Question

For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question.

Football Player Numbers Listed below are the jersey numbers of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII.

What do the results tell us? 89 91 55 7 20 99 25 81 19 82 60

Short answer

(a). The mean is 57.1 approximately.

(b). The median is 60.0.

(c). The mode is none.

(d). The midrange is 53.0.

The values make no sense as the data is categorical.

Step-by-step solution

Given information

The 11 observations for jersey numbers are 89, 91, 55, 7, 20, 99, 25, 81, 19, 82, and 60.

Compute mean

(a).

The formula for mean is stated as follows:

$\overline{x}=\frac{\sum x}{n}$, where xrepresents the observations and nis the count of the observations.

The number of observations (n) is11.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{89+91+...+60}{11} \\ =\frac{628}{11} \\ =57.0909 \\ \approx 57.1 \end{gathered}$$

Thus, the mean value is approximately 57.1.

Compute median

(b).

The formula for the median is stated as follows:

• If n is odd, the middlemost observation is the median.
• If n is even, the average of the two middle observations is the median.

Arrange the observations in ascending order.

7,19,20,25,55,60,81,82,89,91,99

The middlemost observation is 60.

The median is given as:

$M=60$

Thus, the median is 60.0.

Compute mode

(c).

Mode is the observation with the highest frequency.

Since all observations are repeated once, there is no mode.

Compute midrange

(d).

The formula for midrange is stated as:

$\text{Midrange}=\frac{\text{Minimum}\text{value}+\text{Maximum}\text{value}}{2}$

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{\text{7}+\text{99}}{2} \\ =\frac{106}{2} \\ =53 \end{gathered}$$

Thus, the midrange is 53.0.

Interpretation of the results

The data records jersey numbers of 11 players, which can be categorized as nominal scaled data. Measures like mean, median, and midrange do not make any sense.

On the other hand, mode reveals the categorical value that repeats the maximum number of times. In this case, there is no modal value. Thus, there is no such jersey number that repeats itself.