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 =

question help: video 1 video 2

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Explanation:

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

First, find the sum of the data points. The data is \(19, 10, 13, 24, 28, 27, 20\). The number of data points \(n = 7\).
Sum \(S=19 + 10+13 + 24+28 + 27+20=141\)
Mean \(\bar{x}=\frac{S}{n}=\frac{141}{7}\approx20.1429\)

Step2: Calculate the squared differences from the mean

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

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

Step3: Calculate the sum of squared differences

Sum of squared differences \(SS=\sum_{i = 1}^{n}(x_i-\bar{x})^2=1.3062 + 102.8889+51.0204+14.8776+61.7347+47.0204+0.0204 = 278.8686\)

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

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

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

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

Answer:

\(6.82\)