Q.Find median of 14, 3, 1, 7, 10.

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

The median is resistant to outliers. A single extreme value — like a billionaire walking into a room of 100 people — can distort the mean wildly, but the median barely flinches. That's why reports on income or house prices often quote the median: it tells you what a "typical" person or house is like, without the rich or the mansions skewing the picture.

In your exams, you'll also encounter the median for grouped data (frequency distributions). The idea is the same — find the value that splits the total frequency in half — but the calculation uses a formula involving cumulative frequency and class intervals. That's a natural next step once this core idea is solid.