Question

The mean of 15 observations is 8. If the mean of the values of first eight observations is 7 and that of the values of last eight observations is 10, then what is the value of the 8th observation?

Answer 1

Let 15 observations be $x_1,x_2,x_3,\dots ,x_{15}$

$Mean=\frac{Sumofvaluesofallobservations}{Numberofobservations}$

$\Rightarrow$Sum of values of all observations = Mean x Number of observations
Sum of values of all observations = 15 $\times$ 8 = 120.
i.e., $x_1+x_2+x_3+\dots +x_{15}=120$----(1)
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Mean of first 8 observations = 7
Hence, the sum of first 8 observation = 7 $\times$ 8 = 56
$x_1+x_2+x_3+\dots +x_8=56$----(2)
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Mean of last 8 observations = 10
Hence, the sum of the last 8 observation = 10 $\times$8 = 80
$x_8+x_9+x_{10}+\dots +x_{15}=80$-----(3)
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Adding 2 and 3 we get,
$(x_1+x_2+x_3+,\dots ,+x_8)+(x_8+x_9+x_{10}+,\dots ,+x_{15})=56+80$

$\Rightarrow x_1+x_2+x_3+\dots +x_{15}+x_8=136$
substituting 1 in the above equation, we get,
$120+x_8=136$
$\Rightarrow x_8=136-120=16$

Hence, the value of $x_8$ is 16
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