Question

The mean of a set of observation is $a$. If each observation is multiplied by $b$ and each product is decreased by $c$ then the mean of new set of observation is

• A. $\frac{a}{b}+c$
• B. $ab-c$
• C. $\frac{a}{b}-c$
• D. $ab+c$

Answer 1

Answer: (B)

$ab-c$

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

Given that mean of a set of observations is $a$

Let the number of observations be $n$

Let the observations be $x_1,x_2,x_3,...,x_n$

Therefore, $a=\frac{x_1+x_2+...+x_n}{n}$

$⟹x_1+x_2+...+x_n=an$ -------(1)

Given that each observation is multiplied by $b$ and the product is decreased by $c$

Therefore the new observations are $b{x}_1-c,b{x}_2-c,b{x}_3-c,...,b{x}_n-c$

Therefore, the new mean is $\frac{b{x}_1-c+b{x}_2-c+...+b{x}_n-c}{n}$

$⟹\text{new mean}=\frac{b{x}_1+b{x}_2+...+b{x}_n-nc}{n}$

$⟹\text{new mean}=\frac{b(x_1+x_2+...+x_n)-nc}{n}$

$⟹\text{new mean}=\frac{b\left(an\right)-nc}{n}$ (from (1))

$⟹\text{new mean}=\frac{n(ab-c)}{n}$

$⟹\text{new mean}=ab-c$