To determine the variance for the number of miles Fiona biked last week, we need to follow these steps:
1. Calculate the differences between each data point and the mean: Each individual data point is subtracted from the mean. Given the data points [tex]\(4, 7, 4, 10, 5\)[/tex] and the mean [tex]\(\mu = 6\)[/tex]:
[tex]\[
(4 - 6), (7 - 6), (4 - 6), (10 - 6), (5 - 6)
\][/tex]
2. Square these differences: After calculating the differences, we square each result:
[tex]\[
(4 - 6)^2, (7 - 6)^2, (4 - 6)^2, (10 - 6)^2, (5 - 6)^2
\][/tex]
This results in:
[tex]\[
(4 - 6)^2 = 4, \quad (7 - 6)^2 = 1, \quad (4 - 6)^2 = 4, \quad (10 - 6)^2 = 16, \quad (5 - 6)^2 = 1
\][/tex]
3. Sum these squared differences: Add all the squared differences together:
[tex]\[
4 + 1 + 4 + 16 + 1 = 26
\][/tex]
4. Divide by the number of data points to find the variance: Since we are calculating the population variance (denoted as [tex]\(\sigma^2\)[/tex]), we divide by the number of data points, [tex]\( n = 5 \)[/tex]:
[tex]\[
\sigma^2 = \frac{26}{5} = 5.2
\][/tex]
From the options provided, the correct choice that represents the equation for the variance is:
[tex]\[
\sigma^2 = \frac{(4-6)^2 + (7-6)^2 + (4-6)^2 + (10-6)^2 + (5-6)^2}{5}
\][/tex]
