Question

The following table shows the marks obtained by 90 students in a certain examination. Calculate the average marks per students by step deviation method.

Marks	10−20	20−30	30−40	40−50	50−60	60−70	70−80	80−90
No. of students	7	13	20	25	10	8	6	1

Answer 1

Class Interval (Marks)	Mid Values (m)	Students (f)	d=m−A	$\mathit{d}\mathbf{\text{'}}\mathbf{=}\frac{\mathit{d}}{\mathit{i}}\mathbf{=}\frac{\mathit{d}}{\mathbf{10}}$	fd'
10 − 20 20 − 30 30 − 40 40 − 50 50 − 60 60 − 70 70 − 80 80 − 90	15 25 35 $\boxed{45=A}$ 55 65 75 85	7 13 20 25 10 8 6 1	−30 −20 −10 0 10 20 30 40	−3 −2 −1 0 1 2 3 4	−21 −26 −20 0 10 16 18 4
		$\mathit{\Sigma }\mathit{f}$=90			$\mathit{\Sigma }\mathit{f}\mathit{d}\mathbf{\text{'}}$=−19

Calculating mean marks using step-deviation method:

$$\begin{gathered} \overline{\mathrm{X}}=A+\frac{\Sigma fd\text{'}}{\Sigma f}\times i \\ or,\overline{\mathrm{X}}=45+\frac{\left(-19\right)}{90}\times 10 \\ or,\overline{\mathrm{X}}=45-\frac{19}{9} \\ \Rightarrow \overline{\mathrm{X}}=42.89\mathrm{Marks} \end{gathered}$$

Hence, the mean of the above series is 42.89 marks.