Question

Using step-deviation method, calculate standard deviation of the following series:

Marks	0−10	10−20	20−30	30−40	40−50	50−60	60−70	70−80
Number of Students	5	10	20	40	30	20	10	4

Answer 1

Marks (X)	Mid value (m)	Frequency (f)	fm	Deviation from mean value $\left(\mathit{x}\mathbf{=}\mathit{X}\mathbf{-}\overline{\mathbf{X}}\right)$ $\overline{\mathit{X}}$=39.39	x2	fx2
0 − 10 10 − 20 20 − 30 30 − 40 40 − 50 50 − 60 60 − 70 70 − 80	5 15 25 35 45 55 65 75	5 10 20 40 30 20 10 4	25 150 500 1400 1350 1100 650 300	−34.39 −24.39 −14.39 −4.39 5.61 15.61 25.61 35.61	1182.67 594.87 207.07 19.27 31.47 243.67 655.87 1268.07	5913.35 5948.7 4141.4 770.8 944.1 4873.4 6558.7 5072.28
		Σf = 139	Σfm = 5475			Σfx2 = 34222.73

$$\begin{gathered} \mathrm{Mean}\left(\overline{\mathrm{X}}\right)=\frac{\mathrm{\Sigma }fm}{N}=\frac{5475}{139}=39.39 \\ \mathrm{Standard}\mathrm{deviation}(\mathrm{\sigma })=\sqrt{\frac{\mathrm{\Sigma }f{x}^2}{N}}=\sqrt{\frac{34222.73}{139}}=\sqrt{246.21}=15.7 \end{gathered}$$

Hence, the standard deviation of the above series is 15.7