Q.Distinguish between univariate and bivariate frequency distribution.

From Everyday Intuition to Precise Meaning

Imagine you're a market researcher studying how much people spend on groceries each month. You ask 100 households, and you get 100 different numbers — ₹2,300, ₹4,150, ₹3,700, and so on. That raw list is a mess. To make sense of it, you group the data into ranges: ₹2,000–₹3,000, ₹3,000–₹4,000, ₹4,000–₹5,000, and so on. Now you have a grouped frequency distribution.

But here's the problem: once you've grouped the data, you no longer know the exact spending of any household. You only know that, say, 30 households fall in the ₹3,000–₹4,000 range. So when you want to calculate the average spending, what single value do you use to represent all those 30 households? You can't use the actual numbers — you've lost them. You need a representative value for each group.

That representative value is the class midpoint (also called the class mark).

The Precise Definition

The class midpoint is the value exactly halfway between the lower limit and the upper limit of a class interval. For a class interval with lower limit $L$ and upper limit $U$, the midpoint $m$ is:

For the class ₹3,000–₹4,000, the midpoint is $\frac{3000 + 4000}{2} = ₹3,500$. This single number stands in for every household in that group when you do further calculations.

The Assumption That Makes It Work

When you use the class midpoint to represent a group, you are making a critical assumption: the observations within each class are uniformly distributed around the midpoint. In plain language, you assume that for every household spending ₹3,100, there's another spending ₹3,900, so the average of all 30 households really is ₹3,500.

This assumption is almost never perfectly true in real data. But it's the best we can do without the original numbers. The wider the class interval, the more questionable this assumption becomes. A class of ₹3,000–₹5,000 (midpoint ₹4,000) is a much rougher approximation than a class of ₹3,000–₹3,500 (midpoint ₹3,250).

Why It Matters in Economics

The class midpoint is the bridge between raw data and meaningful analysis. Here's where you'll actually use it:

1. Calculating the Mean from Grouped Data

When you have a frequency distribution and want the mean of the entire dataset, you cannot add up the original values (you don't have them). Instead, you:

1. Find the midpoint $m_i$ of each class.
2. Multiply each midpoint by its frequency $f_i$.
3. Sum these products.
4. Divide by the total frequency $N$.

The formula for the mean of grouped data is:

Where:

• $\bar{x}$ = estimated mean
• $f_i$ = frequency of the $i$-th class
• $m_i$ = midpoint of the $i$-th class
• $N$ = total number of observations ($\sum f_i$)

2. Constructing Ogives and Frequency Polygons

When you draw a frequency polygon, you plot the frequency against the midpoint of each class, not the limits. The midpoint becomes the x-coordinate of each point you connect. Similarly, for a less-than ogive, you use the upper limits; for a more-than ogive, you use the lower limits — but the midpoint is what you use for the polygon.

3. Calculating Other Measures …