Question

Find the mean, variance and standard deviation for the following data:
(i) 2, 4, 5, 6, 8, 17.
(ii) 6, 7, 10, 12, 13, 4, 8, 12.
(iii) 227, 235, 255, 269, 292, 299, 312, 321, 333, 348.
(iv) 15, 22, 27, 11, 9, 21, 14, 9.

Answer 1

(i) 2,4,5,6,8,17

$\mathrm{Mean}=\overline{X}=\frac{2+4+5+6+8+17}{6}=\frac{42}{6}=7$

$x_i$	$\left(x_i-\overline{X}\right)=\left(x_{\mathit{i}}-7\right)$	${\left(x_i-7\right)}^2$
2	-5	25
4	-3	9
5	-2	4
6	-1	1
8	1	1
17	10	100
		$\sum _{\mathit{}i=1}^6{\left(x_i\mathit{-}\overline{X}\right)}^{\mathit{2}}=140$

$n=6$

$$\begin{gathered} \mathrm{Variance}\left(X\right)=\frac{{\displaystyle \sum _{i=1}^6}{\left(x_i-\overline{X}\right)}^2}{n} \\ =\frac{140}{6} \\ =23.33 \end{gathered}$$


$$\begin{gathered} \mathrm{Standard}\mathrm{deviation}=\sqrt{\mathrm{Variance}(X}) \\ =\sqrt{23.33} \\ =4.83 \end{gathered}$$



(ii) 6,7,10,12,13,4,8,12

$$\begin{gathered} \mathrm{Mean}=\frac{6+7+10+12+13+4+8+12}{8} \\ =\frac{72}{8} \\ =9 \end{gathered}$$

$x_i$	$\left(x_i-\overline{X}\right)=\left(x_i-9\right)$	${\left(x_{\mathit{i}}\mathit{-}\overline{\mathit{X}}\right)}^2$
6	-3	9
7	-2	4
10	1	1
12	3	9
13	4	16
4	$-$5	25
8	$-$1	1
12	3	9
		$\sum _{i=1}^8{\left(x_{\mathit{i}}-\overline{X}\right)}^2=74$

$n=8$
$$\begin{gathered} \therefore \mathrm{Variance}\left(X\right)=\frac{{\displaystyle \sum _{i=1}^8}{\left(x_i-\overline{X}\right)}^2}{n} \\ =\frac{74}{8} \\ =9.25 \\ \mathrm{Standard}\mathrm{deviation}=\sqrt{\mathrm{Variance}\left(X\right)} \\ =\sqrt{9.25} \\ =3.04 \end{gathered}$$


(iii) 227,235,255,269,292,299,312,321,333,348,


$$\begin{gathered} \mathrm{Mean}=\frac{227+235+255+269+292+299+312+321+333+348}{10} \\ =\frac{2891}{10}=289.1 \end{gathered}$$

$x_i$	$\left(x_{\mathit{i}}-\overline{X}\right)=\left(x_{\mathit{i}}-289.1\right)$	${\left(x_{\mathit{i}}-\overline{X}\right)}^2$
227	− 62.1	3856.41
235	− 54.1	2926.81
255	− 34.1	1162.81
269	− 20.1	404.01
292	2.9	8.41
299	9.9	98.01
312	22.9	524.41
321	31.9	1017.61
333	43.9	1927.21
348	58.9	3469.21
		$\sum _{i=1}^{10}{\left(x_i-\overline{x}\right)}^2=15394.9$

$$\begin{gathered} n=10 \\ \therefore \mathrm{Variance}\left(X\right)=\frac{{\displaystyle \sum _{i=1}^{10}}{\left(x_i-\overline{X}\right)}^2}{n} \\ =\frac{15394.9}{10} \\ =1539.49 \\ \mathrm{Standard}\mathrm{deviation}=\sqrt{\mathrm{Variance}\left(X\right)} \\ =\sqrt{1539.49} \\ =39.24 \end{gathered}$$



(iv) 15,22,27,11,9,21,14,9

$\mathrm{Mean}=\frac{15+22+27+11+9+21+14+9}{8}=\frac{128}{8}=16$

$x_i$	$\left(x_i\mathit{-}\overline{X}\right)\mathit{=}\left(x_i\mathit{-}16\right)$	${\left(x_i\mathit{-}\overline{X}\right)}^{\mathit{2}}$
15	−1	1
22	6	36
27	11	121
11	5	25
9	−7	49
21	5	25
14	−2	4
9	−7	49
		$\sum _{i=1}^8{\left(x_{\mathit{i}}-\overline{X}\right)}^2=310$

$$\begin{gathered} \therefore n=8 \\ \mathrm{Variance}\left(X\right)=\frac{{\displaystyle \sum _{i=1}^8}{\left(x_i-\overline{x}\right)}^2}{n} \\ =\frac{310}{8} \\ =38.75 \\ \mathrm{Standard}\mathrm{deviation}=\sqrt{\mathrm{Variance}\left(X\right)} \\ =\sqrt{38.75} \\ =6.22 \end{gathered}$$