Question

Calculate standard deviation of the following series:

Size	0−5	5−10	10−15	15−20	20−25	25−30	30−35	35−40
Frequency	2	5	7	13	21	16	8	3

Answer 1

Size	Mid Value m	Frequency	Deviation from assumed Average (dx = m − A)	$$\begin{gathered} \mathit{d}\mathit{x}\mathbf{\text{'}}\mathbf{=}\frac{\mathit{d}\mathit{x}}{\mathrm{C}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{C}\mathbf{=}\mathbf{5} \end{gathered}$$	fdx'	fdx'2
0 − 5 5 − 10 10 − 15 15 − 20 20 − 25 25 − 30 30 − 35 35 − 40	2.5 7.5 12.5 17.5(A) 22.5 27.5 32.5 37.5	2 5 7 13 21 16 8 3	−15 −10 −5 0 5 10 15 20	−3 −2 −1 0 1 2 3 4	−6 −10 −7 0 21 32 24 12	18 20 7 0 21 64 72 48
		N = 75			Σfdx' = 66	Σfdx'2 = 250

Calculating Standard deviation through Step deviation method:

$$\begin{gathered} \mathrm{Standard}\mathrm{deviation}(\mathrm{\sigma })=\sqrt{\frac{\mathrm{\Sigma }fdx{\text{'}}^2}{N}-{\left(\frac{\mathrm{\Sigma }fdx\text{'}}{N}\right)}^2}\times C \\ \mathrm{or},\mathrm{\sigma }=\sqrt{\frac{\mathrm{\Sigma }fdx{\text{'}}^2}{N}-{\left(\frac{\mathrm{\Sigma }fdx\text{'}}{N}\right)}^2}\times 5 \\ \mathrm{or},\mathrm{\sigma }=\sqrt{\frac{250}{75}-{\left(\frac{66}{75}\right)}^2}\times 5 \\ \mathrm{or},\mathrm{\sigma }=\sqrt{3.33-0.7744}\times 5 \\ \mathrm{or},\mathrm{\sigma }=\sqrt{2.5556}\times 5 \\ \mathrm{or},\mathrm{\sigma }=1.598\times 5 \\ \Rightarrow \mathrm{\sigma }=7.99 \end{gathered}$$

Hence, standard deviation of the above series is 7.99.