Q.OR Find out median value of the following frequency distribution : Wage Rate ₹ 0-10, 10-20, 20-30, 30-40, 40-50; No. of Workers 22, 38, 46, 35, 20.

Median Calculation in Economics

You already know the median from everyday life. If five friends have pocket money of ₹50, ₹100, ₹150, ₹200, and ₹5000, the "middle" value — ₹150 — tells you more about the group than the average (which gets pulled up to ₹1100 by that one high value). That's the median: the value that splits a sorted list into two equal halves.

The Precise Meaning

The median is the positional average — it depends only on the order of values, not on their magnitude. For any set of data arranged in ascending order:

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

Why It Matters in Economics

Economics deals with skewed distributions — most people earn less than the average, most households own less than the average wealth. The median gives you the typical experience. When the government reports "median household income," it's telling you what the person exactly in the middle earns. Half the households earn less, half earn more. That's far more meaningful than the mean, which a handful of billionaires can distort.

Calculating the Median from Raw Data

Example 1 (odd n): Incomes of 5 workers: ₹8000, ₹12000, ₹15000, ₹25000, ₹60000.

Arrange in ascending order (already done). $n=5$, so median is the $\frac{5+1}{2}=3$rd value = ₹15000.

Example 2 (even n): Incomes of 6 workers: ₹8000, ₹12000, ₹15000, ₹25000, ₹60000, ₹200000.

$n=6$. The $\frac{6}{2}=3$rd value is ₹15000, the $4$th value is ₹25000. Median = $\frac{15000+25000}{2}=$ ₹20000.

The Median for Grouped (Continuous) Data

In economics, you rarely get raw numbers — you get frequency distributions like "income between ₹10000-₹20000: 50 people." Here, the median lies inside a class interval. The formula is:

Where:

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

Step-by-step logic:

1. Find $\frac{N}{2}$.
2. Build a cumulative frequency column.
3. Identify the median class — the first class where cumulative frequency $\geq \frac{N}{2}$.
4. Plug into the formula.

A Worked Example

$N=100$, so $\frac{N}{2}=50$. The cumulative frequency first reaches 50 in the 20000-30000 class (cumulative freq = 60). That's the median class.

$L=20000$, $cf=30$ (cumulative frequency before this class), $f=30$, $h=10000$.

$\text{Median} = 20000 + \frac{50-30}{30} \times 10000 = 20000 + \frac{20}{30} \times 10000 = 20000 + 6666.67 = \textbf{₹26666.67}$ …