## Weight Distributions of Super Bowl Players

How much do Howard Green and Ryan Pickett weigh (both of Green Bay Packers)? How much does Chris Kemoeatu weigh (Pittsburgh Steelers)? These are the heaviest players in the two teams for the Super Bowl 2011. Their weights listed at the official team websites are 340 pounds (Howard Green and Ryan Pickett) and 344 pounds (Chris Kemoeatu). Counting only the active players in the rosters, the weight distributions of the players of these two teams are very similar. Each team has exactly 13 three hundred pounders. The total weights of the two teams are very similar (13,171 pounds for Green Bay, 13,149 pounds for Steelers). The mean weights are 248.51 pounds (Green Bay) and 248.09 pounds (Steelers). We present a comparison between the two weight distributions using descriptive statistics, a general approach to examine data using graphs and numerical summaries. For more information about any statistical notions discussed here see  or your favorite text on introductory statistics.

Descriptive Statistcs
The general strategies for data analysis are: 1. start with a graphical display of the data, 2. look for the overall pattern and note any deviations from that pattern, and 3, use numerical summaries to describe certain aspects of the data (see ).

To display one-variable data such as weight, we can use histogram, stemplot, or boxplot. In this post, we use stemplot to display the data. Once we have a graph, we describe the distribution of the data by noting its shape, center and spread. Then look for outliers (individual values that fall outside of the overall pattern). Next we discuss the numerical summaries of center and spread in more details.

We first graphically compare the weights in pound. The following shows a frequency distribution of the weights and a back-to-back stemplot (Figure 1) of the weights of the two Super Bowl teams. In the stemplot, the stems represent the hundred digits and the leaves represent the ten digits. The one digits are rounded. $\displaystyle \begin{pmatrix} \text{Weight}&\text{ }&\text{Steelers}&\text{Packers} \\\text{ }&\text{ }&\text{ }&\text{ } \\\text{150 to 199}&\text{ }&7&4 \\\text{200 to 249}&\text{ }&23&21 \\\text{250 to 299}&\text{ }&9&15 \\\text{300 to 349}&\text{ }&14&13 \\\text{ }&\text{ }&\text{ }&\text{ } \\\text{Total}&\text{ }&53&53 \end{pmatrix}$ The frequency distribution indicates that the weight classes of 150 to 199 and 200 to 249 are a little more crowded for the Steelers. This makes the Steelers distribution just a tad more skewed than the one for the Packers. Looking at the Stemplot (Figure 1), we see that the distributions for both teams essentially the same overall pattern. Both distributions are roughly symmetric.

To summarize, the overall patterns of the data are: Steelers. Shape: The distribution is roughly symmetric. Center: The midpoint of the distribution is 239 (the 27th leaf in the stemplot). Spread: The spread is from 180 to 344 pounds. There are no obvious outliers or other striking deviations from the overall pattern. Packers. Shape: The distribution is roughly symmetric. Center: The midpoint of the distribution is 247 (the 27th leaf in the stemplot). Spread: The spread is from 184 to 340 pounds. There are no obvious outliers or other striking deviations from the overall pattern.

There are two types of measures to describe the distributions of weights, the measures of center and the measures of spread. A measure of center is a numerical summary that attempts to describe what a typical data value might look like. A measure of spread is a numerical summary that describes the degree to which the data are spread out.

Two common measures of center are mean and median. Both teams have essentially the same mean weight ( $\overline{x}=248.09$ pounds for the Steelers and $\overline{x}=248.51$ for the Packers). The median weights for the two teams are 239 pounds (Steelers) and 247 pounds (Packers). The smaller Steelers median indicates that the distribution for the Steelers is a little skewed in relation to the Packers. Since both distributions are roughly symmetric, so the mean is the preferred measure of center.

A measure of center alone is an incomplete description of the distribution. One numerical summary that contain both a measure of center (median) and a measure of spread (IQR) is the 5-number summary. The 5-number summary for the weight distribution of the Steelers is 180, 206, 239, 299, 344. The IQR implicitly stated in the 5-number summary is $IQR=299-206=93$. The middle 50% of the Steelers players have weights between 206 pounds (first quartile) and 299 pounds (the third quartile). The 5-number summary for the weight distribution of the Packers is 184, 207, 247, 295, 340 with $IQR=295-207=88$. The middle 50% of the Packers players have weights between 207 pounds (first quartile) and 295 pounds (the third quartile).

Since we are dealing with roughly symmetric distributions, the appropriate description of the distributions are the combination of the mean and standard deviation. Steelers: $\overline{x}=248.09$ pounds and $s=48.58$ pounds. Packers: $\overline{x}=248.51$ pounds and $s=46.75$ pounds.

One comment about using the standard deviation as the measure of spread (which we are doing in this case). The standard deviation $s$ measures spread about the mean. The higher this number, the more spread out the data are from the mean. Both distributions have essentially the same mean and the same spread.

Based on comparison using graph and numerical summaries, both teams are equal in terms of body masses. Many biological measurements on speciments from the same species and gender have symmetric distributions (see page 16 of ). The weight distribution of the super bowl players in 2011 is an illustration.

The Data
The weights are obtained from the official websites (Green Bay, Steelers). The weights are for both teams are sorted in ascending order (from left to right).

Weights in Pounds (Green Bay) $\displaystyle \begin{pmatrix} 184&191&191&194&195&186&198&200&200&200 \\202&203&207&207&208&213&215&216&207&218 \\219&225&225&236&238&245&247&247&248&248 \\250&250&252&254&255&255&255&263&285&290 \\300&305&305&308&308&314&316&318&318&320 \\337&340&340&\text{ }&\text{ } \end{pmatrix}$

Weights in Pounds (Steelers) $\displaystyle \begin{pmatrix} 180&180&185&186&190&190&191&195&199&200 \\200&205&205&207&208&208&209&209&216&225 \\229&230&231&233&234&235&239&241&242&243 \\250&252&256&260&262&265&270&280&285&298 \\300&304&304&305&305&315&318&319&324&325 \\325&338&344&\text{ }&\text{ } \end{pmatrix}$

Reference

1. Moore. D. S., McCabe G. P., Craig B. A., Introduction to the Practice of Statistics, 6th ed., W. H. Freeman and Company, New York, 2009
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