Solved on Feb 11, 2024

Find the coefficient of variation for a set of $10$ systolic and diastolic blood pressure measurements (in $\mathrm{mm} \mathrm{Hg}$ ). The coefficient of variation for the systolic measurements is $\square \%$ .

STEP 1

Assumptions
1. The data provided are two samples of blood pressure measurements: systolic and diastolic.
2. Each sample contains 10 measurements.
3. The coefficient of variation (CV) is defined as the ratio of the standard deviation (SD) to the mean (M), usually expressed as a percentage.
4. The formula for the coefficient of variation is: $CV = \left( \frac{SD}{M} \right) \times 100\%$

STEP 2

First, we will calculate the mean of the systolic measurements. The mean is the sum of all measurements divided by the number of measurements.
$M_{systolic} = \frac{\sum systolic\, measurements}{number\, of\, measurements}$

STEP 3

Calculate the sum of the systolic measurements.
$\sum systolic\, measurements = 117 + 127 + 157 + 96 + 158 + 121 + 116 + 136 + 125 + 118$

STEP 4

Compute the sum obtained in STEP_3.
$\sum systolic\, measurements = 1171$

STEP 5

Now, calculate the mean of the systolic measurements using the sum from STEP_4.
$M_{systolic} = \frac{1171}{10}$

STEP 6

Compute the mean for the systolic measurements.
$M_{systolic} = 117.1$

STEP 7

Next, we will calculate the standard deviation of the systolic measurements. The standard deviation is a measure of the amount of variation or dispersion in a set of values.
$SD_{systolic} = \sqrt{\frac{\sum (x_i - M_{systolic})^2}{number\, of\, measurements}}$

STEP 8

Calculate the sum of the squared differences from the mean for the systolic measurements.
$\sum (x_i - M_{systolic})^2 = (117 - 117.1)^2 + (127 - 117.1)^2 + \ldots + (118 - 117.1)^2$

STEP 9

Compute the sum of the squared differences from STEP_8.
$\sum (x_i - M_{systolic})^2 = 0.01 + 98.01 + 1588.81 + \ldots + 0.81$

STEP 10

Compute the sum obtained in STEP_9.
$\sum (x_i - M_{systolic})^2 = 2944.9$

STEP 11

Now, calculate the standard deviation for the systolic measurements using the sum from STEP_10.
$SD_{systolic} = \sqrt{\frac{2944.9}{10}}$

STEP 12

Compute the standard deviation for the systolic measurements.
$SD_{systolic} = \sqrt{294.49}$

STEP 13

Compute the square root obtained in STEP_12.
$SD_{systolic} = 17.16$

STEP 14

Calculate the coefficient of variation for the systolic measurements using the mean and standard deviation.
$CV_{systolic} = \left( \frac{17.16}{117.1} \right) \times 100\%$

STEP 15

Compute the coefficient of variation for the systolic measurements.
$CV_{systolic} = \left( \frac{17.16}{117.1} \right) \times 100\% \approx 14.7\%$

STEP 16

Repeat the process for the diastolic measurements. First, calculate the mean of the diastolic measurements.
$M_{diastolic} = \frac{\sum diastolic\, measurements}{number\, of\, measurements}$

STEP 17

Calculate the sum of the diastolic measurements.
$\sum diastolic\, measurements = 78 + 75 + 76 + 52 + 89 + 88 + 58 + 64 + 70 + 81$

STEP 18

Compute the sum obtained in STEP_17.
$\sum diastolic\, measurements = 731$

STEP 19

Now, calculate the mean of the diastolic measurements using the sum from STEP_18.
$M_{diastolic} = \frac{731}{10}$

STEP 20

Compute the mean for the diastolic measurements.
$M_{diastolic} = 73.1$

STEP 21

Next, calculate the standard deviation of the diastolic measurements.
$SD_{diastolic} = \sqrt{\frac{\sum (x_i - M_{diastolic})^2}{number\, of\, measurements}}$

STEP 22

Calculate the sum of the squared differences from the mean for the diastolic measurements.
$\sum (x_i - M_{diastolic})^2 = (78 - 73.1)^2 + (75 - 73.1)^2 + \ldots + (81 - 73.1)^2$

STEP 23

Compute the sum of the squared differences from STEP_22.
$\sum (x_i - M_{diastolic})^2 = 24.01 + 3.61 + 8.41 + \ldots + 62.41$

STEP 24

Compute the sum obtained in STEP_23.
$\sum (x_i - M_{diastolic})^2 = 1258.9$

STEP 25

Now, calculate the standard deviation for the diastolic measurements using the sum from STEP_24.
$SD_{diastolic} = \sqrt{\frac{1258.9}{10}}$

STEP 26

Compute the standard deviation for the diastolic measurements.
$SD_{diastolic} = \sqrt{125.89}$

