Question

The relationship between Median M, second quartile, the ${5^{{\text{th}}}}$ Decile ${{\text{D}}_5}$ and ${50^{{\text{th}}}}$ percentile ${{\text{P}}_{50}}$ of a set of observation is
$
  \left( {\text{A}} \right){\text{M > }}{{\text{Q}}_2} > {{\text{D}}_5} > {{\text{P}}_{50}} \\
  \left( {\text{B}} \right){\text{M = }}{{\text{Q}}_2}{\text{ = }}{{\text{D}}_5}{\text{ = }}{{\text{P}}_{50}} \\
  \left( {\text{C}} \right){\text{M < }}{{\text{Q}}_2}{\text{ < }}{{\text{D}}_5}{\text{ < }}{{\text{P}}_{50}} \\
  \left( {\text{D}} \right){\text{M < }}{{\text{Q}}_2}{\text{,}}{{\text{D}}_5} > {{\text{P}}_{50}} \\
 $

Answer 1

Hint: since the question of relation is given to us, it is clear that we have to compare all the terms. The best way to solve this problem is by defining down each term and listing down its properties. From this step it will be easy to determine the relation between each of the given terms.

Complete step by step answer:
To start with the solution of the given problem we will start determining the inequalities by listing down the definition of the given terms and then compare them with each other.
Starting with median, Median determines the middle value of the data arranged in an ascending order. Median is said to be the measure of central tendency.
Second quartile denoted by ${{\text{Q}}_2}$. It is also used for determining the middle value of the data given or dividing the data into two equal parts.
Deciles divide the data set into 10 equal parts. The fifth decile here represents the median.
The definition of percentile goes as the number where a certain percentage of score falls below that number. ${50^{{\text{th}}}}$ percentile determines the score which falls below 50% which is the same as that of a median.
Hence from the above observation we can confidently say that all the 4 terms have the same properties and hence also have the same values.

Hence option $\left( {\text{B}} \right){\text{M = }}{{\text{Q}}_2}{\text{ = }}{{\text{D}}_5}{\text{ = }}{{\text{P}}_{50}}$is the right answer.

Note: above, we have solved the problem by comparing all the definitions of the given terms. There can also be another way where you can actually list all of the terms and write their formulas. By comparing the formulas and simplifying them you will come to a conclusion that all the four terms possess the same value.