Question

question 2 of 11, step 1 of 2
consider the following sets of sample data:
a: 20,952, 20,558, 22,177, 21,617, 21,190, 21,749, 20,372, 20,009, 20,057, 20,591, 21,992, 21,288, 21,377, 20,132
b: 15, 38, 15, 33, 19, 13, 19, 36, 34, 17, 39
step 1 of 2: for each of the above sets of sample data, calculate the coefficient of variation, cv. round to one decimal place.
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cv for data set a:
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cv for data set b:
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Explanation:

Step 1: Calculate Mean for Data Set A

First, sum all values in Data Set A:
\(20952 + 20558 + 22177 + 21617 + 21190 + 21749 + 20372 + 20009 + 20057 + 20591 + 21992 + 21288 + 21377 + 20132\)
Let's compute the sum:
\(20952 + 20558 = 41510\)
\(41510 + 22177 = 63687\)
\(63687 + 21617 = 85304\)
\(85304 + 21190 = 106494\)
\(106494 + 21749 = 128243\)
\(128243 + 20372 = 148615\)
\(148615 + 20009 = 168624\)
\(168624 + 20057 = 188681\)
\(188681 + 20591 = 209272\)
\(209272 + 21992 = 231264\)
\(231264 + 21288 = 252552\)
\(252552 + 21377 = 273929\)
\(273929 + 20132 = 294061\)

Number of values (\(n\)) = 14.
Mean (\(\bar{x}\)) = \(\frac{294061}{14} \approx 21004.3571\)

Step 2: Calculate Standard Deviation for Data Set A

Use the formula for sample standard deviation: \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}\)

First, compute \((x_i - \bar{x})^2\) for each \(x_i\):
For \(20952\): \((20952 - 21004.3571)^2 \approx (-52.3571)^2 \approx 2741.26\)
For \(20558\): \((20558 - 21004.3571)^2 \approx (-446.3571)^2 \approx 199235.6\)
For \(22177\): \((22177 - 21004.3571)^2 \approx (1172.6429)^2 \approx 1375096.6\)
For \(21617\): \((21617 - 21004.3571)^2 \approx (612.6429)^2 \approx 375331.3\)
For \(21190\): \((21190 - 21004.3571)^2 \approx (185.6429)^2 \approx 34463.2\)
For \(21749\): \((21749 - 21004.3571)^2 \approx (744.6429)^2 \approx 554493.0\)
For \(20372\): \((20372 - 21004.3571)^2 \approx (-632.3571)^2 \approx 400875.4\)
For \(20009\): \((20009 - 21004.3571)^2 \approx (-995.3571)^2 \approx 990736.8\)
For \(20057\): \((20057 - 21004.3571)^2 \approx (-947.3571)^2 \approx 897506.4\)
For \(20591\): \((20591 - 21004.3571)^2 \approx (-413.3571)^2 \approx 170864.1\)
For \(21992\): \((21992 - 21004.3571)^2 \approx (987.6429)^2 \approx 975437.5\)
For \(21288\): \((21288 - 21004.3571)^2 \approx (283.6429)^2 \approx 80453.3\)
For \(21377\): \((21377 - 21004.3571)^2 \approx (372.6429)^2 \approx 138862.7\)
For \(20132\): \((20132 - 21004.3571)^2 \approx (-872.3571)^2 \approx 761007.0\)

Sum these squared deviations:
\(2741.26 + 199235.6 + 1375096.6 + 375331.3 + 34463.2 + 554493.0 + 400875.4 + 990736.8 + 897506.4 + 170864.1 + 975437.5 + 80453.3 + 138862.7 + 761007.0\)