STEP 27

Compute the square root obtained in STEP_26.
$SD_{diastolic} = 11.22$

STEP 28

Calculate the coefficient of variation for the diastolic measurements using the mean and standard deviation.
$CV_{diastolic} = \left( \frac{11.22}{73.1} \right) \times 100\%$

STEP 29

Compute the coefficient of variation for the diastolic measurements.
$CV_{diastolic} = \left( \frac{11.22}{73.1} \right) \times 100\% \approx 15.3\%$
The coefficient of variation for the systolic measurements is $14.7\%$ and for the diastolic measurements is $15.3\%$ . Comparing the two, the diastolic measurements have a slightly higher coefficient of variation, indicating a slightly higher relative variability in the diastolic blood pressure measurements compared to the systolic measurements.

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Find the coefficient of variation for a set of 101010 systolic and diastolic blood pressure measurements (in mmHg\mathrm{mm} \mathrm{Hg}mmHg). The coefficient of variation for the systolic measurements is □%\square \%□%.

STEP 1

STEP 2

STEP 3

Calculate the sum of the systolic measurements.∑systolic measurements=117+127+157+96+158+121+116+136+125+118 \sum systolic\, measurements = 117 + 127 + 157 + 96 + 158 + 121 + 116 + 136 + 125 + 118 ∑systolicmeasurements=117+127+157+96+158+121+116+136+125+118

STEP 4

Compute the sum obtained in STEP_3.∑systolic measurements=1171 \sum systolic\, measurements = 1171 ∑systolicmeasurements=1171

STEP 5

Now, calculate the mean of the systolic measurements using the sum from STEP_4.Msystolic=117110 M_{systolic} = \frac{1171}{10} Msystolic​=101171​

STEP 6

Compute the mean for the systolic measurements.Msystolic=117.1 M_{systolic} = 117.1 Msystolic​=117.1

STEP 7

STEP 8

STEP 9

Compute the sum of the squared differences from STEP_8.∑(xi−Msystolic)2=0.01+98.01+1588.81+…+0.81 \sum (x_i - M_{systolic})^2 = 0.01 + 98.01 + 1588.81 + \ldots + 0.81 ∑(xi​−Msystolic​)2=0.01+98.01+1588.81+…+0.81

STEP 10

Compute the sum obtained in STEP_9.∑(xi−Msystolic)2=2944.9 \sum (x_i - M_{systolic})^2 = 2944.9 ∑(xi​−Msystolic​)2=2944.9

STEP 11

Now, calculate the standard deviation for the systolic measurements using the sum from STEP_10.SDsystolic=2944.910 SD_{systolic} = \sqrt{\frac{2944.9}{10}} SDsystolic​=102944.9​​

STEP 12

Compute the standard deviation for the systolic measurements.SDsystolic=294.49 SD_{systolic} = \sqrt{294.49} SDsystolic​=294.49​

STEP 13

Compute the square root obtained in STEP_12.SDsystolic=17.16 SD_{systolic} = 17.16 SDsystolic​=17.16

STEP 14

Calculate the coefficient of variation for the systolic measurements using the mean and standard deviation.CVsystolic=(17.16117.1)×100% CV_{systolic} = \left( \frac{17.16}{117.1} \right) \times 100\% CVsystolic​=(117.117.16​)×100%

STEP 15

Compute the coefficient of variation for the systolic measurements.CVsystolic=(17.16117.1)×100%≈14.7% CV_{systolic} = \left( \frac{17.16}{117.1} \right) \times 100\% \approx 14.7\% CVsystolic​=(117.117.16​)×100%≈14.7%

STEP 16

STEP 17

Calculate the sum of the diastolic measurements.∑diastolic measurements=78+75+76+52+89+88+58+64+70+81 \sum diastolic\, measurements = 78 + 75 + 76 + 52 + 89 + 88 + 58 + 64 + 70 + 81 ∑diastolicmeasurements=78+75+76+52+89+88+58+64+70+81

STEP 18

Compute the sum obtained in STEP_17.∑diastolic measurements=731 \sum diastolic\, measurements = 731 ∑diastolicmeasurements=731

STEP 19

Now, calculate the mean of the diastolic measurements using the sum from STEP_18.Mdiastolic=73110 M_{diastolic} = \frac{731}{10} Mdiastolic​=10731​

STEP 20

Compute the mean for the diastolic measurements.Mdiastolic=73.1 M_{diastolic} = 73.1 Mdiastolic​=73.1

STEP 21

Next, calculate the standard deviation of the diastolic measurements.SDdiastolic=∑(xi−Mdiastolic)2number of measurements SD_{diastolic} = \sqrt{\frac{\sum (x_i - M_{diastolic})^2}{number\, of\, measurements}} SDdiastolic​=numberofmeasurements∑(xi​−Mdiastolic​)2​​

