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Question Find the coefficient of variation for a set of 1010 systolic and diastolic blood pressure measurements (in mmHg\mathrm{mm} \mathrm{Hg}). The coefficient of variation for the systolic measurements is □%\square \%.

Studdy Solution

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=(SDM)×100% 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.
Msystolic=∑systolic measurementsnumber of measurements M_{systolic} = \frac{\sum systolic\, measurements}{number\, of\, measurements}

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

STEP 4

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

STEP 5

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

STEP 6

Compute the mean for the systolic measurements.
Msystolic=117.1 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.
SDsystolic=∑(xi−Msystolic)2number of measurements 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.
∑(xi−Msystolic)2=(117−117.1)2+(127−117.1)2+…+(118−117.1)2 \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.
∑(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

STEP 10

Compute the sum obtained in STEP_9.
∑(xi−Msystolic)2=2944.9 \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.
SDsystolic=2944.910 SD_{systolic} = \sqrt{\frac{2944.9}{10}}

STEP 12

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

STEP 13

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

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\%

STEP 16

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

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

STEP 18

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

STEP 19

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

STEP 20

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

STEP 22

Calculate the sum of the squared differences from the mean for the diastolic measurements.
∑(xi−Mdiastolic)2=(78−73.1)2+(75−73.1)2+…+(81−73.1)2 \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.
∑(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

STEP 24

Compute the sum obtained in STEP_23.
∑(xi−Mdiastolic)2=1258.9 \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.
SDdiastolic=1258.910 SD_{diastolic} = \sqrt{\frac{1258.9}{10}}

STEP 26

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

STEP 27

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

STEP 29

Compute the coefficient of variation for the diastolic measurements.
CVdiastolic=(11.2273.1)×100%≈15.3% 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%14.7\% and for the diastolic measurements is 15.3%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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