Question

From the following figures, find the standard deviation and the coefficient of variation:

Marks	0−10	10−20	20−30	30−40	40−50	50−60	60−70	70−80
No. of persons	5	10	20	40	30	20	10	4

Answer 1

Marks (X)	No. of Persons (f)	Mid-Values (m)	fm	m2	fm2
0−10	5	5	25	25	125
10−20	10	15	150	225	2250
20−30	20	25	500	625	12500
30−40	40	35	1400	1225	49000
40−50	30	45	1350	2025	60750
50−60	20	55	1100	3025	60500
60−70	10	65	650	4225	42250
70−80	4	75	300	5625	22500
	Σf=139		Σfm=5475		Σfm2=249875

$$\begin{gathered} Mean,\overline{X}=\frac{\Sigma fm}{\Sigma f}=\frac{5475}{139}=39.39 \\ \mathrm{Standard}\mathrm{deviation},\left(\sigma \right)=\sqrt{\frac{\Sigma f{m}^2}{\Sigma f}-{\left(\overline{X}\right)}^2} \\ or,\left(\sigma \right)=\sqrt{\frac{249875}{139}-{\left(39.39\right)}^2} \\ or,\left(\sigma \right)=\sqrt{1797.66-1551.57} \\ or,\left(\sigma \right)=\sqrt{246.09} \\ \Rightarrow \left(\sigma \right)=15.69\mathrm{marks} \\ \mathrm{Cofficient}\mathrm{of}\mathrm{Variation}=\frac{\left(\sigma \right)}{\overline{X}}\times 100=\frac{15.69}{39.39}\times 100=39.83\% \end{gathered}$$