To calculate the variance of the weekly salaries of the employees at the local bank, we need to follow a series of mathematical steps. Here are the salaries given:
- Anja: \[tex]$245
- Raz: \$[/tex]300
- Natalie: \[tex]$325
- Mic: \$[/tex]465
- Paul: \$100
First, we calculate the mean (average) of the salaries.
[tex]\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
\][/tex]
where [tex]\( x_i \)[/tex] are the individual data points (salaries), and [tex]\( n \)[/tex] is the number of data points.
[tex]\[
\bar{x} = \frac{245 + 300 + 325 + 465 + 100}{5} = \frac{1435}{5} = 287
\][/tex]
Next, we calculate the variance using the formula:
[tex]\[
s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}
\][/tex]
[tex]\[
s^2 = \frac{(245 - 287)^2 + (300 - 287)^2 + (325 - 287)^2 + (465 - 287)^2 + (100 - 287)^2}{5 - 1}
\][/tex]
Calculate each squared deviation:
[tex]\[
(245 - 287)^2 = (-42)^2 = 1764
\][/tex]
[tex]\[
(300 - 287)^2 = (13)^2 = 169
\][/tex]
[tex]\[
(325 - 287)^2 = (38)^2 = 1444
\][/tex]
[tex]\[
(465 - 287)^2 = (178)^2 = 31684
\][/tex]
[tex]\[
(100 - 287)^2 = (-187)^2 = 34969
\][/tex]
Now sum these squared deviations:
[tex]\[
1764 + 169 + 1444 + 31684 + 34969 = 70030
\][/tex]
Finally, divide by the number of data points minus one (degrees of freedom):
[tex]\[
s^2 = \frac{70030}{4} = 17507.5
\][/tex]
Hence, the variance for the given data set is:
[tex]\[
s^2 = 17507.5
\][/tex]
