Question

Find out standard deviation of the distribution of population in 104 villages of a Tehsil, as given below by step-deviation method.

Distribution of Population

Population	No. of Villages
0−200 200−400 400−1,000 1,000−2,000 2,000−5,000	10 28 42 18 6

Answer 1

Population	Mid Value (m)	No. of villages (f)	Deviation assumed from average ${\mathit{d}}_{\mathit{x}}\mathbf{=}\mathit{m}\mathbf{-}\mathit{A}$	$d_x\text{'}=\frac{d_x}{i}=\frac{d_x}{40}$	$\mathit{f}{\mathit{d}}_{\mathbf{x}}\mathbf{\text{'}}\mathbf{=}\mathit{f}\mathbf{\times }{\mathit{d}}_{\mathbf{x}}\mathbf{\text{'}}$	$\mathit{f}{\mathit{d}}_{\mathbf{x}}{\mathbf{\text{'}}}^{\mathbf{2}}\mathbf{=}\mathit{f}\mathbf{\times }{\mathit{d}}_{\mathbf{x}}{\mathbf{\text{'}}}^{\mathbf{2}}$
0-200 200-400 400-1000 1000-2000 2000-5000	100 300 $\boxed{\mathbf{700}\mathbf{=}\mathbf{A}}$ 1500 3500	10 28 42 18 6	-600 -400 0 800 2800	-15 -10 0 20 70	-150 -280 0 360 420	2250 2800 0 7200 29400
		N=104			$\sum _{}f{d}_x\text{'}$=$-$350	$\sum _{}f{d}_x{\text{'}}^2$=41650

Calculating Standard deviation through Step deviation method:

$$\begin{gathered} \mathrm{Standard}\mathrm{Deviation}\left(\mathrm{\sigma }\right)=\sqrt{\frac{\mathrm{\Sigma }f{d}_x{\text{'}}^2}{N}-{\left(\frac{\mathrm{\Sigma }f{d}_x\text{'}}{N}\right)}^2}\times i \\ \mathrm{or},\mathrm{\sigma }=\sqrt{\frac{41650}{104}-{\left(\frac{-350}{104}\right)}^2}\times 40 \\ \mathrm{or},\mathrm{\sigma }=\sqrt{400.480-11.296}\times 40 \\ \mathrm{or},\mathrm{\sigma }=\sqrt{389.184}\times 40 \\ \mathrm{or},\mathrm{\sigma }=19.72775\times 40 \\ \Rightarrow \mathrm{\sigma }=789.11 \end{gathered}$$

Hence, standard deviation of the above series is 789.11