Q.(OR) Calculate Mean, Median and Mode from above data : Mid Value 1, 2, 3, 4, 5; Frequency 10, 20, 30, 20, 10.

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 First

Imagine five families on your street with monthly incomes: ₹15,000, ₹18,000, ₹22,000, ₹25,000, and ₹1,00,000. The average (mean) income is ₹36,000. But does that feel like a fair description of the street? Not really — that one high earner pulled the average up. The median income is ₹22,000, which tells you that half the families earn less than ₹22,000 and half earn more. That ₹22,000 is a much better picture of what a "typical" family on your street earns.

This is why economists prefer the median for income, housing prices, and consumption data. Averages get distorted by extreme values — a few billionaires can make a country's average income look healthy while most people struggle. The median resists that distortion.

The Precise Meaning

The median is the value of the middle observation when all observations are arranged in ascending or descending order.

For an ungrouped frequency distribution (raw data):

• If the number of observations $n$ is odd, the median is the value of the $\left(\frac{n+1}{2}\right)^{\text{th}}$ observation.
• If $n$ is even, the median is the average of the $\left(\frac{n}{2}\right)^{\text{th}}$ and $\left(\frac{n}{2}+1\right)^{\text{th}}$ observations.

For a grouped frequency distribution (data in class intervals), the median is found using:

Where:

• $L$ = lower limit of the median class (the class where the cumulative frequency first reaches or exceeds $\frac{N}{2}$)
• $N$ = total frequency (total number of observations)
• $cf$ = cumulative frequency of the class preceding the median class
• $f$ = frequency of the median class
• $h$ = class width (size of the interval)

Why This Formula Works

Think of it as interpolation. You know that the $\frac{N}{2}^{\text{th}}$ observation lies somewhere inside the median class. The term $\frac{N}{2} - cf$ tells you how many observations into that class you need to go. Dividing by $f$ gives the fraction of the class width you must travel, and multiplying by $h$ converts that fraction into actual units. Adding it to $L$ gives you the exact position.

A Worked Example

Consider this distribution of daily wages (in ₹) for 50 workers:

First, find cumulative frequencies: 8, 20, 35, 45, 50. Here $N = 50$, so $\frac{N}{2} = 25$. The class where cumulative frequency first reaches or exceeds 25 is 140-160 (cumulative frequency 35). So:

• $L = 140$
• $cf = 20$ (cumulative frequency before the median class)
• $f = 15$
• $h = 20$

So the median daily wage is ₹146.67. Half the workers earn less than this, half earn more.

Why It Matters in Economics

The median appears in three critical contexts:

Income and wage analysis. Government reports on household income almost always cite the median, not the mean. When you hear "the median household income in India is ₹X," that X is the income level that splits the population into two equal halves. It tells you what the "typical" household earns, unaffected by the Ambanis and Adanis at the top. …