Q.Which of the following is a partition value ?

Median Calculation in Economics

You already know the median from everyday life. When your teacher says "half the class scored above 65% and half below," that 65% 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 thousands of rupees): 20, 25, 30, 35, 200. The average (mean) income is (20+25+30+35+200)/5 = 62. That suggests a "typical" village earns ₹62,000 — but four out of five villages earn less than that. The one rich village pulled the average up. The median, however, is the middle value when you arrange them in order: 20, 25, 30, 35, 200. The median is 30. That ₹30,000 is far more representative of what most villages actually earn.

This is why economists prefer the median for income, housing prices, and consumption data. Averages get distorted by extreme values (a few billionaires or a few very poor households). The median tells you about the person in the middle — the "typical" economic agent.

The Precise Definition

For any set of data arranged in ascending order, the median is the value that divides the data into two equal parts. Exactly half the observations lie below it, and half above.

For ungrouped data (raw numbers):

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

For grouped data (frequency distributions), which you'll meet in economics problems, the formula is:

Where:

• $L$ = lower limit of the median class (the class interval where the median lies)
• $N$ = 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)

How It Works: A Step-by-Step Example

Suppose you have the following monthly expenditure data for 50 households:

Step 1: Find $N/2$. Here $N = 50$, so $N/2 = 25$.

Step 2: Build the cumulative frequency column:

• Up to 2000: 5
• Up to 3000: 5+10 = 15
• Up to 4000: 15+20 = 35
• Up to 5000: 35+10 = 45
• Up to 6000: 45+5 = 50

Step 3: Find the median class. The class where cumulative frequency first reaches or exceeds 25 is 3000–4000 (cumulative frequency 35). So:

• $L = 3000$
• $cf = 15$ (cumulative frequency before this class)
• $f = 20$
• $h = 1000$

Step 4: Plug into the formula:

Half the households spend less than ₹3,500 per month, and half spend more.

Why It Matters in Economics

The median appears in three critical contexts: …