Question

find the variance of the data. 12, 53, 141, 219, 500
\\( \overline { x } = 185 \\)
variance \\( ( \sigma ^ { 2 } ) = \\) ?

Explanation:

Step1: Calculate each data point minus the mean

$$\begin{align*} 12 - 185&=- 173\\ 53 - 185&=-132\\ 141 - 185&=-44\\ 219 - 185&=34\\ 500 - 185&=315 \end{align*}$$

Step2: Square the results from Step1

$$\begin{align*} (-173)^2&=29929\\ (-132)^2&=17424\\ (-44)^2&=1936\\ 34^2&=1156\\ 315^2&=99225 \end{align*}$$

Step3: Find the average of the squared results (since variance formula for a population is $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}$ and here $n = 5$)

$$\begin{align*} \sigma^{2}&=\frac{29929+17424 + 1936+1156+99225}{5}\\ &=\frac{149670}{5}\\ &=29934 \end{align*}$$

Answer:

$29934$