What you'll learn
This topic covers frequency polygons and histograms with equal class widths — ways of displaying grouped continuous data. In this guide you will learn how to draw a frequency polygon, how to plot a histogram with equal classes, how to read and interpret them, and how to compare distributions. These are core statistics skills for handling grouped data.
Key terms and definitions
Grouped data — data sorted into class intervals.
Class interval — a range of values (e.g. 10 ≤ x < 20).
Midpoint — the middle value of a class, used for plotting.
Frequency polygon — a line graph joining the midpoints of each class.
Histogram — a bar chart for continuous data with no gaps between bars.
Core concepts
Histograms with equal class widths
A histogram displays grouped continuous data with bars touching (no gaps), because the data is continuous. When the class widths are equal, the bar height is simply the frequency. The horizontal axis is a continuous scale, not separate categories.
Drawing a frequency polygon
A frequency polygon is drawn by plotting the frequency against the midpoint of each class and joining the points with straight lines. The midpoint is the average of the class boundaries (for 10 ≤ x < 20 it is 15). Polygons are good for showing the shape of a distribution.
Reading midpoints
To find a midpoint, add the lower and upper boundaries and divide by 2. Plotting at midpoints (not boundaries) is essential for frequency polygons and for estimating the mean of grouped data.
Interpreting the shape
The shape of a polygon or histogram tells you about the data: where the peak (mode class) is, whether it is symmetric or skewed, and how spread out it is. Use this to describe the distribution.
Comparing distributions
To compare two data sets, draw both frequency polygons on the same axes and compare their peaks and spreads. A polygon shifted right has higher values; a narrower polygon shows more consistent data. Always interpret in context.
Worked examples
Example 1: Midpoint
Find the midpoint of the class 20 ≤ x < 30.
(20 + 30) ÷ 2 = 25.
Example 2: Bar height
With equal class widths, a class has frequency 12. What is the bar height?
12 — for equal widths, height equals frequency.
Example 3: Plotting point
A class 0 ≤ x < 10 has frequency 7. Where is the polygon point?
At midpoint 5, height 7: (5, 7).
Common mistakes and how to avoid them
Leaving gaps in a histogram. Continuous data bars must touch.
Plotting at boundaries. Frequency polygons use midpoints.
Joining with curves. Use straight lines between points.
Wrong midpoint. Average the lower and upper boundaries.
Comparing without context. Relate peaks and spreads to the situation.
Exam technique for Frequency Polygons and Histograms
Draw touching bars for continuous data with equal widths.
Plot frequency against midpoint for polygons.
Join points with straight lines.
Read the shape for mode class, symmetry and spread.
Overlay polygons on the same axes to compare, in context.
Quick revision summary
For grouped continuous data, a histogram with equal class widths has touching bars (no gaps) whose height equals the frequency. A frequency polygon plots the frequency against the midpoint of each class and joins the points with straight lines; the midpoint is the average of the class boundaries (10 ≤ x < 20 → 15). Plotting at midpoints, not boundaries, is essential and is also used when estimating the mean of grouped data. The shape reveals the mode class (the peak), symmetry or skew, and spread. To compare distributions, draw both polygons on the same axes and compare peaks and spreads, always interpreting in context. The common errors are leaving gaps in a histogram, plotting at boundaries instead of midpoints, joining with curves, and comparing without context. Use touching bars for equal widths, plot frequency against midpoint, join with straight lines, and compare peaks and spreads on shared axes.