Question

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

Explanation:

Step1: Recall Variance Formula

Variance ($\sigma^2$ for population, $s^2$ for sample) is calculated as the average of the squared differences from the Mean. For a population, variance $\sigma^2=\frac{\sum (x_i - \mu)^2}{N}$, where $N$ is the number of data points, $\mu$ is the mean, and $x_i$ are the data values. For a sample, $s^2=\frac{\sum (x_i - \bar{x})^2}{n - 1}$, but here we check the options. The data points are 4, 7, 4, 10, 5 (5 data points), mean $\mu = 6$.

Step2: Analyze Each Option

• Option 1: Denominator 6 (incorrect, $N = 5$).
• Option 2: Formula for standard deviation (has square root), and incorrect term $(6 - 6)^2$ (not a data point).
• Option 3: Formula for standard deviation (square root) and wrong denominator (4, should be $n - 1$ or $N$, but here $N = 5$).
• Option 4: $\sigma^2=\frac{(4 - 6)^2+(7 - 6)^2+(4 - 6)^2+(10 - 6)^2+(5 - 6)^2}{5}$, which matches population variance formula (sum of squared deviations from mean, divided by number of data points $N = 5$).

Answer:

$\boldsymbol{\sigma^2=\frac{(4 - 6)^2+(7 - 6)^2+(4 - 6)^2+(10 - 6)^2+(5 - 6)^2}{5}}$ (the fourth option, assuming the last option is $\sigma^2=\frac{(4 - 6)^2+(7 - 6)^2+(4 - 6)^2+(10 - 6)^2+(5 - 6)^2}{5}$)