Question

Medians of 15, 28, \[72.56\] , 44, 32, 31, 43 and 51 are 43.
A.True
B.False
C.Neither
D.Either

Answer 1

Hint: Here we have to check whether the given statement is true or false. We will first arrange the terms in ascending order and then we will find the number of terms present. If the total number of terms is odd then we will apply the formula of the median for an odd number of terms. If the number of terms is even then we will apply the formula of the median for even numbers of terms. Then we will choose the option according to the answer.

Formula used:
\[{\rm{Median}} = \dfrac{{\left[ {{{\left( {\dfrac{n}{2}} \right)}^{th}}{\rm{term}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}{\rm{term}}} \right]}}{2}\], where \[n\] is the number of terms present.

Complete step-by-step answer:
Here we have to check whether the given statement is true or false.
We need to find the median of the given set of the numbers.
The given terms are:- 15, 28, 72.56, 44, 32, 31, 43 and 51.
Now, we will first arrange the given numbers in ascending order.
15, 28, 31, 32, 43, 44, 51, 72.56
Now, we will find the number of terms present here.
There are a total 8 terms present here.
We can see that the number of terms is even.
Now, substituting the number of terms, \[n = 8\] in the formula \[{\rm{Median}} = \dfrac{{\left[ {{{\left( {\dfrac{n}{2}} \right)}^{th}}{\rm{term}} + {{\left( {\dfrac{n}{2} + 1} \right)}^{th}}{\rm{term}}} \right]}}{2}\], we get.
Median \[ = \dfrac{{\left[ {{{\left( {\dfrac{8}{2}} \right)}^{th}}{\rm{term}} + {{\left( {\dfrac{8}{2} + 1} \right)}^{th}}{\rm{term}}} \right]}}{2}\]
On further simplifying the terms, we get
\[ \Rightarrow \] Median \[ = \dfrac{{\left[ {{4^{th}}{\rm{term}} + {5^{th}}{\rm{term}}} \right]}}{2}\]
We know that the fourth term is 32 and fifth term is 43.
On substituting these values here, we get
\[ \Rightarrow \] Median \[ = \dfrac{{\left( {32 + 43} \right)}}{2}\]
On adding the numbers, we get
\[ \Rightarrow \] Median \[ = \dfrac{{75}}{2} = 37.5\]
Therefore, the required median is equal to \[37.5\].
Thus, the given statement is not correct.
Hence, the correct option is option B.

Note: We know that the median is defined as the middle value of the given list of numbers or data. Here, we need to keep in mind that the formula of the median for an even number of terms and an odd number of terms is different. That’s why we have applied the formula of the median number of terms. Before finding the median we need to arrange the given set in either descending order or ascending order. If we will arrange the given set we will get the wrong answer.