Histograms

A histogram shows grouped continuous data in bars with no gaps. The classes can have different widths, so the height of each bar is the frequency density, and the area of each bar is the frequency.

Key facts

  • Frequency density = frequency ÷ class width. So frequency = frequency density × class width.
  • The area of a bar is its frequency; the bars touch because the data are continuous.
  • The modal class has the highest frequency density, which is not always the highest frequency.

Worked examples

Example 1

In a histogram, the bar for 20 < a ≤ 40 has a frequency density of 1.5. How many values are in that class? Estimate how many are between 25 and 40.

  1. Find the class width. 40 − 20 = 20
  2. Frequency: 1.5 × 20 = 30
  3. From 25 to 40 is a width of 15. 1.5 × 15 = 22.5

Answer: 30; about 22.5

Example 2

The class 0 < t ≤ 10 has a frequency of 20 and the class 10 < t ≤ 40 has a frequency of 30. Find the frequency density of each class.

  1. First class: width 10. 20 ÷ 10 = 2
  2. Second class: width 30. 30 ÷ 30 = 1

Answer: 2 and 1

Example 3

Which is the modal class: 0 < x ≤ 10 with a frequency of 12, or 10 < x ≤ 30 with a frequency of 18?

  1. Divide each frequency by its class width. 12 ÷ 10 = 1.2, 18 ÷ 20 = 0.9
  2. The modal class has the highest frequency density, even though its frequency is lower.

Answer: 0 < x ≤ 10 (frequency density 1.2, not 0.9)

Example 4

A histogram has a bar from 0 to 10 with a frequency density of 3 and a bar from 10 to 25 with a frequency density of 2. How many values are there altogether?

  1. Frequency = frequency density × class width. 3 × 10 = 30, 2 × 15 = 30
  2. Add the frequencies. 30 + 30 = 60

Answer: 60