STEP 22

STEP 23

Compute the sum of the squared differences from STEP_22.∑(xi−Mdiastolic)2=24.01+3.61+8.41+…+62.41 \sum (x_i - M_{diastolic})^2 = 24.01 + 3.61 + 8.41 + \ldots + 62.41 ∑(xi​−Mdiastolic​)2=24.01+3.61+8.41+…+62.41

STEP 24

Compute the sum obtained in STEP_23.∑(xi−Mdiastolic)2=1258.9 \sum (x_i - M_{diastolic})^2 = 1258.9 ∑(xi​−Mdiastolic​)2=1258.9

STEP 25

Now, calculate the standard deviation for the diastolic measurements using the sum from STEP_24.SDdiastolic=1258.910 SD_{diastolic} = \sqrt{\frac{1258.9}{10}} SDdiastolic​=101258.9​​

STEP 26

Compute the standard deviation for the diastolic measurements.SDdiastolic=125.89 SD_{diastolic} = \sqrt{125.89} SDdiastolic​=125.89​

STEP 27

Compute the square root obtained in STEP_26.SDdiastolic=11.22 SD_{diastolic} = 11.22 SDdiastolic​=11.22

STEP 28

Calculate the coefficient of variation for the diastolic measurements using the mean and standard deviation.CVdiastolic=(11.2273.1)×100% CV_{diastolic} = \left( \frac{11.22}{73.1} \right) \times 100\% CVdiastolic​=(73.111.22​)×100%

STEP 29

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Find the coefficient of variation for a set of $10$ systolic and diastolic blood pressure measurements (in $\mathrm{mm} \mathrm{Hg}$ ). The coefficient of variation for the systolic measurements is $\square \%$ .

Calculate the sum of the systolic measurements.
$\sum systolic\, measurements = 117 + 127 + 157 + 96 + 158 + 121 + 116 + 136 + 125 + 118$

Compute the sum obtained in STEP_3.
$\sum systolic\, measurements = 1171$

Now, calculate the mean of the systolic measurements using the sum from STEP_4.
$M_{systolic} = \frac{1171}{10}$

Compute the mean for the systolic measurements.
$M_{systolic} = 117.1$

Compute the sum of the squared differences from STEP_8.
$\sum (x_i - M_{systolic})^2 = 0.01 + 98.01 + 1588.81 + \ldots + 0.81$

Compute the sum obtained in STEP_9.
$\sum (x_i - M_{systolic})^2 = 2944.9$

Now, calculate the standard deviation for the systolic measurements using the sum from STEP_10.
$SD_{systolic} = \sqrt{\frac{2944.9}{10}}$

Compute the standard deviation for the systolic measurements.
$SD_{systolic} = \sqrt{294.49}$

Compute the square root obtained in STEP_12.
$SD_{systolic} = 17.16$

Calculate the coefficient of variation for the systolic measurements using the mean and standard deviation.
$CV_{systolic} = \left( \frac{17.16}{117.1} \right) \times 100\%$

Compute the coefficient of variation for the systolic measurements.
$CV_{systolic} = \left( \frac{17.16}{117.1} \right) \times 100\% \approx 14.7\%$

Calculate the sum of the diastolic measurements.
$\sum diastolic\, measurements = 78 + 75 + 76 + 52 + 89 + 88 + 58 + 64 + 70 + 81$

Compute the sum obtained in STEP_17.
$\sum diastolic\, measurements = 731$

Now, calculate the mean of the diastolic measurements using the sum from STEP_18.
$M_{diastolic} = \frac{731}{10}$

Compute the mean for the diastolic measurements.
$M_{diastolic} = 73.1$

Next, calculate the standard deviation of the diastolic measurements.
$SD_{diastolic} = \sqrt{\frac{\sum (x_i - M_{diastolic})^2}{number\, of\, measurements}}$

Compute the sum of the squared differences from STEP_22.
$\sum (x_i - M_{diastolic})^2 = 24.01 + 3.61 + 8.41 + \ldots + 62.41$

Compute the sum obtained in STEP_23.
$\sum (x_i - M_{diastolic})^2 = 1258.9$

Now, calculate the standard deviation for the diastolic measurements using the sum from STEP_24.
$SD_{diastolic} = \sqrt{\frac{1258.9}{10}}$

Compute the standard deviation for the diastolic measurements.
$SD_{diastolic} = \sqrt{125.89}$

Compute the square root obtained in STEP_26.
$SD_{diastolic} = 11.22$

Calculate the coefficient of variation for the diastolic measurements using the mean and standard deviation.
$CV_{diastolic} = \left( \frac{11.22}{73.1} \right) \times 100\%$