Certainly! Let's start by understanding the steps involved in solving this problem.
We are given a set of numbers:
[tex]\[ 9, 24, 4, 12, 15, 22, 18, 10, 7, 10 \][/tex]
We need to calculate the variance [tex]\(\sigma^2\)[/tex] of this set of numbers. Variance measures how much the numbers in a data set deviate from the mean (average) of the set.
### Step-by-Step Solution
1. Calculate the Mean (Average) of the Data Set:
The mean [tex]\(\mu\)[/tex] is calculated by summing all the numbers in the data set and dividing by the total number of elements.
[tex]\[
\mu = \frac{9 + 24 + 4 + 12 + 15 + 22 + 18 + 10 + 7 + 10}{10}
\][/tex]
Given the result, the mean is:
[tex]\[
\mu = 13.1
\][/tex]
2. Compute Each Squared Deviation from the Mean:
For each number [tex]\(x_i\)[/tex] in the data set, compute the squared deviation [tex]\((x_i - \mu)^2\)[/tex]:
[tex]\[
\begin{align*}
(9 - 13.1)^2 &= (-4.1)^2 = 16.81 \\
(24 - 13.1)^2 &= (10.9)^2 = 118.81 \\
(4 - 13.1)^2 &= (-9.1)^2 = 82.81 \\
(12 - 13.1)^2 &= (-1.1)^2 = 1.21 \\
(15 - 13.1)^2 &= (1.9)^2 = 3.61 \\
(22 - 13.1)^2 &= (8.9)^2 = 79.21 \\
(18 - 13.1)^2 &= (4.9)^2 = 24.01 \\
(10 - 13.1)^2 &= (-3.1)^2 = 9.61 \\
(7 - 13.1)^2 &= (-6.1)^2 = 37.21 \\
(10 - 13.1)^2 &= (-3.1)^2 = 9.61 \\
\end{align*}
\][/tex]
3. Sum of Squared Deviations:
Sum all the squared deviations:
[tex]\[
16.81 + 118.81 + 82.81 + 1.21 + 3.61 + 79.21 + 24.01 + 9.61 + 37.21 + 9.61 = 382.9
\][/tex]
4. Calculate the Variance:
Finally, divide the sum of squared deviations by the number of elements to get the variance [tex]\(\sigma^2\)[/tex]:
[tex]\[
\sigma^2 = \frac{382.9}{10} = 38.29
\][/tex]
### Conclusion
The mean of the given set of numbers is [tex]\(\mathbf{13.1}\)[/tex], the sum of the squared deviations from the mean is [tex]\(\mathbf{382.9}\)[/tex], and the variance is [tex]\(\mathbf{38.29}\)[/tex].
