Question

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

Short answer

The sample coefficient of variation for pulse rates of males is 16.2%.

The sample coefficient of variation for pulse rates of females is 11.2%.

There seems to be a difference in the variation of male and female pulse rates. The male pulse rates have a greater variation than the female pulse rates.

Step-by-step solution

Given information

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

Coefficient of variation

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.

Calculation of sample means 

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.

Calculations for sample standard deviations

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.

Calculations for sample coefficients of variation

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%.

Compare the coefficient of variation measures

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.