Question

If the median of $46,64,88,40,x,76,33,91,56,32,{\text{ and }}91{\text{ is }}58$, find $x$.

Answer 1

Hint: We will first arrange the given data in either ascending or descending order. Then, we will count the number of values in the data. Next, we will use the appropriate formula of median and find the value of $x$.

Formula used:
Median\[ = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\] observation, where $n$ is the total number in the series.

Complete step-by-step answer:
Let us first arrange the given data in the increasing order of their values.
This is done as $32,33,40,46,56,64,76,88,91,91$.
Now, we have to place $x$ in the given data. We do not know the value of $x$, so we will try placing it in the beginning of the arrangement. Hence, we get the arrangement as
 $x,32,33,40,46,56,64,76,88,91,91$.
We see that the number of values in the data is 11.
So, substituting $n = 11$ in the formula \[{\text{Median}} = {\left( {\dfrac{{n + 1}}{2}} \right)^{th}}\]observation, we get
 \[{\text{Median}} = {\left( {\dfrac{{11 + 1}}{2}} \right)^{th}}\]
Adding the terms in the numerator, we get
\[ \Rightarrow {\text{Median}} = {\left( {\dfrac{{12}}{2}} \right)^{th}} = {6^{th}}\] observation
From the arrangement, we see that the ${6^{th}}$ observation is 56.
But we have been given that the median is 58. So, the arrangement is not appropriate.
Let us try placing $x$ at the end of the arrangement. Thus, we get the arrangement as $32,33,40,46,56,64,76,88,91,91,x$.
Now, let us check the median from this arrangement. We have already found out that the median is the ${6^{th}}$ observation.
Here, the ${6^{th}}$ observation is 64. But this is not the same as 58. So, this arrangement is not appropriate either.
Now, let us try placing $x$ in the centre of the arrangement.
In this case, the arrangement becomes $32,33,40,46,56,x,64,76,88,91,91$.
Since the median is the ${6^{th}}$ observation, we get $x$ as the ${6^{th}}$ observation. The median of the data is given as 58.
Therefore, we have $x = 58$.

Note: We can also approach the above problem without using the trial-and-error method. The median is given as 58. And we know that the median is the middle value of the data. Now, 58 lies between 56 and 64. Thus, we must place $x$ in the center of the arrangement. We know that the formula of the median of even numbers of observations and the odd number of observations is different. Therefore, we need to count the number of observations first before using the formula.