All revision notes topics

HistogramsCambridge IGCSE Maths: Revision notes

Section 1

What is a histogram?

A histogram is a diagram used to display grouped continuous data, especially when the class widths are not all equal. Unlike a bar chart, a histogram has:

  • No gaps between bars (the data is continuous)
  • Bars whose area, not height, represents the frequency
  • A vertical axis labelled 'Frequency density', never 'Frequency'

Histograms are used when class widths differ, because plotting raw frequency as height would make wide classes look misleadingly tall.

Key termshistogramfrequency density
Common mistake

Do not label the vertical axis 'Frequency' — examiners specifically require 'Frequency density'.

Section 2

Why do we need frequency density?

If class widths are unequal, plotting frequency directly on the height would distort the picture: a class twice as wide would look twice as tall even if it represents the same 'density' of data. By plotting frequency density instead, the area of each bar correctly represents the frequency, so bars can be fairly compared regardless of class width.

Key termsclass width
Think of it like this

Think of frequency density like population density on a map — it tells you how 'packed' the data is per unit of width, not the raw total.

Section 3

How do you calculate frequency density?

The key formula is:

frequency density = frequency ÷ class width

Steps:

  1. Find the class width (upper class boundary − lower class boundary)
  2. Divide the frequency for that class by its class width
  3. This gives the height to plot for that bar

To reverse the process (find frequency from a histogram): frequency = frequency density × class width

Key termsfrequency density
Example

A class 10–25 (width 15) has frequency 30. Frequency density = 30 ÷ 15 = 2. The bar is drawn with height 2 over the interval from 10 to 25.

Section 4

How do you draw a histogram?

  1. Draw a grouped frequency table, adding a column for class width and a column for frequency density
  2. Calculate frequency density for every class using the formula above
  3. Draw bars with no gaps between them, using class boundaries on the horizontal axis
  4. Make sure each bar's height equals its frequency density, so its area equals its frequency
  5. Label the vertical axis 'Frequency density' and the horizontal axis with the variable being measured
Key termsclass boundary

Section 5

How do you interpret a histogram (find frequency)?

To find the frequency of a class from a drawn histogram:

  1. Read off the frequency density (bar height) for that class
  2. Read off the class width from the horizontal axis
  3. Multiply: frequency = frequency density × class width

To find frequencies for parts of a bar (e.g. only part of a class interval), assume the data is spread evenly within that class and use proportional area.

Exam tip

When a question asks for the number of values within a certain range that doesn't align with class boundaries, examiners expect area (density × width) of the relevant portion, assuming uniform spread within the class.

Must Know

  • Histograms display grouped continuous data, especially with unequal class widths
  • Vertical axis is always 'Frequency density', never 'Frequency'
  • Frequency density = frequency ÷ class width
  • Frequency = frequency density × class width (rearranged)
  • The area of each bar represents frequency, not the height alone
  • Bars have no gaps between them, positioned using class boundaries

That's the notes covered.

Carry on to the next subtopic.