To find the correct equation that shows the variance for the number of miles Fiona biked last week, we need to understand the formula for variance.
The variance of a dataset is calculated as:
[tex]\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \][/tex]
Where:
- [tex]\( \sigma^2 \)[/tex] is the variance.
- [tex]\( N \)[/tex] is the number of data points.
- [tex]\( x_i \)[/tex] represents each data point.
- [tex]\( \mu \)[/tex] is the mean of the data points.
Given the data points: [tex]\( 4, 7, 4, 10, 5 \)[/tex] and the mean [tex]\( \mu = 6 \)[/tex], we will plug in these values into the variance formula.
1. Calculate each deviation from the mean and square it:
[tex]\[
(4 - 6)^2 = (-2)^2 = 4
\][/tex]
[tex]\[
(7 - 6)^2 = (1)^2 = 1
\][/tex]
[tex]\[
(4 - 6)^2 = (-2)^2 = 4
\][/tex]
[tex]\[
(10 - 6)^2 = (4)^2 = 16
\][/tex]
[tex]\[
(5 - 6)^2 = (-1)^2 = 1
\][/tex]
2. Sum these squared deviations:
[tex]\[
4 + 1 + 4 + 16 + 1 = 26
\][/tex]
3. Since we are calculating the population variance (using [tex]\( N = 5 \)[/tex]):
[tex]\[
\sigma^2 = \frac{26}{5} = 5.2
\][/tex]
Thus, the equation that represents the variance correctly is:
[tex]\[ \sigma^2 = \frac{(4-6)^2+(7-6)^2+(4-6)^2+(10-6)^2+(5-6)^2}{5} \][/tex]
Therefore, the correct equation is:
[tex]\[ \sigma^2 = \frac{(4-6)^2+(7-6)^2+(4-6)^2+(10-6)^2+(5-6)^2}{5} \][/tex]
