Question

In order to make the computation of the arithmetic mean of a set of $50$ numbers simpler each observation is subtracted from $53$ and the arithmetic mean of the set of differences is found to be $-3.5$. The arithmetic mean of the set of given numbers is

• A. $53.07$
• B. $52.93$
• C. $56.50$
• D. $49.50$

Answer 1

The correct option is D $49.50$

Mean is the sum of the values divided by the total number of values.

Given that arithmetic mean of the $50$ numbers when each observation is subtracted by $53$ is $-3.5$

Let the $50$ numbers be $x_1,x_2,...,x_{50}$

Therefore, $-3.5=\frac{(x_1-53)+(x_2-53)+...+(x_{50}-53)}{50}$

$⟹x_1+x_2+...+x_{50}-50\times 53=-3.5\times 50$

$⟹x_1+x_2+...+x_{50}=2475$ ------(1)

Mean of $50$ numbers is $\frac{x_1+x_2+...+x_{50}}{50}$

$⟹mean=\frac{x_1+x_2+...+x_{50}}{50}$

$⟹mean=\frac{2475}{50}$ (from (1))

$⟹mean=49.5$