To find the variance of the given salaries, we will follow a detailed, step-by-step approach:
### Step 1: List the Salaries
We have the following salaries:
- Employee: [tex]$245
- Anja: $[/tex]300
- Raz: [tex]$325
- Natalie: $[/tex]465
- Mic: $100
### Step 2: Calculate the Mean
First, we calculate the mean (average) of the salaries.
[tex]\[
\bar{x} = \frac{245 + 300 + 325 + 465 + 100}{5}
\][/tex]
Summing these salaries:
[tex]\[
245 + 300 + 325 + 465 + 100 = 1435
\][/tex]
Next, we divide the sum by the number of salaries (5):
[tex]\[
\bar{x} = \frac{1435}{5} = 287.0
\][/tex]
### Step 3: Calculate the Squared Differences from the Mean
Next, we for each salary, calculate the squared difference from the mean:
[tex]\[
\left(x_1 - \bar{x}\right)^2 = (245 - 287.0)^2 = (-42)^2 = 1764
\][/tex]
[tex]\[
\left(x_2 - \bar{x}\right)^2 = (300 - 287.0)^2 = (13)^2 = 169
\][/tex]
[tex]\[
\left(x_3 - \bar{x}\right)^2 = (325 - 287.0)^2 = (38)^2 = 1444
\][/tex]
[tex]\[
\left(x_4 - \bar{x}\right)^2 = (465 - 287.0)^2 = (178)^2 = 31684
\][/tex]
[tex]\[
\left(x_5 - \bar{x}\right)^2 = (100 - 287.0)^2 = (-187)^2 = 34969
\][/tex]
### Step 4: Sum the Squared Differences
Sum these squared differences:
[tex]\[
1764 + 169 + 1444 + 31684 + 34969 = 70030
\][/tex]
### Step 5: Calculate the Variance
Finally, the variance is given by the sum of the squared differences divided by [tex]\(n-1\)[/tex], where [tex]\(n\)[/tex] is the number of salaries (5 in this case).
[tex]\[
s^2 = \frac{70030}{5 - 1} = \frac{70030}{4} = 17507.5
\][/tex]
Therefore, the variance for the given data is:
[tex]\[
\boxed{17507.5}
\][/tex]