Let's sum step-by-step:
\(2741.26 + 199235.6 = 201976.86\)
\(201976.86 + 1375096.6 = 1577073.46\)
\(1577073.46 + 375331.3 = 1952404.76\)
\(1952404.76 + 34463.2 = 1986867.96\)
\(1986867.96 + 554493.0 = 2541360.96\)
\(2541360.96 + 400875.4 = 2942236.36\)
\(2942236.36 + 990736.8 = 3932973.16\)
\(3932973.16 + 897506.4 = 4830479.56\)
\(4830479.56 + 170864.1 = 5001343.66\)
\(5001343.66 + 975437.5 = 5976781.16\)
\(5976781.16 + 80453.3 = 6057234.46\)
\(6057234.46 + 138862.7 = 6196097.16\)
\(6196097.16 + 761007.0 = 6957104.16\)

Now, \(n - 1 = 13\).
Variance (\(s^2\)) = \(\frac{6957104.16}{13} \approx 535161.86\)
Standard Deviation (\(s\)) = \(\sqrt{535161.86} \approx 731.547\)

Step 3: Calculate Coefficient of Variation (CV) for Data Set A

CV formula: \(CV = \frac{s}{\bar{x}} \times 100\%\)

\(CV = \frac{731.547}{21004.3571} \times 100\% \approx 3.5\%\) (rounded to one decimal place)

Step 4: Calculate Mean for Data Set B

Sum all values in Data Set B:
\(15 + 38 + 15 + 33 + 19 + 13 + 19 + 36 + 34 + 17 + 39\)
Sum: \(15+38=53\); \(53+15=68\); \(68+33=101\); \(101+19=120\); \(120+13=133\); \(133+19=152\); \(152+36=188\); \(188+34=222\); \(222+17=239\); \(239+39=278\)

\(n = 11\).
Mean (\(\bar{x}\)) = \(\frac{278}{11} \approx 25.2727\)

Step 5: Calculate Standard Deviation for Data Set B

Compute \((x_i - \bar{x})^2\) for each \(x_i\):
For \(15\): \((15 - 25.2727)^2 \approx (-10.2727)^2 \approx 105.53\)
For \(38\): \((38 - 25.2727)^2 \approx (12.7273)^2 \approx 162.00\)
For \(15\): \((15 - 25.2727)^2 \approx 105.53\)
For \(33\): \((33 - 25.2727)^2 \approx (7.7273)^2 \approx 59.69\)
For \(19\): \((19 - 25.2727)^2 \approx (-6.2727)^2 \approx 39.34\)
For \(13\): \((13 - 25.2727)^2 \approx (-12.2727)^2 \approx 150.62\)
For \(19\): \((19 - 25.2727)^2 \approx 39.34\)
For \(36\): \((36 - 25.2727)^2 \approx (10.7273)^2 \approx 115.08\)
For \(34\): \((34 - 25.2727)^2 \approx (8.7273)^2 \approx 76.17\)
For \(17\): \((17 - 25.2727)^2 \approx (-8.2727)^2 \approx 68.43\)
For \(39\): \((39 - 25.2727)^2 \approx (13.7273)^2 \approx 188.44\)

Sum these squared deviations:
\(105.53 + 162.00 + 105.53 + 59.69 + 39.34 + 150.62 + 39.34 + 115.08 + 76.17 + 68.43 + 188.44\)

Sum step-by-step:
\(105.53 + 162.00 = 267.53\)
\(267.53 + 105.53 = 373.06\)
\(373.06 + 59.69 = 432.75\)
\(432.75 + 39.34 = 472.09\)
\(472.09 + 150.62 = 622.71\)
\(622.71 + 39.34 = 662.05\)
\(662.05 + 115.08 = 777.13\)
\(777.13 + 76.17 = 853.30\)
\(853.30 + 68.43 = 921.73\)
\(921.73 + 188.44 = 1110.17\)

\(n - 1 = 10\).
Variance (\(s^2\)) = \(\frac{1110.17}{10} = 111.017\)
Standard Deviation (\(s\)) = \(\sqrt{111.017} \approx 10.536\)

Step 6: Calculate Coefficient of Variation (CV) for Data Set B

\(CV = \frac{10.536}{25.2727} \times 100\% \approx 41.7\%\) (rounded to one decimal place)

Answer:

CV for Data Set A: \(\boldsymbol{3.5\%}\)
CV for Data Set B: \(\boldsymbol{41.7\%}\)