Question

calculate the sample standard deviation of the quantitative data shown. round to two decimal places.
x
19
10
13
24
28
27
20
sample standard deviation =
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Explanation:

Step1: Calculate the mean ($\bar{x}$)

The data set is \(19, 10, 13, 24, 28, 27, 20\). The number of data points \(n = 7\).
The mean \(\bar{x}=\frac{19 + 10 + 13 + 24 + 28 + 27 + 20}{7}=\frac{141}{7}\approx20.1429\)

Step2: Calculate the squared differences from the mean

For each data point \(x_i\), calculate \((x_i-\bar{x})^2\):

• \((19 - 20.1429)^2\approx(-1.1429)^2\approx1.3062\)
• \((10 - 20.1429)^2\approx(-10.1429)^2\approx102.8889\)
• \((13 - 20.1429)^2\approx(-7.1429)^2\approx51.0204\)
• \((24 - 20.1429)^2\approx(3.8571)^2\approx14.8776\)
• \((28 - 20.1429)^2\approx(7.8571)^2\approx61.7347\)
• \((27 - 20.1429)^2\approx(6.8571)^2\approx47.0204\)
• \((20 - 20.1429)^2\approx(-0.1429)^2\approx0.0204\)

Step3: Calculate the sum of squared differences

Sum these squared differences: \(1.3062+102.8889 + 51.0204+14.8776+61.7347+47.0204+0.0204 = 278.8686\)

Step4: Calculate the sample variance ($s^2$)

Sample variance \(s^2=\frac{\sum(x_i - \bar{x})^2}{n - 1}=\frac{278.8686}{7 - 1}=\frac{278.8686}{6}\approx46.4781\)

Step5: Calculate the sample standard deviation ($s$)

Sample standard deviation \(s=\sqrt{s^2}=\sqrt{46.4781}\approx6.82\)

Answer:

\(6.82\)