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. Blood Pressure A sample of blood pressure measurements is taken from Data Set 1 “Body Data” in Appendix B, and those values (mm Hg) are listed below. The values are matched so that 10 subjects each have a systolic and diastolic measurement.

Systolic: 118 128 158 96 156 122 116 136 126 120

Diastolic: 80 76 74 52 90 88 58 64 72 82

Short answer

The coefficient of variation for systolic blood pressure is equal to 14.6%.

The coefficient of variation for systolic blood pressure is equal to 16.9%.

The variation in the systolic and diastolic measurements of blood pressure is approximately equal.

Step-by-step solution

Given information

The measurement of systolic blood pressure and diastolic blood pressure are given for 10 subjects in two samples.

Meaning of coefficient of variation

The coefficient of variation measures theamount of variation present in a set of values interms of percentage. It is calculated using the given formula:

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

Here,

$s$ is the sample standard deviation;

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

Computation of variance measure for systolic measurements

Sample 1 shows the systolic blood pressures. The values of the sample mean are calculated as follows:

$$\begin{gathered} {\overline{x}}_1=\frac{\left(118+128+158+96+156+122+116+136+126+120\right)}{10} \\ =\frac{1276}{10} \\ =127.6 \end{gathered}$$

The sample standard deviation is calculated as follows:

$$\begin{gathered} s_1=\sqrt{\frac{{\sum }_{i=1}^n{\left(x_i-{\overline{x}}_1\right)}^2}{n_1-1}} \\ =\sqrt{\frac{{\sum }_{i=1}^{10}{\left(x_i-127.6\right)}^2}{10-1}} \\ =18.6 \end{gathered}$$

Thus, the sample mean is equal to 127.6 mm Hg, and the sample standard deviation is equal to 18.6 mm Hg.

The coefficient of variation for systolic blood pressure is as follows:

$$\begin{gathered} C{V}_1=\frac{s_1}{{\overline{x}}_1}\times 100 \\ =\frac{18.6}{127.6}\times 100 \\ =14.6\% \end{gathered}$$

Therefore, the coefficient of variation for systolic blood pressure is equal to 14.6%.

Computation of variance measure for diastolic measurements

Sample 2 shows the diastolic blood pressures. The values of the sample mean is calculated as follows:

$$\begin{gathered} {\overline{x}}_2=\frac{\left(80+76+74+52+90+88+58+64+72+82\right)}{10} \\ =\frac{736}{10} \\ =73.6 \end{gathered}$$

The sample standard deviation is calculated as follows:

$$\begin{gathered} s_2=\sqrt{\frac{{\sum }_{i=1}^n{\left(x_i-{\overline{x}}_2\right)}^2}{n_2-1}} \\ =\sqrt{\frac{{\left(80-73.6\right)}^2+{\left(76-73.6\right)}^2+...+{\left(82-73.6\right)}^2}{10-1}} \\ =12.5 \end{gathered}$$

Thus, the sample mean is equal to 73.6 mm Hg, and the sample standard deviation is equal to 12.5 mm Hg.

The coefficient of variation for diastolic blood pressure is calculated as follows:

$$\begin{gathered} C{V}_2=\frac{s_2}{{\overline{x}}_2}\times 100 \\ =\frac{12.5}{73.6}\times 100 \\ =16.9\% \end{gathered}$$

Therefore, the coefficient of variation for diastolic blood pressure is equal to 16.9%.

Comparison for two coefficients of variation for two samples

The coefficients of variation for the two samples are quite close.

Therefore, the variation in the two kinds of blood pressure measurements to the respective mean values is approximately the same.