Question

What is the standard deviation of the data given below?

$10,28,13,18,29,30,22,23,25,and32$

Answer 1

Determine the standard deviation.

Given data:

$10,28,13,18,29,30,22,23,25,and32$

Here is the total number of observations$\left(n\right)$ are$10$.

Hence,$\sum _{i=1}^{10}{x}_i=10+28+13+18+29+30+22+23+25+32=230$.

Hence, The Mean:

$\mu =\frac{{\displaystyle \sum _{i=1}^{10}}{x}_i}{n}=\frac{230}{10}=23$.

$x_i$	$x_i-\mu$	$(x_i-\mu {)}^2$
$10$	$10–23=–13$	$169$
$28$	$28-23=5$	$25$
$13$	$13-23=-10$	$100$
$18$	$18-23=-5$	$25$
$29$	$29-23=6$	$36$
$30$	$30-23=0$	$49$
$22$	$22-23=-1$	$1$
$23$	$23-23=0$	$0$
$25$	$25-23=2$	$4$
$32$	$32-23=9$	$81$
$\Sigma (x_i-\mu {)}^2$		$490$

Use the formula to calculate the standard deviation.

$$\begin{gathered} \sqrt{\left[\right(\frac{1}{n}\left)\Sigma \right(x_i–\mu {)}^2]}=\sqrt{\left[\frac{1}{10}\times 490\right]} \\ \Rightarrow =\sqrt{49} \\ \Rightarrow =7 \end{gathered}$$

Hence, the standard deviation of the given data is $7$.