurate estimates.
Berry (2000) provides such estimates based on NCAA Men’s Basketball tournament results from
1986-2000. A former student, Jared Fostveit (2008), developed his own method as part of a senior
honors project. He assumed that the strength of teams would decrease by a factor of 0<
<1 as the
seed increased, and that the probability of winning would be based on relative strengths

By translating this into a logistic regression model with the difference in seeds as the
predictor for who wins the game, Jared came up with a way to estimate
based on past tournament
results. When fit to the past NCAA basketball tournaments, the estimate was , so a #1
seed should beat the #16 seed about 93% of the time, but only have 67% chance of beating a #4

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seed. He also looked at 20 years of data from the World Matchplay Golf Championships which has
a similar structure of 64 players seeded into four 16-player brackets. For the golf data the estimated
, so he concluded that the NCAA March Madness was much more predictable than
matchplay golf.

ASSESSING PLAYER PERFORMANCE

A Multiple Regression Project

Students looking for their own data for a multiple regression project are often drawn to
sports examples. A common theme is to try to model how one statistic (e.g., home runs hit by MLB
players in a season) might depend on other statistics (batting average, RBI, games played, age,
weight, position,...). These projects often introduce interesting discussions, for example on the
effects of multicollinearity when many of the predictors are strongly related to each other. The
regression output in Figure 4 was produced from data on all (n=157) MLB players in the 2009
season with at least 500 plate appearances (Note: Students often restrict their “sample” to a single
favorite team in which case playing time becomes a dominating factor in many models). We
observe the curious fact that in this model both batting average (AVG) and number of at bats (AB)
have significant negative coefficients in the model when runs batted in (RBI) is included. On their
own AVG has little linear relationship with HR (r=-0.0008) and AB is mildly significant and
positively correlated with HR (r=0.173, p-value=0.03).

Figure 4. Multiple regression to predict home runs for MLB players

Empirical Bayes – Stein Estimators

Efron & Morris (1975, 1977) and Everson (2007) have written about the use of James-
Stein estimators in an empirical Bayesian approach to parameter estimation. Each illustrates the
method with a sports-based example - using early season player/team statistics (MLB player
batting averages for Efron & Morris, NBA team scoring averages for Everson) to predict full
season results. The gist of the method is to develop a posterior distribution for a parameter (batting
average or team scoring average) based on a weighted combination of the early season
performance of the individual player/team and the distribution of the other players/teams in the
league. An undergraduate senior honors student, Joe Cleary (2008), adapted this method to predict
save percentages for NHL goalies. He started with the observed save percentage for each regular
NHL goalie over the first ten games. The basic idea of the Stein estimator is to adjust the individual
save proportions from the initial games ( ) to a weighted average that takes into account the
distribution of all the goalies ( ) over a full season (for example, the previous NHL season).

Thus a goalie who starts the season “hot” with several strong games will find his estimated
full season proportion scaled back a bit towards the rest of the league and one who struggles in the
first few outings will be helped. The weighting factor ( ) is different for each goalie and depends
on the number of shots faced and results in the early ten games. Figure 5 shows a plot of initial

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estimates and confidence bounds based on the Stein estimates along with traditional proportion
confidence intervals and the final season save proportions for each of the goalies. The Stein
estimates tended to be closer to the full season save proportions (sum of squared errors less than
half the size of the errors based on sample proportions over the first ten games) and the Stein
intervals are narrower but still as accurate.

Figure 5. Stein and traditional estimates and confidence bounds for NHL save proportions

CONCLUSION
While recognizing that all students are not equally enthusiastic about sports-based
examples, projects and activities, they do provide a good opportunity for motivating some students
and giving them real experience in collecting data and performing statistical analyses to address
questions of interest to them. The widespread availability of sports data on the web makes it
feasible to carry out projects such as those outlined in this paper and many more that would be
suggested by students/sports fans in these and similar areas.

REFERENCES
Berry, S. (2000) My Triple Crown. Chance, 13(3).
Cleary, J. (2008). Using Stein Estimation to Predict Performance. Senior thesis.
Efron, B., & Morris, C. (1975). Data Analysis Using Stein’s Estimator. JASA, 70, 311-319.
Efron, B., & Morris, C. (1977). Stein’s Paradox in Statistics. Scientific America, 236(5), 119-127.
Everson, P. (2007). Stein’s Paradox Revisited. Chance, 20(3), 49-56.
Fostveit, J. (2008). Madness of March: More Predictable than Golf. Senior thesis.
James, B. (1981). Baseball Abstract 1981. Self published.

WEB SOURCES
Many links related to this paper can be found at it.stlawu.edu/~rlock/icots8sports.html
2008-9 NBA data: espn.go.com/nba/standings?year=2009
2009 NFL data: www.pro-football-reference.com/years/2009/games.htm
2008-9 NHL data: espn.go.com/nhl/standings?year=2009
2009 MLB batting: www.cbssports.com/mlb/stats/playersort/regularseason/yearly/MLB/ALL
LeBron James’ 2009 splits: www.basketball-reference.com/players/j/jamesle01/splits/2009/
LeBron James’ 2009 games: www.basketball-reference.com/players/j/jamesle01/gamelog/2009/
2009 NY Yankees data: www.baseball-reference.com/teams/tgl.cgi?team=NYY&t=b&year=2009

International Association of Statistical Education (IASE) www.stat.auckland.ac.nz/~iase/