To find the standard deviation for the given set of data [tex]\(\{17, 26, 23, 29, 17, 4, 21, 6, 19\}\)[/tex], we'll follow a systematic approach involving the calculation of the mean, variance, and standard deviation. Here are the steps in detail:
1. Calculate the Mean:
The mean (average) of a set of data is calculated by summing all the values and then dividing by the number of data points.
[tex]\[
\text{Mean} = \frac{\sum \text{data values}}{\text{number of data values}}
\][/tex]
For our dataset:
[tex]\[
\text{Mean} = \frac{17 + 26 + 23 + 29 + 17 + 4 + 21 + 6 + 19}{9} = \frac{162}{9} = 18.0
\][/tex]
2. Calculate the Variance:
Variance is the average of the squared differences between each data point and the mean. First, we find each difference, square it, and then average those squared differences.
[tex]\[
\text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n}
\][/tex]
Where [tex]\( x_i \)[/tex] represents each data point and [tex]\( n \)[/tex] is the number of data points.
Using the given mean of 18, we calculate the squared differences:
[tex]\[
\begin{align*}
(17 - 18)^2 & = 1 \\
(26 - 18)^2 & = 64 \\
(23 - 18)^2 & = 25 \\
(29 - 18)^2 & = 121 \\
(17 - 18)^2 & = 1 \\
(4 - 18)^2 & = 196 \\
(21 - 18)^2 & = 9 \\
(6 - 18)^2 & = 144 \\
(19 - 18)^2 & = 1 \\
\end{align*}
\][/tex]
Summing these squared differences gives:
[tex]\[
1 + 64 + 25 + 121 + 1 + 196 + 9 + 144 + 1 = 562
\][/tex]
Then, the variance is:
[tex]\[
\text{Variance} = \frac{562}{9} = 62.44444444444444
\][/tex]
3. Calculate the Standard Deviation:
The standard deviation is the square root of the variance.
[tex]\[
\text{Standard Deviation} = \sqrt{62.44444444444444} \approx 7.9021797274197985
\][/tex]
Therefore, the standard deviation for the given set of data is approximately 7.9. Thus, the correct answer is:
B. 7.9
