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Mathematics

The numbers 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.

Measures of Central Tendency

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Answer

Here, n = 6, which is even.

Median=n2th term+(n2+1)th term2=62th term+(62+1)th term2=3rd term + 4th term2=10+122=222=11.\text{Median} = \dfrac{\dfrac{n}{2}\text{th term} + \Big(\dfrac{n}{2} + 1\Big)\text{th term}}{2} \\[1em] = \dfrac{\dfrac{6}{2}\text{th term} + \Big(\dfrac{6}{2} + 1\Big)\text{th term}}{2} \\[1em] = \dfrac{\text{3rd term + 4th term}}{2} \\[1em] = \dfrac{10 + 12}{2} \\[1em] = \dfrac{22}{2} \\[1em] = 11.

Sum of observations = 6 + 8 + 10 + 12 + 13 + x = x + 49.

Mean = Sum of observationsNo. of observations=x+496\dfrac{\text{Sum of observations}}{\text{No. of observations}} = \dfrac{x + 49}{6}.

As, mean = median

⇒x+496=11\Rightarrow \dfrac{x + 49}{6} = 11

⇒ x + 49 = 66

⇒ x = 66 - 49 = 17.

Hence, x = 17.

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