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

Hurricanes Listed below are the numbers of Atlantic hurricanes that occurred in each year. The data 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? 8 9 8 7 9 15 5 6 8 4 12 7 8 2

Short answer

(a) The mean is 7.7 hurricanes.

(b) The median is 8.0 hurricanes.

(c) The mode is 8 hurricanes.

(d) The midrange is 8.5 hurricanes.

The pattern of change for the number of hurricanes each year is not identified using the measures of the center.

Step-by-step solution

Given information

The number of hurricanes each year, starting from the year 2000, is recorded below.

8, 9, 8, 7, 9, 15, 5, 6, 8, 4, 12, 7, 8, 2

Compute mean

(a)

The formula for the mean is given as:

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

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{8+9+...+2}{14} \\ =\frac{108}{14} \\ \approx 7.7143 \end{gathered}$$

Thus, the mean value is approximately 7.7 hurricanes.

Compute median

The method to compute the median is given below:

• Arrange the data in ascending or descending order.
• The median is equal to the middlemost observation in the case of an odd data set (n).
• The median is equal to the average of the two middle observations in the case of an even data set (n).

The number of observations is14.

Arrange the observations in ascending order.

2
4
5
6
7
7
8
8
8
8
9
9
12
15

The middlemost observations are 8 and 8.

The median is given as:

$$\begin{gathered} M=\frac{8+8}{2} \\ =8 \end{gathered}$$

Thus, the median is 8.0 hurricanes.

Compute mode

(c)

Mode is(are) the observation(s) with the maximum frequency value(s).

The frequency distributions for the number of hurricanes are:

Number of hurricanes	Frequency
2	1
4	1
5	1
6	1
7	2
8	4
9	2
12	1
15	1

The highest frequency is obtained for 8.

Thus, the mode of the data is 8 hurricanes.

Compute midrange

(d)

The formula for the midrange is given below:

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{2+15}{2} \\ =\frac{17}{2} \\ =8.5 \end{gathered}$$

Thus, the midrange is 8.5 hurricanes.

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

The values are recorded for each year. It implies that the data is time-based. The measures of the center do not identify the pattern in which the recorded number of hurricanes changes.

It is important to identify the pattern to reveal the prediction of the subsequent periods.