Certainly! Let's go through the process of calculating variance step-by-step.
### Given Data Set:
(For the purpose of this explanation, let's assume a sample data set. If you have a specific data set in mind, replace the following values with those.)
[tex]\[
\text{Data} = [4, 8, 6, 5, 3, 8, 9, 10, 2, 6]
\][/tex]
### Step 1: Calculate the Mean ([tex]\(\mu\)[/tex])
The mean is the average of all data points. It is calculated by summing up all the values and then dividing by the number of values.
[tex]\[
\mu = \frac{\sum x_i}{N}
\][/tex]
Where [tex]\(N\)[/tex] is the number of data points, and [tex]\(x_i\)[/tex] are the data points.
[tex]\[
\mu = \frac{4 + 8 + 6 + 5 + 3 + 8 + 9 + 10 + 2 + 6}{10} = \frac{61}{10} = 6.1
\][/tex]
### Step 2: Calculate the Numerator for Variance
The numerator is the sum of the squared differences between each data point and the mean ([tex]\(\mu\)[/tex]).
[tex]\[
\text{Numerator} = (x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2
\][/tex]
Let's calculate each term:
[tex]\[
\begin{align*}
(4 - 6.1)^2 & = (-2.1)^2 = 4.41 \\
(8 - 6.1)^2 & = (1.9)^2 = 3.61 \\
(6 - 6.1)^2 & = (-0.1)^2 = 0.01 \\
(5 - 6.1)^2 & = (-1.1)^2 = 1.21 \\
(3 - 6.1)^2 & = (-3.1)^2 = 9.61 \\
(8 - 6.1)^2 & = (1.9)^2 = 3.61 \\
(9 - 6.1)^2 & = (2.9)^2 = 8.41 \\
(10 - 6.1)^2 & = (3.9)^2 = 15.21 \\
(2 - 6.1)^2 & = (-4.1)^2 = 16.81 \\
(6 - 6.1)^2 & = (-0.1)^2 = 0.01
\end{align*}
\][/tex]
Sum these values up to get the numerator:
[tex]\[
\text{Numerator} = 4.41 + 3.61 + 0.01 + 1.21 + 9.61 + 3.61 + 8.41 + 15.21 + 16.81 + 0.01 = 63.91
\][/tex]
### Step 3: Calculate the Denominator for Variance
The denominator is simply [tex]\(N\)[/tex], the number of data points:
[tex]\[
\text{Denominator} = N = 10
\][/tex]
### Step 4: Calculate the Variance ([tex]\(\sigma^2\)[/tex])
Now, we can calculate the variance using the formula:
[tex]\[
\sigma^2 = \frac{\text{Numerator}}{\text{Denominator}} = \frac{63.91}{10} = 6.391
\][/tex]
### Final Answers
- The numerator evaluates to: [tex]\( 63.91 \)[/tex]
- The denominator evaluates to: [tex]\( 10 \)[/tex]
- The variance equals: [tex]\( 6.391 \)[/tex]
