Question

The arithmetic mean of n observations is $\overline{x}$. If the sum of (n – m) observations is A, then the mean of remaining m observations is

• A. $\frac{n\overline{x}+A}{n-m}$
• B. $\frac{n\overline{x}-A}{m}$
• C. $\frac{n\overline{x}-A}{n-m}$
• D. $\frac{n\overline{x}+A}{m}$

Answer 1

The correct option is B $\frac{n\overline{x}-A}{m}$

Arithmetic mean of n observations = $\overline{x}$
If $x_1+x_2+.....,x_n$ are n observations then $\frac{x_1+x_2+.....+x_n}{n}=\overline{x}$
$\Rightarrow x_1+x_2+.....+x_n=n\overline{x}$ ...........(i)
Also given that the sum of (n-m) observations is A.
$\therefore x_1+x_2+.....+x_{n-m}=A$ .........(ii)
From (i)
$x_1+....+x_{n-m}+x_{n-m+1}+.....+x_n=n\overline{x}$
$\Rightarrow A+x_{n-m+1}+.....+x_n=n\overline{x}$
$\Rightarrow x_{n-m+1}+x_{n-m+2}+.....+x_n=n\overline{x}-A$ [from (i) and (ii)]
$\therefore$ Mean of last m observations,
$\frac{x_{n-m+1}+x_{n-m+2}+.....+x_n}{m}=\frac{n\overline{x}-A}{m}$
Hence, the correct answer is option b.