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Chapter 9: Problem 479

From the sample of data \(5,8,2,1\), compute the standard deviation of the sample.

Short Answer

Expert verified
The sample standard deviation of the data set \(5, 8, 2, 1\) is approximately \(3.16\).

Step by step solution

01

Calculate the mean (\(\bar{x}\)) of the data set

First, we need to calculate the mean of the data set. To do this, we add the data points together and divide the result by the number of data points: \[\bar{x} = \frac{1}{4}(5+8+2+1) = \frac{16}{4} = 4\]
02

Find the deviation of each data point from the mean

Next, we find the deviation of each data point from the mean by subtracting the mean from each data point: \[5-4 = 1\] \[8-4 = 4\] \[2-4 = -2\] \[1-4 = -3\]
03

Square each deviation and sum them up

Now, we square the deviations and sum them up: \[(1)^2 + (4)^2 + (-2)^2 + (-3)^2 = 1 + 16 + 4 + 9 = 30\]
04

Divide the sum by \(n-1\) (degrees of freedom)

We divide the sum of the squared deviations by \(n-1\) (the degrees of freedom), where n is the number of data points: \[\frac{30}{4-1} = \frac{30}{3} = 10\]
05

Take the square root of the result

Finally, we take the square root of the result to get the sample standard deviation: \[s = \sqrt{10} \approx 3.16\] The sample standard deviation of the data set is approximately 3.16.

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