In Exercises 21–24, find the coefficient of variation for each of the two samples; then compare the variation. (The same data were used in Section 3-1.) 21.

Pulse Rates Listed below are pulse rates (beats per minute) from samples of adult males and females (from Data Set 1 “Body Data” in Appendix B). Does there appear to be a difference?

Male: 86 72 64 72 72 54 66 56 80 72 64 64 96 58 66

Female: 64 84 82 70 74 86 90 88 90 90 94 68 90 82 80

The two samples represent the male pulse rates and female pulse rates. The sample sizes are equal to 15 each.

When samples are measured in different units, the coefficient of variation is used to compare theamount of variation present in the samples. It is expressed as the following:

$C.V.=\frac{s}{\overline{x}}\times 100$

, where

$s$is the sample standard deviation;

$\overline{x}$is the sample mean.

The mean pulse rate for males is given as follows:

$$\begin{gathered} {\overline{x}}_1=\frac{{\sum }_{i=1}^{n_1}{x}_{1i}}{n_1} \\ =\frac{\left(86+72+...+66\right)}{15} \\ =69.5 \end{gathered}$$

Thus, the mean pulse rate for males is equal to 69.5 beats per minute.

The mean pulse rate for females is given as follows:

$$\begin{gathered} {\overline{x}}_1=\frac{{\sum }_{i=1}^{n_1}{x}_{1i}}{n_1} \\ =\frac{\left(86+72+...+66\right)}{15} \\ =69.5 \end{gathered}$$

Thus, the mean pulse rate for females is equal to 82.1 beats per minute.

The standard deviation for pulse rates of males is computed as follows:

$$\begin{gathered} s_1=\sqrt{\frac{{\sum }_{i=1}^n{\left(x_{1i}-{\overline{x}}_1\right)}^2}{n_1-1}} \\ =\sqrt{\frac{{\left(86-127.6\right)}^2+{\left(72-127.6\right)}^2+...+{\left(66-127.6\right)}^2}{15-1}} \\ =11.3 \end{gathered}$$

Thus, the standard deviation of the pulse rate for males is equal to 11.3 beats per minute.

The standard deviation for female pulse rates is computed as follows:

$$\begin{gathered} s_2=\sqrt{\frac{{\sum }_{i=1}^{n_2}{\left(x_{2i}-{\overline{x}}_2\right)}^2}{n_2-1}} \\ =\sqrt{\frac{{\left(64-73.6\right)}^2+{\left(84-73.6\right)}^2+...+{\left(80-73.6\right)}^2}{15-1}} \\ =9.2 \end{gathered}$$

Thus, the standard deviation of the pulse rate for females is equal to 9.2 beats per minute.

The coefficient of variation for male pulse rates is computed as follows:

$$\begin{gathered} C{V}_1=\frac{s_1}{{\overline{x}}_1}\times 100 \\ =\frac{11.3}{69.5}\times 100 \\ =16.2\% \end{gathered}$$

Therefore, the coefficient of variation for male pulse rates is equal to 16.2%.

The coefficient of variation for female pulse rates is as follows:

$$\begin{gathered} C{V}_2=\frac{s_2}{{\overline{x}}_2}\times 100 \\ =\frac{9.2}{82.1}\times 100 \\ =11.2\% \end{gathered}$$

Therefore, the coefficient of variation for female pulse rates is equal to 11.2%.

The coefficient of variation for pulse rates in males is greater than the coefficient of variation pulse rates in females.

Therefore, it can be said that pulse rates for males have a greater variation than the pulse rates for females, as they differ significantly.