Q.Define Mode, Median and Quartile.

Median Calculation in Economics

You already know the median from everyday life. When your teacher says "half the class scored above 70% and half below," that 70% is the median. It's the middle point — the value that splits a group into two equal halves. In economics, this simple idea becomes a powerful tool for understanding income, wages, prices, and inequality.

The Intuition: Why Not Just Average?

Imagine five villages with annual incomes (in ₹ lakhs): 2, 3, 4, 5, 100. The average (mean) is ₹22.8 lakhs. Does that sound like a typical village? No — one rich village pulls the average up. The median, however, is the middle value when you arrange them in order: 2, 3, 4, 5, 100. The median is ₹4 lakhs. That tells you what a "typical" village actually earns. Half the villages earn less than ₹4 lakhs, half earn more.

This is why economists prefer the median for income, housing prices, and consumption data — it resists distortion by extreme values.

The Precise Definition

The median is the value that divides a data set into two equal halves when arranged in ascending or descending order.

For ungrouped data (raw numbers), the process is:

1. Arrange all values in ascending order.
2. Find the middle position: if there are $n$ values, the median is the $\frac{n+1}{2}$th value when $n$ is odd. When $n$ is even, the median is the average of the $\frac{n}{2}$th and $\frac{n}{2}+1$th values.

For Grouped Data (Frequency Distribution)

In economics, you rarely get raw numbers. You get data grouped into classes — like "income between ₹10,000–₹20,000: 50 families." Here, the median lies inside a class interval. The formula becomes:

Where:

• $L$ = lower limit of the median class (the class where the cumulative frequency crosses $\frac{N}{2}$)
• $N$ = total frequency (sum of all frequencies)
• $cf$ = cumulative frequency of the class before the median class
• $f$ = frequency of the median class
• $h$ = class width (size of the interval)

Step-by-Step Example

Suppose you have this income data for 100 households:

Step 1: Find $N/2 = 100/2 = 50$.

Step 2: Build cumulative frequencies: 10, 30, 60, 85, 100. The 50th household lies in the class 10000–15000 (since cumulative frequency 30 is below 50, and 60 crosses it). This is the median class.

Step 3: Identify values: …