Calculate the variance of the following set of data: [tex]$3, 33, 303, 233, 3, 73, 83, 63$[/tex]

Use [tex]$s^2=\frac{\sum\left(x_i-\bar{x}\right)^2}{(N-1)}$[/tex].

A. 110
B. 12,084
C. 103
D. 12,333



Answer :

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]