What you'll learn
Histograms with unequal class widths use frequency density to represent grouped data fairly. In this guide you will learn how a histogram differs from a bar chart, how to calculate and use frequency density, how to draw a histogram with unequal class widths, how to read frequencies from one, and how to estimate values such as the median. This is a higher-tier statistics topic.
Key terms and definitions
Histogram — a diagram for continuous grouped data where area represents frequency.
Frequency density — frequency divided by class width; the height of each bar.
Class width — the size of a data interval (upper bound minus lower bound).
Frequency — the number of data values in a class.
Continuous data — data that can take any value in a range (e.g. height, time).
Core concepts
Histograms versus bar charts
A histogram is used for continuous grouped data. Unlike a bar chart, the bars have no gaps, and crucially the area of each bar (not its height) represents the frequency. This matters when classes have unequal widths, where heights alone would be misleading.
Frequency density
To draw a histogram with unequal class widths, plot frequency density on the vertical axis:
frequency density = frequency ÷ class width
This makes each bar's area equal to its frequency, so wider classes are not given unfair visual weight. For example, a class of width 10 with frequency 30 has frequency density 30 ÷ 10 = 3.
Drawing the histogram
For each class, work out the class width and the frequency density, then draw a bar spanning the class interval with height equal to the frequency density. The bars touch, and the horizontal axis is a continuous scale.
Reading frequencies from a histogram
To find a frequency from a histogram, calculate the area of the bar:
frequency = frequency density × class width
You can find the frequency for part of a bar by taking the appropriate proportion of its width — useful for estimating how many values fall in a given range.
Estimating statistics
Histograms can be used to estimate the total frequency (sum of all areas) and the median or other values by finding where a certain cumulative area is reached. Since the data is grouped, these are estimates, not exact values.
Worked examples
Example 1: Frequency density
A class 20 ≤ x < 30 has frequency 24. Find the frequency density.
Class width = 10; frequency density = 24 ÷ 10 = 2.4.
Example 2: Frequency from a bar
A bar has frequency density 5 over a class width of 4. What is the frequency?
Frequency = density × width = 5 × 4 = 20.
Example 3: Comparing classes
Why use frequency density rather than frequency for the bar heights?
Because classes have unequal widths, so using area = frequency (via frequency density) represents the data fairly.
Common mistakes and how to avoid them
Plotting frequency as height. With unequal widths, plot frequency density, not frequency.
Forgetting area = frequency. Frequency is the area of the bar, not its height.
Wrong class width. Class width = upper bound − lower bound; check the inequalities.
Leaving gaps between bars. Histogram bars touch (continuous data).
Treating estimates as exact. Grouped-data statistics from histograms are estimates.
Exam technique for Histograms
Use frequency density (frequency ÷ class width) for the heights.
Remember area = frequency when reading values.
Calculate class widths carefully from the boundaries.
Draw touching bars on a continuous axis.
Estimate frequencies for ranges using proportions of bar areas.
Quick revision summary
A histogram displays continuous grouped data with touching bars, where the area of each bar represents the frequency. With unequal class widths, plot frequency density = frequency ÷ class width on the vertical axis, so that area equals frequency and wider classes aren't over-weighted. To draw one, find each class width and frequency density, then draw a bar over the class interval at that height. To read a frequency, calculate the area = frequency density × class width, and take a proportion of a bar's width to estimate frequencies for part of a range. Histograms also let you estimate totals and the median, but since the data is grouped these are estimates. The common mistakes are plotting frequency instead of frequency density, forgetting that area (not height) gives frequency, miscalculating class widths, leaving gaps, and treating estimates as exact. Use frequency density for heights, remember area = frequency, compute class widths carefully, and read values via bar areas.