Question

The mean and variance of $7$ observations are $8$ and $16$, respectively. If five observations are $2,4,10,12,14$ then the absolute difference of the remaining two observations is

• A. $1$
• B. $4$
• C. $3$
• D. $2$

Answer 1

The correct option is D

$2$

Explanation for the correct option:

Step 1: Find the relation between the remaining two observations.

Since, it is given that the mean of $7$ observations is $8$ and five observations are $2,4,10,12,14$.

Assume that, the remaining two observations are $x_1$ and $x_2$.

Therefore,

$$\begin{gathered} \frac{{\displaystyle \underset{i=1}{\overset{7}{\sum \left(x\right)}}}}{7}=8 \\ \Rightarrow \underset{i=1}{\overset{7}{\sum \left(x\right)}}=56 \\ \Rightarrow 2+4+10+12+14+x_1+x_2=56 \\ \Rightarrow x_1+x_2=14...1 \end{gathered}$$

Also, it is given that the variance of $7$ observations is $16$.

$$\begin{gathered} \frac{{\displaystyle \underset{i=1}{\overset{7}{\sum \left(x^2\right)}}}}{7}-8^2=16 \\ \Rightarrow \frac{\underset{i=1}{\overset{7}{\sum \left(x^2\right)}}}{7}=80 \\ \Rightarrow \underset{i=1}{\overset{7}{\sum \left(x^2\right)}}=560 \\ \Rightarrow {\left(2\right)}^2+{\left(4\right)}^2+{\left(10\right)}^2+{\left(12\right)}^2+{\left(14\right)}^2+{\left(x_1\right)}^2+{\left(x_2\right)}^2=560 \\ \Rightarrow {\left(x_1\right)}^2+{\left(x_2\right)}^2=100...2 \end{gathered}$$

Step 2: Find the absolute difference between the remaining two observations.

Since, we know that ${(x_1+x_2)}^2-2x_1{x}_2=x_1^2+x_2^2$

From equation $1$ and equation $2$.

$$\begin{gathered} {\left(14\right)}^2-2x_1{x}_2=100 \\ \Rightarrow 196-2x_1{x}_2=100 \\ \Rightarrow 2x_1{x}_2=96...3 \end{gathered}$$

Since, ${\left(x_1-x_2\right)}^2={(x_1+x_2)}^2-4x_1{x}_2$.

From equation $1$ and equation $3$.

$$\begin{gathered} {\left(x_1-x_2\right)}^2={\left(14\right)}^2-2·96 \\ \Rightarrow {\left(x_1-x_2\right)}^2=196-192 \\ \Rightarrow {\left(x_1-x_2\right)}^2=4 \\ \Rightarrow \left|x_1-x_2\right|=2 \end{gathered}$$

Therefore, the absolute difference between the remaining two observations is $2$.

Hence, option $D$ is the correct answer.