Question

Compute mean marks from the data given below by: (i) Direct method, (ii) Short-cut Method, and (iii) Step Deviation Method.

Marks	5	15	25	35	45	55	65
Students	4	6	10	20	10	6	4

Answer 1

Marks (X)	Students (f)	fX	d=X−A	fd	$\mathit{d}\mathbf{\text{'}}\mathbf{=}\frac{\mathit{d}}{\mathit{i}}\mathbf{=}\frac{\mathit{d}}{\mathbf{10}}$	fd'
5 15 25 $\boxed{35=A}$ 45 55 65	4 6 10 20 10 6 4	20 90 250 700 450 330 260	$-$30 −20 −10 0 10 20 30	−120 −120 −100 0 100 120 120	−3 −2 −1 0 1 2 3	−12 −12 −10 0 10 12 12
	N=Σf= 60	ΣfX= 2100		$\mathbf{\Sigma }\mathit{f}\mathit{d}$=0		$\mathbf{\Sigma }\mathit{f}\mathit{d}\mathbf{\text{'}}$=0

i) Calculating mean marks using direct method:

$$\begin{gathered} \overline{X}=\frac{\Sigma fX}{\Sigma f} \\ \mathrm{or},\overline{X}=\frac{2100}{60} \\ \Rightarrow \overline{X}=35\mathrm{marks} \end{gathered}$$

(ii) Calculating mean marks using short cut method:

$\overline{X}=A+\frac{\Sigma fd}{\Sigma f}$

Here,

A represents assumed mean.
d represents the deviation of the values from the assumed mean.

Here, we take 35 as the assumed mean. So, we take deviations of each item in the series from 35.

$$\begin{gathered} \overline{X}=35+\frac{0}{60} \\ \Rightarrow \overline{X}=35\mathrm{marks} \end{gathered}$$

(iii) Calculating mean marks using step-deviation method:

$\overline{X}=A+\frac{\Sigma fd\text{'}}{\Sigma f}\times i$

$$\begin{gathered} \overline{X}=35+\frac{0}{60}\times 10 \\ \Rightarrow \overline{X}=35\mathrm{marks} \end{gathered}$$

Hence, the mean marks of the above data is 35