The mean of 6 distinct observations
Question:

The mean of 6 distinct observations is $6.5$ and their variance is $10.25$. If 4 out of 6 observations are 2 , 4,5 and 7 , then the remaining two observations are:

  1. 10, 11

  2. 3, 18

  3. 8, 13

  4. 1, 20


Correct Option: 1

Solution:

Let other two numbers be $\mathrm{a},(21-\mathrm{a})$

Now,

$10.25=\frac{\left(4+16+25+49+a^{2}+(21-a)^{2}\right)}{6}-(6.5)^{2}$

(Using formula for variance)

$\Rightarrow 6(10.25)+6(6.5)^{2}=94+a^{2}+(21-a)^{2}$

$\Rightarrow a^{2}+(21-a)^{2}=221$

$\therefore a=10$ and $(21-a)=21-10=11$

So, remaining two observations are 10,11 .

$\Rightarrow$ Option (1) is correct.

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