To find the variance of the given data set [tex]\([100, 100, 120, 120, 180]\)[/tex], follow these detailed steps:
1. Calculate the Mean:
The mean (average) is calculated by summing all the values and then dividing by the number of values.
[tex]\[
\text{Mean} = \frac{100 + 100 + 120 + 120 + 180}{5}
\][/tex]
The sum of the values is [tex]\(620\)[/tex].
[tex]\[
\text{Mean} = \frac{620}{5} = 124
\][/tex]
2. Determine each deviation from the Mean:
Subtract the mean from each data point:
[tex]\[
100 - 124 = -24, \quad 100 - 124 = -24, \quad 120 - 124 = -4, \quad 120 - 124 = -4, \quad 180 - 124 = 56
\][/tex]
3. Square each deviation:
Calculate the square of each deviation obtained above:
[tex]\[
(-24)^2 = 576, \quad (-24)^2 = 576, \quad (-4)^2 = 16, \quad (-4)^2 = 16, \quad 56^2 = 3136
\][/tex]
4. Calculate the Mean of the Squared Deviations:
Sum all the squared deviations:
[tex]\[
576 + 576 + 16 + 16 + 3136 = 4320
\][/tex]
Divide this sum by the number of data points (5):
[tex]\[
\text{Variance} = \frac{4320}{5} = 864
\][/tex]
Therefore, the variance of the given data set is [tex]\(864\)[/tex].
