Question

the table presents detailed tax data for 10 individuals, including their respective incomes, taxes paid, deductions, and tax refunds.
what is the variance of the incomes in the sample tax data? round your answer to the nearest whole number.
enter your answer in the box.

Explanation:

Step1: Calculate sample mean

First, sum all incomes and divide by sample size $n=10$.
$$\bar{x} = \frac{78340 + 51270 + 114950 + 88560 + 47380 + 104870 + 67490 + 83240 + 91760 + 57210}{10}$$
$$\bar{x} = \frac{785070}{10} = 78507$$

Step2: Compute squared deviations

For each income $x_i$, calculate $(x_i - \bar{x})^2$:

• $(78340-78507)^2 = (-167)^2 = 27889$
• $(51270-78507)^2 = (-27237)^2 = 741854169$
• $(114950-78507)^2 = (36443)^2 = 1328092249$
• $(88560-78507)^2 = (10053)^2 = 101062809$
• $(47380-78507)^2 = (-31127)^2 = 968889129$
• $(104870-78507)^2 = (26363)^2 = 694907769$
• $(67490-78507)^2 = (-11017)^2 = 121374289$
• $(83240-78507)^2 = (4733)^2 = 22401289$
• $(91760-78507)^2 = (13253)^2 = 175642009$
• $(57210-78507)^2 = (-21297)^2 = 453562209$

Step3: Sum squared deviations

Add all the squared values:
$$\sum(x_i - \bar{x})^2 = 27889 + 741854169 + 1328092249 + 101062809 + 968889129 + 694907769 + 121374289 + 22401289 + 175642009 + 453562209$$
$$\sum(x_i - \bar{x})^2 = 4607813810$$

Step4: Calculate sample variance

Divide the sum by $n-1=9$:
$$s^2 = \frac{4607813810}{9} \approx 511979312.22$$

Answer:

511979312