To find the standard deviation of Omar's sample data: [tex]\(13, 17, 9, 21\)[/tex], we will follow these steps.
1. Calculate the mean ([tex]\(\bar{x}\)[/tex]):
The mean is the average of all the numbers in the data set.
[tex]\[
\bar{x} = \frac{13 + 17 + 9 + 21}{4} = \frac{60}{4} = 15
\][/tex]
2. Calculate each deviation from the mean:
We subtract the mean from each number in the data set.
[tex]\[
\begin{align*}
13 - 15 &= -2 \\
17 - 15 &= 2 \\
9 - 15 &= -6 \\
21 - 15 &= 6 \\
\end{align*}
\][/tex]
3. Square each deviation:
Squaring each of the deviations calculated in the previous step.
[tex]\[
\begin{align*}
(-2)^2 &= 4 \\
2^2 &= 4 \\
(-6)^2 &= 36 \\
6^2 &= 36 \\
\end{align*}
\][/tex]
4. Calculate the variance ([tex]\(s^2\)[/tex]):
Variance is the average of these squared deviations. Since this is a sample, we divide by [tex]\(n - 1\)[/tex] (where [tex]\(n\)[/tex] is the number of data points).
[tex]\[
s^2 = \frac{4 + 4 + 36 + 36}{4 - 1}=\frac{80}{3}\approx 26.67
\][/tex]
5. Calculate the standard deviation ([tex]\(s\)[/tex]):
Standard deviation is the square root of the variance.
[tex]\[
s = \sqrt{26.67} \approx 5.16
\][/tex]
So, the standard deviation for Omar's data is approximately [tex]\(5.2\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{5.2}
\][/tex]
