Question

tax data calculating variance
refunds for various taxpayers.
what is the variance of the income in the sample tax data? round answer to the nearest whole number. enter your answer in the box.

Explanation:

Step1: Calculate the mean of the income data

Let the income data be \(x_1 = 75000,x_2 = 50000,x_3 = 90000,x_4 = 120000,x_5 = 45000,x_6 = 110000,x_7 = 65000,x_8 = 80000,x_9 = 95000,x_{10}=55000\).
The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\), where \(n = 10\).
\(\sum_{i=1}^{10}x_i=75000 + 50000+90000+120000+45000+110000+65000+80000+95000+55000 = 785000\).
\(\bar{x}=\frac{785000}{10}=78500\).

Step2: Calculate the squared - differences

\((x_1-\bar{x})^2=(75000 - 78500)^2=(- 3500)^2 = 12250000\).
\((x_2-\bar{x})^2=(50000 - 78500)^2=(-28500)^2 = 812250000\).
\((x_3-\bar{x})^2=(90000 - 78500)^2=(11500)^2 = 132250000\).
\((x_4-\bar{x})^2=(120000 - 78500)^2=(41500)^2 = 1722250000\).
\((x_5-\bar{x})^2=(45000 - 78500)^2=(-33500)^2 = 1122250000\).
\((x_6-\bar{x})^2=(110000 - 78500)^2=(31500)^2 = 992250000\).
\((x_7-\bar{x})^2=(65000 - 78500)^2=(-13500)^2 = 182250000\).
\((x_8-\bar{x})^2=(80000 - 78500)^2=(1500)^2 = 2250000\).
\((x_9-\bar{x})^2=(95000 - 78500)^2=(16500)^2 = 272250000\).
\((x_{10}-\bar{x})^2=(55000 - 78500)^2=(-23500)^2 = 552250000\).

Step3: Calculate the variance

The sample - variance formula is \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\).
\(\sum_{i = 1}^{10}(x_i-\bar{x})^2=12250000+812250000+132250000+1722250000+1122250000+992250000+182250000+2250000+272250000+552250000=5800000000\).
\(s^2=\frac{5800000000}{9}\approx644444444\).

Answer:

644444444