Question

4, 7, 4, 10, 5
the mean is given by $mu = 6$. which equation shows the variance for the number of miles fiona biked last week?
$ s^{2}=\frac{(4 - 6)^{2}+(7 - 6)^{2}+(4 - 6)^{2}+(10 - 6)^{2}+(5 - 6)^{2}}{6}$
$sigma=sqrt{\frac{(4 - 6)^{2}+(7 - 6)^{2}+(4 - 6)^{2}+(10 - 6)^{2}+(5 - 6)^{2}}{5}}$
$sigma=sqrt{\frac{(4 - 6)^{2}+(7 - 6)^{2}+(4 - 6)^{2}+(10 - 6)^{2}+(5 - 6)^{2}}{4}}$
$sigma^{2}=\frac{(4 - 6)^{2}+(7 - 6)^{2}+(4 - 6)^{2}+(10 - 6)^{2}+(5 - 6)^{2}}{5}$

Explanation:

Step1: Recall variance formula

The formula for the variance $s^{2}$ of a sample of data points $x_1,x_2,\cdots,x_n$ with mean $\mu$ is $s^{2}=\frac{\sum_{i = 1}^{n}(x_i-\mu)^2}{n - 1}$ for a sample and $\sigma^{2}=\frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n}$ for a population. Here, we assume population - level variance calculation as no indication of sample is given. We have $n = 5$ data points $x_1=4,x_2 = 7,x_3=4,x_4 = 10,x_5=5$ and $\mu = 6$.

Step2: Identify the correct formula

The variance formula $\sigma^{2}=\frac{(4 - 6)^2+(7 - 6)^2+(4 - 6)^2+(10 - 6)^2+(5 - 6)^2}{5}$ is the correct one for population - level variance.

Answer:

The correct equation for the variance is $\sigma^{2}=\frac{(4 - 6)^2+(7 - 6)^2+(4 - 6)^2+(10 - 6)^2+(5 - 6)^2}{5}$ (the last option in the original multiple - choice list).