A stochastic tip-off: Simulating the NBA playoffs with a graphing calculator

Mathematics and Computer Education, Fall 2002 by Ratliff, Michael I, Martinez-Cruz, Armando M

One goal in the Principles and Standards for School Mathematics (NCTM, 2000) is for students to recognize and value connections within and outside of mathematics. Furthermore, "technology is [considered] essential in teaching and learning mathematics" [1, p. 25]. Technology can help to reach this goal by bringing real-world situations into the mathematics classroom. Because sport topics often generate student interest, we have chosen to simulate a basketball tournament using data from the internet and a graphing calculator. A main objective of this activity is for students to use a combination of stochastic ideas (such as mean, standard deviation, and normal and binomial distributions) to simulate a basketball tournament. We have tried this activity with high school students, and they have found it to be an interesting simulation.

The tournament setup consists of eight teams each with 14 players (although five players are on the court we are considering the real situation which includes back-up players). The players are drafted, that is, selected, from the distribution of heights of NBA players (for simplicity we assume to be normal). Then each team's mean height is used to decide the seeds (the lowest seed, 1, is considered the best team) of a single-elimination tournament. A team is eliminated when it loses a best of seven series. Since there are initially eight teams, there will be four best of seven series in the first round. The four winners will advance to the second round, called the semifinal round. The two winners of the semifinal round will advance to the final round. The winner of the final series is the champion. A simulation of the binomial distribution is used to decide the winner in each best of seven series. We use the TI-83 calculator in the activity, but any calculator with statistical capabilities can be used. Blank worksheets are provided for students, at the end of the paper.

INITIATING THE DISCUSSION

Students and teachers can provide examples of how statistics is used in several fields. If you ask students how statistics is used in sports, they will mention among other things "batting averages, yards per game, goals per game," and so on. Certainly most students are aware that statistics is used extensively in sports. However, at first they might not realize that it is possible to simulate a sporting tournament using statistical ideas. As an example the instructor may illustrate how to simulate a basketball tournament using previously learned statistical concepts and a graphing calculator. Once the objective is clear, students can start talking about the number of teams and players required, the number of games that have to be won to pass to the next round and other related ideas.

Since our simulation is a basketball tournament, ask the students to name some attributes that are needed to be an NBA player. Not surprisingly, height is a popular attribute. Next, ask the students to guess the average height of an NBA player. Most students are willing to make guesses and some of them turn out to be close to the actual mean. To bring the ideas of range and dispersion into the discussion mention that Shawn Bradley is the tallest player in the NBA at 7' 6" and plays as a center for the Dallas Mavericks. In contrast, Earl Boykins at 5' 5" is the shortest player in the league and is a guard for the Los Angeles Clippers. The first step is to "draft" or select the players for each team. We indicate how this is done in the next section. Remember that our tournament has eight teams each with 14 players. Before drafting players for each team we need to select team names. You can select any name, but we suggest that students use either the NBA teams (i.e., Lakers) or university team names (i.e., Buckeyes). In our case we use team names from the NBA.

USING A NORMAL DISTRIBUTION TO DRAFT THE PLAYERS

The drafting of the team members is based on the heights of NBA players. We randomly produce 14 heights from this population using the graphing calculator. Although the NBA population has several characteristics, it seems that height is an important factor in becoming an NBA player. Let's assume that height is the primary attribute of this population. Communicate to the students that this is an arbitrary decision and that in the field of statistical predictions and simulations some reasonable assumptions are necessarily made. Remark that height is a continuous variable.

Figure 1 shows the actual distribution of heights of NBA players. The figure shows that we can assume that the population of heights follows a normal distribution (data for this figure were obtained from the Internet as discussed below), so we need to find its mean and standard deviation. This is the information that the graphing calculator needs to "draft" the heights.

These statistics are found on the web. If students have access to the Internet, have some of them go to the CBS Sportsline webpage (http://wwl.sportsline.com/u/basketball/nba/teams/index.html) and get the actual heights of all NBA players in the 2001-2002 season. Using the calculator we found the mean to be a = 79.2 inches, and standard deviation to be cr = 3.73 inches. We have super-imposed a normal distribution with these parameters over the original data. We see that the drawn normal distribution provides a reasonable fit for the data and thus we use it in our simulation of heights. That is, we are assuming that the population of NBA players' heights actually follows a normal distribution with (mu) = 79.2 inches and sigma = 3.73 inches. Consult, for example, Weiss [2, p. 335] for properties of the normal distribution.