Frequency Density

Statement

In statistics, when data is grouped into classes of different widths, we use frequency density to represent the data fairly. The formula is:

\[ \text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}} \]

Why it works

• A histogram represents data where the area of each bar corresponds to the frequency of the class.
• If all classes had equal width, frequency alone would be enough. But when class widths differ, taller or shorter bars could give a misleading impression.
• By dividing frequency by class width, we ensure the area of each bar (frequency density × class width) correctly represents the frequency.

Recipe (Steps to Apply)

1. Identify the frequency for the class interval.
2. Calculate the class width (upper boundary − lower boundary).
3. Divide frequency by class width.
4. The result is the frequency density, which is plotted as the height of the histogram bar.

Spotting it

Use this formula when drawing or interpreting histograms with unequal class widths.

Common pairings

• Histograms (area = frequency).
• Grouped frequency tables.
• Estimating averages from grouped data.

Mini examples

1. Class interval 10–20, frequency = 8. Class width = 10. Frequency density = 8 ÷ 10 = 0.8.
2. Class interval 30–40, frequency = 15. Class width = 10. Frequency density = 15 ÷ 10 = 1.5.

Pitfalls

• Forgetting to calculate class width before dividing.
• Confusing frequency density with frequency.
• Thinking histogram bar height equals frequency — it actually equals frequency density.

Exam Strategy

• Always check class widths — if unequal, you must use frequency density.
• Show your calculation clearly: class width, then frequency ÷ width.
• Remember: in a histogram, area = frequency, not height.

Worked Micro Example

A class interval 20–30 has frequency 12. Find its frequency density.

Class width = 30 − 20 = 10. Frequency density = 12 ÷ 10 = 1.2.

Summary

Frequency density allows histograms to fairly represent data when class widths differ. The height of each bar is frequency density, while the area of each bar represents the actual frequency.