Use the equation for variance below, along with the given data set, to answer the following questions.

[tex]\[ \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \][/tex]

1. What does the numerator evaluate to? [tex]\(\_\_\_\_\_\_\_\)[/tex]
2. What does the denominator evaluate to? [tex]\(\_\_\_\_\_\_\_\)[/tex]
3. The variance equals: [tex]\(\_\_\_\_\_\_\_\)[/tex]



Answer :

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]