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

Foot Lengths Listed below are foot lengths in inches of randomly selected Army women measured in the 1988 Anthropometric Survey (ANSUR). Are the statistics representative of the current population of all Army women?

10.4 9.3 9.1 9.3 10.0 9.4 8.6 9.8 9.9 9.1 9.1

Short answer

(a) The mean is 9.45 inches.

(b) The median is 9.30 inches.

(c) The mode is 9.1 inches.

(d) The midrange is 9.50 inches.

No, the statistics are not representative of the current population of army women.

Step-by-step solution

Given information

Foot lengths are measured in 1988 ANSUR for 11 randomly selected army women.

10.4, 9.3, 9.1, 9.3, 10.0, 9.4, 8.6, 9.8, 9.9, 9.1, 9.1

The lengths are measured in inches.

Compute mean

(a)

The formula for the mean of the sample data is:

$\overline{x}=\frac{\sum x}{n}$, wherexrepresents the values and nis the count of the values.

Substitute the values in the formula.

$$\begin{gathered} \overline{x}=\frac{10.4+9.3+...+9.1}{11} \\ =\frac{104}{11} \\ \approx 9.45 \end{gathered}$$

Thus, the mean value is approximately 9.45 inches.

Compute median

(b)

The formula for the median is:

• Median is the middlemost observation for an odd set of values.
• It can also be the average of the two middlemost observations for an even set of values.

The number of observations is 11.

Arrange the observations in ascending order.

8.6
9.1
9.1
9.1
9.3
9.3
9.4
9.8
9.9
10
10.4

The middlemost observation is 9.30.

The median is given as:

$M=9.30$

Thus, the median is 9.30 inches.

Compute mode

(c)

The value with the maximum frequency is the mode.

The frequency distributions for the foot lengths are:

Foot lengths	Frequency
8.6	1
9.1	3
9.3	2
94	1
9.8	1
9.9	1
10	1
10.4	1

The highest frequency is obtained for 9.1.

Thus, the mode of the data is 9.1 inches.

Compute midrange

(d)

The formula for midrange is:

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

Substitute the values in the formula.

$$\begin{gathered} \text{Midrange}=\frac{8.6+10.4}{2} \\ =\frac{19}{2} \\ =9.5 \end{gathered}$$

Thus, the midrange is 9.50 inches.

Discuss if the statistics are an appropriate representation of the current army women

The data is known to have been collected in 1988 ANSUR (survey). The data recorded for women then may not represent the women in the army at present.

Thus, the statistics are not considered to be an appropriate representative of the population of army women in the current state.