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Question

The number $$6, 8, 10, 12, 13$$ and $$x$$ are arranged in an ascending order. If the mean of the observations is equal to the median, find the value of $$x$$


Solution

Numbers in ascending order are $$6, 8, 10, 12, 13, x$$.
Mean $$= \dfrac {6 + 8 + 10 + 12 + 13 + x}{6} = \dfrac {49 + x}{6}$$
No. of terms $$(n) = 6$$ (even)
Median $$= \dfrac {\left (\dfrac {n}{2}\right )^{th} \text{term}+ \left (\dfrac {n}{2} + 1\right )^{th} \text{term}}{2}$$
Median $$= \dfrac {\left (\dfrac {6}{2}\right )^{th} \text{term} + \left (\dfrac {6}{2} + 1\right )^{th} \text{term}}{2} = \dfrac {3^{rd} + 4^{th}}{2}$$
$$= \dfrac {10 + 12}{2} = \dfrac {22}{2} = 11$$
According to given condition
$$\dfrac {49 + x}{6} = 11$$
$$\Rightarrow 49 + x = 66$$

$$\Rightarrow x = 17$$

Mathematics

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