For the data set [tex]$7, 5, 10, 11, 12$[/tex], the mean, [tex]$\bar{x}$[/tex], is 9. What is the standard deviation?

The formula for the sample standard deviation is [tex]$s = \sqrt{\frac{1}{n-1} \sum (x-\bar{x})^2}$[/tex].
Use the table to help you.
[tex]\[
\begin{array}{|c|c|c|}
\hline
X & (x-\bar{x}) & (x-\bar{x})^2 \\
\hline
7 & -2 & 4 \\
\hline
5 & -4 & 16 \\
\hline
10 & 1 & 1 \\
\hline
11 & 2 & 4 \\
\hline
12 & 3 & 9 \\
\hline
\multicolumn{3}{|c|}{\text{Sum} = 34} \\
\hline
\end{array}
\][/tex]

Round your answer to the nearest tenth.

A. 2.6
B. 8.5
C. 2.9
D. 6.8



Answer :

To find the standard deviation of the given data set [tex]\(7, 5, 10, 11, 12\)[/tex] with the mean [tex]\(\bar{x} = 9\)[/tex], follow these steps:

1. Calculate the deviations from the mean:

For each data point [tex]\(x\)[/tex], subtract the mean [tex]\(\bar{x} = 9\)[/tex]:
- For [tex]\(7\)[/tex]: [tex]\(7 - 9 = -2\)[/tex]
- For [tex]\(5\)[/tex]: [tex]\(5 - 9 = -4\)[/tex]
- For [tex]\(10\)[/tex]: [tex]\(10 - 9 = 1\)[/tex]
- For [tex]\(11\)[/tex]: [tex]\(11 - 9 = 2\)[/tex]
- For [tex]\(12\)[/tex]: [tex]\(12 - 9 = 3\)[/tex]

2. Square each deviation:

- For [tex]\(-2\)[/tex]: [tex]\((-2)^2 = 4\)[/tex]
- For [tex]\(-4\)[/tex]: [tex]\((-4)^2 = 16\)[/tex]
- For [tex]\(1\)[/tex]: [tex]\(1^2 = 1\)[/tex]
- For [tex]\(2\)[/tex]: [tex]\(2^2 = 4\)[/tex]
- For [tex]\(3\)[/tex]: [tex]\(3^2 = 9\)[/tex]

3. Sum of the squared deviations:

Calculate the sum of [tex]\((x - \bar{x})^2\)[/tex]:
[tex]\[ 4 + 16 + 1 + 4 + 9 = 34 \][/tex]

4. Determine [tex]\(n - 1\)[/tex]:

The number of data points [tex]\(n\)[/tex] is 5, therefore [tex]\(n - 1 = 5 - 1 = 4\)[/tex].

5. Calculate the sample standard deviation:

Use the formula [tex]\(s = \sqrt{\frac{1}{n-1} \Sigma (x - \bar{x})^2}\)[/tex]:
[tex]\[ s = \sqrt{\frac{34}{4}} = \sqrt{8.5} \approx 2.9154759474226504 \][/tex]

6. Round to the nearest tenth:

[tex]\(s\)[/tex] rounded to the nearest tenth is [tex]\(2.9\)[/tex].

So, the standard deviation for the data set is [tex]\(2.9\)[/tex]. Hence, the correct answer is:

C. 2.9