Reading Histograms Without Confusing Frequency and Frequency Density

A histogram can look like an ordinary bar chart, but it represents grouped continuous data in a different way. The most important distinction is this: when class widths are unequal, bar height does not directly represent frequency. Bar area does.

This guide develops frequency density from first principles, shows how to recover frequencies from histogram bars, and explains why a wide low bar can contain more observations than a narrow tall bar.

Bar chart and histogram are not interchangeable

A bar chart is usually used for discrete or categorical data, with gaps between categories. A histogram is used for grouped continuous data, so adjacent class intervals touch.

More importantly, histogram area represents frequency. When every class has equal width, height happens to be proportional to frequency, which can hide the deeper rule.

Frequency density

frequency density = frequency ÷ class width

Rearranging gives:

frequency = frequency density × class width

That second relationship explains the geometry: bar area = height × width = frequency density × class width = frequency.

Worked example 1

A class interval 10 ≤ x < 15 has frequency 20. Its width is 5.

frequency density = 20 ÷ 5 = 4

The histogram bar has height 4.

Worked example 2: unequal widths

Class A has interval width 5 and frequency density 6. Class B has width 10 and frequency density 4.

Frequency A = 5 × 6 = 30
Frequency B = 10 × 4 = 40

Class A has the taller bar, but Class B contains more observations. This is exactly why height alone cannot be read as frequency.

Recovering a missing scale

Sometimes a histogram gives one known class frequency and asks for others. Use the known bar to establish the vertical scale or frequency-density value, then apply area reasoning to the remaining bars.

For example, if a width-4 bar has frequency 12, its density is 3. If the drawn height corresponding to density 3 is known, the vertical scale can be calibrated.

Class boundaries matter

The width of 20 ≤ x < 30 is 10, not the number of integer labels inside it. Histograms represent intervals on a continuous scale, so class width is found from boundary difference.

Common errors

Reading height as frequency when widths differ. Use area, not height alone.

Using frequency density as frequency. Density is frequency per unit class width.

Misreading class width. Subtract the boundaries.

Adding frequency densities to find total frequency. Convert each bar back to frequency first.

Treating touching bars as proof of equal widths. Histograms touch because the variable is continuous, not because the intervals are equal.

A reliable reading protocol

1. Read each class interval carefully.
2. Find each class width.
3. Read or determine the frequency density.
4. Calculate frequency as width × density.
5. Check that total frequencies match any stated total.

Practice

1. A class interval has width 6 and frequency 24. Find its frequency density.
2. A histogram bar has width 8 and density 3.5. Find its frequency.
3. Bar A has width 4 and height 7. Bar B has width 10 and height 3, where height represents frequency density. Which class has greater frequency?
4. Explain why the tallest bar need not contain the most data.

Answers

1. 4. 2. 28. 3. A has frequency 28; B has frequency 30, so B is greater. 4. Frequency is represented by bar area; a wider lower bar can have greater area than a narrow tall bar.

Connected routes

Use Mean, Median and Mode and Range and Spread for numerical summaries, then continue to Box Plots and Distribution. Return to the Mathematics Learning Hub for the wider route.