Question

employee | weekly salary
arja | $245
raz | $300
natalie | $325
mic | $465
paul | $100
what is the variance for the data?
variance:
$s^2 = \frac{\left(x_1 - \bar{x}\
ight)^2 + \left(x_2 - \bar{x}\
ight)^2 + \dots + \left(x_n - \bar{x}\
ight)^2}{n - 1}$
\bigcirc 118.35
\bigcirc 132.32
\bigcirc 14,006
\bigcirc 17,507.5

Explanation:

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

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

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

• For Arja: $(245 - 287)^2 = (-42)^2 = 1764$
• 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: $1764 + 169 + 1444 + 31684 + 34969 = 70030$.

Step4: Calculate the variance

Using the formula $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$, substitute $n = 5$:
$s^2 = \frac{70030}{5 - 1} = \frac{70030}{4} = 17507.5$? Wait, no—wait, recalculate the sum:
Wait, $245+300=545; 545+325=870; 870+465=1335; 1335+100=1435$. Mean is 287. Correct.
Now, $(245-287)^2 = (-42)^2 = 1764$
$(300-287)^2 = 13^2 = 169$
$(325-287)^2 = 38^2 = 1444$
$(465-287)^2 = 178^2 = 31684$
$(100-287)^2 = (-187)^2 = 34969$
Sum: $1764 + 169 = 1933; 1933 + 1444 = 3377; 3377 + 31684 = 35061; 35061 + 34969 = 70030$.
Then $n - 1 = 4$, so $70030 / 4 = 17507.5$? Wait, but the options include 17,507.5. Wait, maybe I miscalculated? Wait, no—wait, the formula given is for sample variance ($n-1$), but let's check again. Wait, the data: Arja ($245$), Raz ($300$), Natalie ($325$), Mic ($465$), Paul ($100$). Yes. So sum of squares: 1764 + 169 + 1444 + 31684 + 34969. Let's add 1764 + 169 = 1933; 1933 + 1444 = 3377; 3377 + 31684 = 35061; 35061 + 34969 = 70030. Then 70030 / 4 = 17507.5. Wait, but the option is 17,507.5. So that's the answer. Wait, but earlier I thought maybe a miscalculation, but no—let's check the options. The last option is 17,507.5. So that's correct.

Answer:

17,507.5 (Option: 17,507.5)