Sure, let's calculate each part step by step. We'll work with the general data set [tex]\( x = \{ x_1, x_2, x_3, \ldots, x_N \} \)[/tex].
### Step 1: Calculate the Mean ([tex]\(\mu\)[/tex])
The mean [tex]\(\mu\)[/tex] of the data set is given by:
[tex]\[
\mu = \frac{1}{N} \sum_{i=1}^{N} x_i
\][/tex]
### Step 2: Calculate the Numerator
The numerator of the variance formula is:
[tex]\[
\sum_{i=1}^{N} (x_i - \mu)^2
\][/tex]
### Step 3: Calculate the Denominator
The denominator of the variance formula is simply:
[tex]\[
N
\][/tex]
### Step 4: Calculate the Variance ([tex]\(\sigma^2\)[/tex])
The variance is given by:
[tex]\[
\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}
\][/tex]
Let's put this into practice with a specific example. Consider a data set [tex]\( x = \{ 2, 4, 4, 4, 5, 5, 7, 9 \} \)[/tex].
1. Calculate the mean ([tex]\(\mu\)[/tex]):
[tex]\[
\mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5
\][/tex]
2. Calculate the numerator:
[tex]\[
\sum_{i=1}^{N} (x_i - \mu)^2 = (2 - 5)^2 + (4 - 5)^2 + (4 - 5)^2 + (4 - 5)^2 + (5 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (9 - 5)^2
\][/tex]
[tex]\[
= (-3)^2 + (-1)^2 + (-1)^2 + (-1)^2 + (0)^2 + (0)^2 + (2)^2 + (4)^2
\][/tex]
[tex]\[
= 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16
\][/tex]
[tex]\[
= 32
\][/tex]
3. Calculate the denominator:
[tex]\[
N = 8
\][/tex]
4. Calculate the variance ([tex]\(\sigma^2\)[/tex]):
[tex]\[
\sigma^2 = \frac{32}{8} = 4
\][/tex]
### Summary
- Numerator: [tex]\( 32 \)[/tex]
- Denominator: [tex]\( 8 \)[/tex]
- Variance: [tex]\( 4 \)[/tex]
