Question

Following 12 observations are arranged in ascending order of order as follows. 2,3,4,5,9, x, x+1, 14,16,18,19,20. Find "x" .If the median of data is 10.5​

Answer 1

★ How to do :-

Here, we are given with twelve observations in which they are arranged in ascending order. We are given with ten of those observations, but two of those observations are replaced by a variable x. We are asked to find the value of that variable x. First, we should find the value which is in given data that should be replaced by the median value by using the formula of median given. Then, we apply the median value and that given equation and find the value of x by shifting the numbers from one hand side to the other. So, let's solve!!

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➤ Solution :-

${\sf \leadsto \underline{\boxed{\sf Median = \dfrac{n}{2} + 1th \: \: term}}}$

Substitute the value of n.

${\tt \leadsto \dfrac{12}{2} + 1th \: \: term}$

Simplify the fraction first.

${\tt \leadsto 6 + 1th \: \: term}$

Add the values obtained now.

${\tt \leadsto 7th \: \: term}$

Here, we can see that the 7th term is x+1, which is equal to the median 10.5. So,

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Value of x :-

${\tt \leadsto x + 1 = 10.5}$

Shift the number 1 from LHS to RHS, changing it's sign.

${\tt \leadsto x = 10.5 - 1}$

Subtract the values on RHS.

${\tt \leadsto x = 9.5}$

So, the sixth observation is 9.5 and the seventh observation is 10.5, which is the median value.

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${\red{\underline{\boxed{\bf So, \: the \: value \: of \: x = 9.5}}}}$

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$\dashrightarrow$ Formulas of median :-

${\to \sf {Median}_{(Even \: observations)} = \dfrac{n}{2} + 1th \: \: term}$

${\to \sf {Median}_{(Odd \: observations)} = \dfrac{n + 1}{2} th \: \: term}$

In these formulas, the letter n means the 'number of observations' given in the data.