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 …