Question

The mean of a set of numbers is $\overline{x}.$ If each number is decreased by $\lambda ,$ the mean of the new set is

• A. $\overline{x}$
• B. $\overline{x}+\lambda$
• C. $\lambda -\overline{x}$
• D. $\overline{x}-\lambda$

Answer 1

The correct option is D $\overline{x}-\lambda$

Let initial dist. is $x_1,x_2,x_3,.............x_n$
given mean of this distribution is $=\frac{\sum x_i}{n}=\overline{x}$ (i)
Now after decreasing $\lambda$ from each terms new distribution becomes,
$x_1-\lambda ,x_2-\lambda ,.........,x_n-\lambda$
Hence new mean $=\frac{\sum (x_i-\lambda )}{n}=\frac{\sum x_i-\sum \lambda }{n}=\frac{\sum x_i-n\lambda }{n}=\overline{x}-\lambda$