Frequency Polygon: From Intuition to Definition

Imagine you're looking at a histogram of marks scored by students in a class. The bars rise and fall — tall where many students scored, short where few did. Now picture this: instead of staring at the rectangles, you take a pen and connect the midpoints of the tops of those bars. That line you just drew is a frequency polygon.

Why would you do that? Because a single line is often easier to read than a stack of bars. It shows the shape of the data — where most values cluster, how the distribution tapers off — in one smooth sweep. It's like tracing the silhouette of the histogram.

The Precise Construction

A frequency polygon is a line graph that represents the frequencies of different class intervals. Here's how you build it, step by step:

1. Start with a grouped frequency distribution. You have class intervals (e.g., 0–10, 10–20, ...) and their frequencies.

2. Find the class mark (midpoint) of each interval.

   For an interval $a$–$b$, the class mark is $\frac{a+b}{2}$.

3. Plot points: For each class mark on the x-axis, go up to the corresponding frequency on the y-axis. Place a dot there.

4. Add two imaginary classes — one before the first interval and one after the last — each with frequency zero. Their class marks are the midpoints of those non-existent intervals. This "anchors" the polygon to the x-axis at both ends.

5. Connect the dots with straight line segments, in order.

A Concrete Example

Suppose you have this data:

Step 1: Find class marks:

$5, 15, 25, 35$

Step 2: Add imaginary classes:

Before: $(-10)$–$0$, class mark $-5$, frequency $0$

After: $40$–$50$, class mark $45$, frequency $0$

Step 3: Plot points:

$(-5,0)$, $(5,5)$, $(15,12)$, $(25,8)$, $(35,3)$, $(45,0)$

Step 4: Connect them in order. …