Question

The following table shows the marks obtained by 60 students. Calculate mean and standard deviation.

Marks (more than)	70	60	50	40	30	20
No. of students	7	18	40	40	55	60

Answer 1

Converting more than cumulative frequency into ordinary continuos series:

Marks X	Frequency (f)	Mid-Values (m)	fm	m2	fm2
20 −30 30 −40 40 −50 50 −60 60 −70 70 −80	60-55=5 55-40=15 40-40=0 40-18=22 18-7=11 7-0=7	25 35 45 55 65 75	125 525 0 1210 715 525	625 1225 2025 3025 4225 5625	3125 18375 0 66550 46475 39375
	Σf = 60		Σfm = 3100		Σfm2 = 173900

$$\begin{gathered} \mathrm{Mean}\left(\overline{X}\right)=\frac{\Sigma fm}{\Sigma f}=\frac{3100}{60}=51.67 \\ \mathrm{Standard}\mathrm{deviation}\left(\sigma \right)=\sqrt{\frac{\Sigma f{m}^2}{\Sigma f}-{\left(\overline{X}\right)}^2} \\ =\sqrt{\frac{173900}{60}-{\left(51.67\right)}^2} \\ =\sqrt{2898.33-2669.79} \\ =\sqrt{228.54} \\ =15.11 \end{gathered}$$

Hence, mean and standard deviation of the above series are 51.67 marks and 15.11 marks respectively.