7. Median and Mode

7.1 Median

Median is a middle value that lies between the higher half and the lower half of a finite list of numbers. The median, unlike mean, is not influenced by outliers.

For $n$ terms arranged in ascending or descending order,

Median $= {\left(\dfrac{n + 1}{2} \right)}^{th}$ term where $\bm{n}$ is odd

Median $=$ Mean of ${\left( \dfrac{n}{2} \right)}^{th}$ term and ${\left( \dfrac{n}{2} + 1 \right)}^{th}$ term where $\bm{n}$ is even

Example 19

What is the median of $87, 54, 32, 47, 9006, 38$?

Solution

The $6$ terms in ascending order are $32, 38, 47, 54, 87, 9006.$

Median $=$ Mean of ${\left( \dfrac{6}{2} \right)}^{th}$ term and ${\left( \dfrac{6}{2} + 1 \right)}^{th} =$ Mean of $3^{\text{rd}}$ and $4^{\text{t}}$ terms

$= \dfrac{47 + 54}{2} = 50.5$

Answer: $50.5$

Note: In this example, $9006$ will be considered an outlier as it is more than $100$ times the largest value.

This might affect the mean but not the median. For instance, increasing the last term from $9006$ to any bigger number does not affect the median, which will remain unchanged at $50.5$.

Example 20

If the mean and median of $5$ natural numbers $1, 4, 6, 11, x$ are equal, then how many different values can $x$ take?
($1$) $0$ ($2$) $1$ ($3$) $2$ ($4$) $3$

Solution

As there are $5$ terms, the median is the ${\left( \dfrac{5 + 1}{2} \right)}^{th} = 3^{\text{rd}}$ term. This $3^{\text{rd}}$ term can be $4, 6$ or $x$.

Mean $= \dfrac{1 + 4 + 6 + 11 + x}{5} = \dfrac{22 + x}{5}$

Scenario 1: Median $= 4$

$\dfrac{22 + x}{5} = 4 $ ⇒ $ x = -2$ (Rejected, as $-2$ is not a natural number)

Scenario 2: Median $= 6$

$\dfrac{22 + x}{5} = 6 $ ⇒ $$ $\bm{x = 8}$

Data set is {$1, 4, 6, 8, 11$}, where $6$ is the median. This is accepted, i.e., $x$ can take only $1$ value.

Scenario 3: Median $= x$

$\dfrac{22 + x}{5} = x $ ⇒ $ \bm{x = 5.5}$ (Rejected, as $5.5$ is not a natural number)

∴ Scenario $2$ is the correct answer.

Answer: ($2$) $1$.

7.2 Mode

For n terms, mode is the term(s) with the maximum number of occurrences.

A set of numbers can have only $1$ mean and $1$ median. However, the set can have one or more modes.

Example 21

What is the mode for $2, 3, 3, 4, 4, 4, 5, 6, 6, 7, 7, 8, 8, 8$?
($1$) $4$ ($2$) $8$ ($3$) $4$ or $8$ ($4$) $4$ and $8$

Solution

$4$ and $8$ occur the most number of times ($3$ times).

∴ This set has $2$ modes, i.e. $4$ and $8$.

Answer: ($4$) $4$ and $8$

Note: Option ($3$) states "$4$ or $8$". This is incorrect as both $4$ and $8$ are modes and not either one of them.

8. Past Questions

Question 1

Consider the set S $=$ {$2, 3, 4$, ..., $2n + 1$}, where $n$ is a positive integer larger than $2007$. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X $-$ Y?
[CAT 2007]

		$0$
		$1$
		$\dfrac{n}{2}$
		$\dfrac{n + 1}{2n}$
		$2008$

Question 2

Rajiv is a student in a business school. After every test he calculates his cumulative average. QT and OB were his last two tests. $83$ marks in QT increased his average by $2$. $75$ marks in OB further increased his average by $1$. Reasoning is the next test, if he gets $51$ in Reasoning, his average will be _____?
[XAT 2008]

		$63$
		$62$
		$61$
		$60$
		$59$

Question 3

Ramesh analysed the monthly salary figures of five vice presidents of his company. All the salary figures (in lakhs) are integers. The mean and the median salary figures are $5$ lakh, and the only mode is $8$ lakh. Which of the options below is the sum (in lakh) of the highest and the lowest salaries?
[XAT 2012]

		$9$
		$10$
		$11$
		$12$
		None of these