Question

If each entry of a data is increased by $5,$ then the arithmetic mean

• A. remains the same
• B. increases by $5$
• C. decreases by $5$
• D. none of the foregoing

Answer 1

The correct option is B increases by $5$

$\Rightarrow$ Let there are $n$ observations are follow :

$a_1,a_2,a_3,......,a_n$

The mean of above observation is given by :

${\overline{X}}_n=\frac{a_1+a_2+a_3+...+a_n}{n}$

$\Rightarrow$ $a_1+a_2+a_3+....+a_n=n{\overline{X}}_n$ -------- ( 1 )

Now each observation is increased by $5$, therefore, the new observations are :

$(a_1+5),(a_2+5),(a_3+5)....+(a_n+5)$

The mean of above observation is given by

$\overline{X}=\frac{(a_1+5)+(a_2+5)+....+(a_n+5)}{n}$

$=\frac{(a_1+a_2+...+a_n)+5\times n}{n}$

$=\frac{n{\overline{X}}_n+5n}{n}$ [ Using ( 1 ) ]

$=\frac{n[{\overline{X}}_n+5]}{n}$

$\therefore$ $\overline{X}={\overline{X}}_n+5$

$\therefore$ The new arithmetic mean also increased by $5$ from the previous arithmetic mean.