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 standard 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 of this quotient. the standard deviation is the the standard deviation of the numbers 2, 3, 4, 4, and 6 is approximately (round to three decimal places as needed.) if all of the sample values are the same, then the standard deviation is

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: Calculate the deviations from the mean

For \(x_1 = 2\), the deviation \(d_1=2 - 3.8=-1.8\); for \(x_2 = 3\), the deviation \(d_2=3 - 3.8=-0.8\); for \(x_3 = 4\), the deviation \(d_3=4 - 3.8 = 0.2\); for \(x_4 = 4\), the deviation \(d_4=4 - 3.8 = 0.2\); for \(x_5 = 6\), the 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: Calculate the sum of the squared deviations

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

Step5: Calculate the variance

The variance \(s^{2}=\frac{\sum_{i = 1}^{n}d_i^{2}}{n - 1}\) (for a sample), here \(n = 5\), so \(s^{2}=\frac{8.8}{4}=2.2\).

Step6: Calculate the standard deviation

The standard deviation \(s=\sqrt{2.2}\approx1.483\).
If all of the 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 standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}=\sqrt{\frac{0}{n - 1}} = 0\).

Answer:

The standard deviation of \(2,3,4,4,6\) is approximately \(1.483\), and if all sample values are the same, the standard deviation is \(0\).