Question

calculate the standard deviation and variance of the sample quantitative data shown, to two decimal places.

x
18.2
20.6
2.6
17.1
15.6
6.9
8.1
2

standard deviation:

variance:

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Explanation:

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

First, sum all the data points: \(18.2 + 20.6 + 2.6 + 17.1 + 15.6 + 6.9 + 8.1 + 2 = 91.1\)
There are \(n = 8\) data points. So the mean is \(\bar{x}=\frac{91.1}{8}=11.3875\)

Step2: Calculate the squared differences from the mean

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

• \((18.2 - 11.3875)^2=(6.8125)^2 = 46.41015625\)
• \((20.6 - 11.3875)^2=(9.2125)^2 = 84.87015625\)
• \((2.6 - 11.3875)^2=(-8.7875)^2 = 77.22015625\)
• \((17.1 - 11.3875)^2=(5.7125)^2 = 32.63265625\)
• \((15.6 - 11.3875)^2=(4.2125)^2 = 17.74515625\)
• \((6.9 - 11.3875)^2=(-4.4875)^2 = 20.14765625\)
• \((8.1 - 11.3875)^2=(-3.2875)^2 = 10.80765625\)
• \((2 - 11.3875)^2=(-9.3875)^2 = 88.12515625\)

Step3: Sum the squared differences

Sum these values: \(46.41015625+84.87015625 + 77.22015625+32.63265625+17.74515625+20.14765625+10.80765625+88.12515625 = 377.95875\)

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

The formula for sample variance is \(s^2=\frac{\sum (x_i - \bar{x})^2}{n - 1}\). Here, \(n = 8\), so \(n-1 = 7\).
\(s^2=\frac{377.95875}{7}\approx53.9941\)

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

The standard deviation is the square root of the variance: \(s=\sqrt{53.9941}\approx7.35\)

Answer:

Standard deviation: \(7.35\)
Variance: \(53.99\)