To find the standard deviation for the set of data [tex]\(\{12, 20, 7, 18, 25, 6, 17, 26, 16\}\)[/tex], follow these steps:
1. Calculate the Mean:
First, find the mean (average) of the data set.
[tex]\[
\text{Mean} = \frac{\sum \text{data points}}{\text{number of data points}}
\][/tex]
Calculate the sum of the data points:
[tex]\[
12 + 20 + 7 + 18 + 25 + 6 + 17 + 26 + 16 = 147
\][/tex]
Then, divide by the number of data points (which is 9):
[tex]\[
\text{Mean} = \frac{147}{9} = 16.3333
\][/tex]
2. Find the Squared Differences from the Mean:
For each data point, subtract the mean and square the result:
[tex]\[
\begin{align*}
(12 - 16.3333)^2 &= 18.7778 \\
(20 - 16.3333)^2 &= 13.4444 \\
(7 - 16.3333)^2 &= 87.1111 \\
(18 - 16.3333)^2 &= 2.7778 \\
(25 - 16.3333)^2 &= 74.7778 \\
(6 - 16.3333)^2 &= 106.7778 \\
(17 - 16.3333)^2 &= 0.4444 \\
(26 - 16.3333)^2 &= 93.4444 \\
(16 - 16.3333)^2 &= 0.1111 \\
\end{align*}
\][/tex]
3. Calculate the Variance:
The variance is the average of these squared differences. Since this is a population (as opposed to a sample), we divide by the number of data points (n = 9):
[tex]\[
\text{Variance} = \frac{18.7778 + 13.4444 + 87.1111 + 2.7778 + 74.7778 + 106.7778 + 0.4444 + 93.4444 + 0.1111}{9} = 44.21
\][/tex]
4. Calculate the Standard Deviation:
Finally, the standard deviation is the square root of the variance:
[tex]\[
\text{Standard Deviation} = \sqrt{44.21} = 6.65
\][/tex]
Therefore, the standard deviation for the given set of data is [tex]\(6.65\)[/tex].
So, the correct answer is:
A. 6.65
