Question

describe the process of calculating a standard deviation. give a simple example of its calculation (such as calculating the standard deviation of the numbers 2, 3, 4, 4, and 6). what is the deviation if all of the sample values are the same? fill in the blanks to complete the process of calculating a standard deviation. compute the mean of the data set. then find the deviation from the mean for every data value by the data value. find the of all the deviations from the mean, and then them together. divide this sum by the the standard deviation is the of this quotient. the standard deviation of the numbers 2, 3, 4, 4, and 6 is approximately 1.483. (round to three decimal places as needed.) if all of the sample values are the same, then the standard deviation is 0.

Explanation:

Step1: Calculate the mean

The data - set is \(2,3,4,4,6\). The mean \(\bar{x}=\frac{2 + 3+4+4+6}{5}=\frac{19}{5}=3.8\)

Step2: Find the deviations from the mean

For \(x_1 = 2\), deviation \(d_1=2 - 3.8=-1.8\); for \(x_2 = 3\), deviation \(d_2=3 - 3.8=-0.8\); for \(x_3 = 4\), deviation \(d_3=4 - 3.8 = 0.2\); for \(x_4 = 4\), deviation \(d_4=4 - 3.8 = 0.2\); for \(x_5 = 6\), deviation \(d_5=6 - 3.8 = 2.2\)

Step3: Square the deviations

\(d_1^{2}=(-1.8)^{2}=3.24\), \(d_2^{2}=(-0.8)^{2}=0.64\), \(d_3^{2}=(0.2)^{2}=0.04\), \(d_4^{2}=(0.2)^{2}=0.04\), \(d_5^{2}=(2.2)^{2}=4.84\)

Step4: Sum the squared deviations

\(\sum_{i = 1}^{5}d_i^{2}=3.24+0.64 + 0.04+0.04+4.84=8.8\)

Step5: Divide by \(n - 1\) (for sample standard - deviation, \(n = 5\))

\(s^{2}=\frac{\sum_{i=1}^{n}d_i^{2}}{n - 1}=\frac{8.8}{4}=2.2\)

Step6: Take the square - root

\(s=\sqrt{2.2}\approx1.483\)

If all sample values are the same, say \(x_1=x_2=\cdots=x_n = k\). The mean \(\bar{x}=k\). The deviation of each data - point from the mean \(x_i-\bar{x}=k - k = 0\) for \(i = 1,2,\cdots,n\). Then the sum of squared deviations \(\sum_{i = 1}^{n}(x_i-\bar{x})^2 = 0\), and the standard deviation \(s = 0\)

Answer:

Fill in the blanks:

• First blank: subtracting the mean from
• Second blank: sum
• Third blank: square - root

If all of the sample values are the same, the standard deviation is \(0\)