Sure, let's calculate the variance of the given set of data step-by-step.
The data set given is: [tex]\(3, 33, 303, 233, 3, 73, 83, 63\)[/tex].
First, we need to find the mean ([tex]\(\bar{x}\)[/tex]) of the data set:
[tex]\[ \bar{x} = \frac{3 + 33 + 303 + 233 + 3 + 73 + 83 + 63}{8} \][/tex]
Adding up all the values:
[tex]\[
3 + 33 + 303 + 233 + 3 + 73 + 83 + 63 = 794
\][/tex]
Now, divide by the number of data points (N = 8):
[tex]\[
\bar{x} = \frac{794}{8} = 99.25
\][/tex]
Next, we calculate the squared differences from the mean for each data point:
[tex]\[
(3 - 99.25)^2 = 9264.0625
\][/tex]
[tex]\[
(33 - 99.25)^2 = 4389.0625
\][/tex]
[tex]\[
(303 - 99.25)^2 = 41514.0625
\][/tex]
[tex]\[
(233 - 99.25)^2 = 17889.0625
\][/tex]
[tex]\[
(3 - 99.25)^2 = 9264.0625
\][/tex]
[tex]\[
(73 - 99.25)^2 = 689.0625
\][/tex]
[tex]\[
(83 - 99.25)^2 = 264.0625
\][/tex]
[tex]\[
(63 - 99.25)^2 = 1314.0625
\][/tex]
Adding up these squared differences:
[tex]\[
9264.0625 + 4389.0625 + 41514.0625 + 17889.0625 + 9264.0625 + 689.0625 + 264.0625 + 1314.0625 = 84587.5
\][/tex]
Now, we calculate the variance using the formula:
[tex]\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{N - 1}
\][/tex]
[tex]\[
s^2 = \frac{84587.5}{8 - 1}
\][/tex]
[tex]\[
s^2 = \frac{84587.5}{7}
\][/tex]
[tex]\[
s^2 = 12083.92857142857
\][/tex]
Therefore, the variance of the data set is approximately 12083.93.
From the options provided, the closest match to our calculated value is:
[tex]\[
\boxed{12,084}
\][/tex]
