Question

Critical Thinking. 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

Firefighter Fatalities Listed below are the numbers of heroic firefighters who lost their lives in the United States each year while lighting forest fires. The numbers are listed in order by year, starting with the year 2000. What important feature of the data is not revealed by any of the measures of center?

20 18 23 30 20 12 24 9 25 15 8 11 15 34

Short answer

(a) The mean is 18.9.

(b) The median is 19.0.

(c) Bimodal; the modal values are 15 and 20.

(d) The midrange is 21.0.

The measures do not reveal the pattern.

Step-by-step solution

Given information

Starting from the year 2000, the number of firefighters who lost their lives in forest fires is recorded as follows:

20, 18, 23, 30, 20, 12, 24, 9, 25, 15, 8, 11, 15, 34

Compute mean

(a)

The mean is computed as follows:

$\overline{x}=\frac{\sum x}{n}$, where x represents the values and n is the count of the values.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{20+18+...+34}{14} \\ =\frac{264}{14} \\ \approx 18.8571 \end{gathered}$$

Thus, the mean value is approximately 18.9.

Compute median

(b)

The median is computed as follows:

• For an odd data set, the middlemost observation is considered to be the median.
• For an even number of values, the average of the two middlemost observations is the median.

The number of observations is 14.

Arrange the observations in ascending order.

8
9
11
12
15
15
18
20
20
23
24
25
30
34

The middlemost observations are 18 and 20.

The median is given as:

$$\begin{gathered} M=\frac{18+20}{2} \\ =\frac{38}{2} \\ =19 \end{gathered}$$

Thus, the median is 19.0.

Compute mode

(c)

Mode is considered to be the most frequent value in the data set.

The frequency distributions for the number of firefighters are:

Number of firefighters	Frequency
8	1
9	1
11	1
12	1
15	2
18	1
20	2
23	1
24	1
25	1
30	1
34	1

The highest frequencies obtained are 15 and 20.

Thus, the data is bimodal, with modes 15 and 20.

Compute midrange

(d)

The midrange is computed as follows:

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{8+34}{2} \\ =21 \end{gathered}$$

Thus, the midrange is 21.0.

Discuss the feature that is not identified by any measure of center

The data is based on years, which means it is a time-based data. The most important feature of time series data is the pattern in which changes occur with the change in time.

Thus, the pattern cannot be identified using any measure of center.