Question

9.the median of the following observations ,arranged in ascending order is 15. 10,12,14,x-3,x,x+2,25. Then find x. *​

Answer 1

Question :

The following Observations are arranged in Ascending Order:

10, 12, 14, x-3, x, x+2 ,25

If the Median is 15, find the Value of x

$\begin {gathered} \end {gathered}$

Solution :

Let the Total Number of Observations be n.

Here, n = 7, which is Odd

$\begin {gathered} \end {gathered}$

$\begin {aligned} \bold {Median} &= \mathrm {Value\: of\: \Big (\dfrac{n + 1}{2}\Big)^{th}\: Observation}\\\\&\Rightarrow Value\: of\: \Big (\dfrac{7 + 1}{2}\Big)^{th}\: Observation\\\\&= Value\: of\: 4^{th}\: Observation\\\\&= \bf x - 3 \end {aligned}$

$\begin {gathered} \end {gathered}$

But, Median = 15 (given)

$\begin {gathered} \end {gathered}$

$\begin {aligned} \therefore\: x - 3 &= 15\\\\x &= 15 + 3 \longrightarrow \boxed {\text{Transposing -3 to the RHS}}\\\\\bold x &= \underline {\underline {\bf 18}}} \end {aligned}$

$\begin {gathered} \end {gathered}$

Hence, the Value of x is 18.

$\begin {gathered} \end {gathered}$

Additional Information :

Median :-

After arranging the given Data in an Ascending or Descending Order of Magnitude, the Value of the Middle - most Observation (s) is called the Median of the Data.

$\begin {gathered} \end {gathered}$

Let the Total Number of Observations be n.

(i) If n is Odd, then Median = Value of $\bf \Big (\dfrac{n + 1}{2}\Big )^{th}$Observation.

(ii) If n is Even, then Median = Mean of $\bf \Big (\dfrac{n}{2}\Big )^{th}$and $\bf \Big (\dfrac{n}{2} + 1\Big )^{th}$Observations.

$\begin {gathered} \end {gathered}$

Mean :-

The Average of a given set of Numbers is called the Mean of the given Numbers.

$\mathtt {Mean = \dfrac{Sum\: of\: Observations}{Number\: of\: Observations}}$

Answer 2

★ How to do :-

Here, we are given with seven observations and the median value of the same data. But, we are not given with the value of three observations in the given data, but it's replaced with a variable x. We are asked to find the value of that variable x. So, first we should find out the value in the given data which replaces with the value of median i.e, 15. So, we will use the formula of the median to find the value and later we can find the the value of x. So, let's solve!!

$\:$

➤ Solution :-

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

Substitute the value of n.

${\tt \leadsto \dfrac{7 + 1}{2}}$

Add the numbers in the numerator.

${\tt \leadsto \dfrac{8}{2} th \: \: term}$

Simplify the fraction.

${\tt \leadsto 4th \: \: term}$

Here, we can see that the 4th term is x-3. So, that represents the value of median.

$\:$

Value of x :-

${\tt \leadsto x - 3 = 15}$

Shift the number 3 from LHS to RHS, changing it's sign.

${\tt \leadsto x = 15 + 3}$

Add the values on RHS.

${\tt \leadsto \pink{\underline{\boxed{\tt x = 18}}}}$

$\Huge\therefore$ The value of x in the following data is 18.

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$\dashrightarrow$ Some related formulas :-

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

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

The letter 'n' represents the number of observations in the given data.