Q.Observe the graph of data given in earlier example of mean where goals scored by footballer Lionel Messi from 2008 to 2017 in all competitions were given (Year 1-10 corresponding to 2008-2017; goals 38, 41, 41, 45, 47, 53, 54, 58, 60, 73 arranged in ascending order for the graph). Since the total number of years is even, locate the median value graphically, noting that the positional average (median) exactly divides the graph into two parts having equal areas.

What is the Median?

Imagine you and four friends are comparing pocket money. The amounts are: ₹20, ₹50, ₹30, ₹100, ₹40. If you line these up from smallest to largest — ₹20, ₹30, ₹40, ₹50, ₹100 — the middle value is ₹40. That's the median: the number that splits the data exactly in half.

The median answers a simple question: "What is the typical value, if we ignore extremes?" In the example above, the average (mean) is ₹48, but one friend gets ₹100, which pulls the average up. The median stays at ₹40, which better represents what most of you actually get.

The Precise Definition

For ungrouped data (a list of numbers):

1. Arrange all values in ascending order (smallest to largest).
2. Let $n$ be the number of observations.
   • If $n$ is odd: the median is the $\left(\frac{n+1}{2}\right)^{\text{th}}$ value.
   • If $n$ is even: the median is the average of the $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2}+1\right)^{\text{th}}$ values.

Example 1 (odd count):

Data: 2, 5, 1, 8, 4

Sorted: 1, 2, 4, 5, 8

$n = 5$, position = $\frac{5+1}{2} = 3$

Median = 4

Example 2 (even count):

Data: 3, 7, 2, 9, 1, 6

Sorted: 1, 2, 3, 6, 7, 9

$n = 6$, positions = $\frac{6}{2} = 3$ and $\frac{6}{2}+1 = 4$

Values at those positions: 3 and 6

Median = $\frac{3+6}{2} = 4.5$

Why the Median Matters …