Question

Following is the monthly income of eight families in a locality
Monthly Income (₹) : 70, 10, 500, 75, 13, 250, 8, 42
Find out arithmetic mean using Direct Method and Short-cut Method.

Answer 1

Families	Monthly income (X)	Deviation from the Assumed average (d = X − A) (A = 75)
1 2 3 4 5 6 7 8	70 10 500 75 (A) 13 250 8 42	−5 −65 425 0 −62 175 −67 −33
(N) = 8	(ΣX)=968	Σd = −232 + 600 = 368

where,
N= Number of items
$\Sigma X$= Total of monthly income of all the families
$\Sigma d$= net sum of deviations

Calculation of Mean by Direct Method
$\overline{X}=\frac{\Sigma X}{N}=\frac{968}{8}=121$

Calculation of mean by short cut method
$$\begin{gathered} \overline{X}=\mathrm{A}+\frac{\Sigma d}{N} \\ \mathrm{or},\overline{X}=75+\frac{368}{8} \\ \mathrm{or},\overline{X}=75+46 \\ \Rightarrow \overline{X}=121 \end{gathered}$$

So, mean monthly income of a family is Rs 121.