Question

if the mean of observations x, x + 2, x + 4,
x + 6 and x + 8 is 11, find :
(i) the value of x;
(ii) the mean of the first three observations.​

Answer 1

Answer

First, we should find the value of x. To find the value of x, we use the formula that is used to calculate the mean.

Value of x :

$\sf \dashrightarrow Mean = \dfrac{Sum \: of \: all \: observations}{Number \: of \: observations}$

$\sf \dashrightarrow 11 = \dfrac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}{5}$

$\sf \dashrightarrow 11 = \dfrac{5x + 2 + 4 + 6 + 8}{5}$

$\sf \dashrightarrow 11 = \dfrac{5x + 12}{5}$

$\sf \dashrightarrow \dfrac{5x + 12}{5} = 11$

$\sf \dashrightarrow 5x + 12 = 11 \times 5$

$\sf \dashrightarrow 5x + 12 = 55$

$\sf \dashrightarrow 5x = 55 - 12$

$\sf \dashrightarrow 5x = 43$

$\sf \dashrightarrow x = \dfrac{43}{5}$

$\sf \dashrightarrow x = 8.6$

Now, we should find the values of first three observations.

Value of first observation :

$\sf \dashrightarrow x = 8.6$

Value of second observation :

$\sf \dashrightarrow x + 2 = 8.6 + 2$

$\sf \dashrightarrow 10.6$

Value of third observation :

$\sf \dashrightarrow x + 4 = 8.6 + 5$

$\sf \dashrightarrow 13.6$

Now, we can find the mean of these three observations.

Mean of first three observations :

$\sf \dashrightarrow Mean = \dfrac{Sum \: of \: all \: observations}{Number \: of \: observations}$

$\sf \dashrightarrow \dfrac{8.6 + 10.6 + 13.6}{3}$

$\sf \dashrightarrow \dfrac{19.2 + 13.6}{3}$

$\sf \dashrightarrow \dfrac{32.8}{3}$

$\sf \dashrightarrow \dfrac{328}{30}$

$\sf \dashrightarrow Mean = 10.93$

Hence, the value of x is 8.6 and the mean of first three observations is 10.93.