Question

Calculate standard deviation of the given below:

Size	3	4	5	6	7	8	9
Frequency	3	7	22	60	85	32	8

Answer 1

Size (X)	Frequency (f)	fX	$\mathit{x}\mathbf{=}\left(\mathit{X}\mathbf{-}\overline{\mathit{X}}\right)$	x2	fx2
3	3	9	3.59	12.89	38.67
4	7	28	2.59	6.71	46.97
5	22	110	1.59	2.53	55.66
6	60	360	0.59	0.35	21
7	85	595	0.41	0 .17	14.45
8	32	256	1.41	1.99	63.68
9	8	72	2.41	5.81	46.48
	N=Σf=217	ΣfX=1430			Σfx2=286.91

$$\begin{gathered} Mean,\overline{X}=\frac{\Sigma fX}{\Sigma f}=\frac{1430}{217}=6.59 \\ \mathrm{Standard}\mathrm{deviation},\left(\sigma \right)=\sqrt{\frac{\Sigma f{x}^2}{N}} \\ or,\left(\sigma \right)=\sqrt{\frac{286.91}{217}} \\ or,\left(\sigma \right)=\sqrt{1.32} \\ \Rightarrow \left(\sigma \right)=1.149 \end{gathered}$$

Hence, standard deviation of the above series is 1.149