Question

the weekly salaries of a sample of employees at the local bank are given in the table.

employee	weekly salary
raz	$300
natalie	$325
mic	$465
paul	$100

what is the variance for the data?
variance: $s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}$
options: 17,507.5; 132.32; 14,006; 118.35

Explanation:

Step1: Calculate the mean ($\bar{x}$)

First, find the sum of the salaries: $245 + 300 + 325 + 465 + 100 = 1435$.
There are $n = 5$ employees, so the mean $\bar{x} = \frac{1435}{5} = 287$.

Step2: Calculate each $(x_i - \bar{x})^2$

• For Anja: $(245 - 287)^2 = (-42)^2 = 1764$
• For Raz: $(300 - 287)^2 = (13)^2 = 169$
• For Natalie: $(325 - 287)^2 = (38)^2 = 1444$
• For Mic: $(465 - 287)^2 = (178)^2 = 31684$
• For Paul: $(100 - 287)^2 = (-187)^2 = 34969$

Step3: Sum the squared differences

Sum these values: $1764 + 169 + 1444 + 31684 + 34969 = 69930$.

Step4: Calculate the variance

Use the formula $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$. Here, $n - 1 = 4$, so $s^2 = \frac{69930}{4} = 17482.5$? Wait, no, wait, I must have miscalculated the sum. Wait, let's recalculate the squared differences:

Wait, Anja: 245-287 = -42, squared is 1764. Correct.

Raz: 300-287=13, squared 169. Correct.

Natalie: 325-287=38, squared 1444. Correct.

Mic: 465-287=178, squared: 178178. Let's calculate 180^2=32400, minus 2180*2 + 2^2= 32400 - 720 +4=31684. Correct.

Paul: 100-287=-187, squared: 187187. 180^2=32400, 7^2=49, 21807=2520. So (180+7)^2=180^2 + 2180*7 +7^2=32400+2520+49=34969. Correct.

Now sum: 1764 + 169 = 1933; 1933 + 1444 = 3377; 3377 + 31684 = 35061; 35061 + 34969 = 70030. Ah, I had a typo earlier. So sum is 70030.

Then variance: 70030 / 4 = 17507.5.

Answer:

17,507.5