Histograms and Frequency Density (Equal and Unequal Width)

Bars that touch

A histogram looks like a bar chart and behaves completely differently. Getting that straight is most of this topic. **A bar chart** shows categories. The bars have gaps between them because there is nothing in between "red" and "blue". The height of a bar is the frequency. **A histogram** shows **continuous** data grouped into classes. The bars touch, because one class ends exactly where the next begins. **And on a histogram the AREA of a bar is the frequency, not its height.** Everything else in this module comes from that one sentence. ⚠️ **Tier note.** Histograms with equal class widths are Higher-tier content. **Frequency density and histograms with unequal widths are Higher only.** If you are entered for Foundation, the second half of this module is beyond what you will be assessed on, though it is worth seeing.

Class, width, density, area

Five terms, and one formula that connects them.

Bar chart or histogram?

They look similar on the page. Everything that matters is different.

When do you need a histogram?

You have the times taken by 200 runners to finish a race, grouped into classes of different widths. Why is a histogram the right choice, and why must the vertical axis be frequency density?

• Time is continuous, so classes touch and a histogram fits; and because the classes differ in width, plotting frequency as height would make the wider classes look bigger than they are
• Only because time is continuous
• Because there are a lot of runners
• Because histograms look neater than bar charts

Area, not height

Here is the whole of frequency density in three lines. **frequency density = frequency ÷ class width** **frequency = frequency density × class width** **area of a bar = frequency** **Why it has to work this way.** Suppose one class is 20 wide holding 40 values, and the next is 5 wide holding 30. If you plotted frequency as height, the first bar would be shorter than the second, even though it contains more data. Frequency density fixes that: 40 ÷ 20 = 2, and 30 ÷ 5 = 6. The second bar IS taller, but the first is four times wider, so it covers more area. **So on a histogram you never read the height on its own.** A tall narrow bar and a short wide one can hold completely different amounts of data, and only the area tells you which.

Calculate the frequency density

A class runs **0 ≤ x < 20** and contains **30** values. What is the frequency density for that class?

Which bar holds the most data?

This histogram has four bars. **Tap the one containing the most data.** Look carefully: the tallest bar is not necessarily the answer, because on a histogram the frequency is the AREA.

Work backwards to the frequency

A bar on a histogram has a frequency density of **2.4** and covers a class of width **15**. How many values are in that class?

What the area tells you

Select the TWO correct statements about a histogram.

• The area of each bar equals the frequency of that class
• The total area of all the bars equals the total frequency of the data set
• The tallest bar always contains the most data
• Gaps between bars show classes with no values in them

Match each class to its width

Histograms in a paragraph

A histogram is used for _____ data, and its bars _____ because one class ends where the next begins. On a histogram the _____ of each bar represents the frequency. When the classes have different widths the vertical axis must show _____, which is calculated as frequency _____ class width. To recover a frequency from a chart you do the reverse and _____ the frequency density by the class width.

Estimate part of a class

A class runs **20 ≤ x < 40** and contains **60** values. Estimate how many of them lie between **20 and 25**. Take the part of the class you want as a fraction of its width, and apply that fraction to the frequency. ⚠️ This assumes the values are spread evenly across the class, which is worth saying in an exam answer.

A worked histogram question

**"The table shows the times taken by 120 people. Draw a histogram, then estimate how many took less than 35 minutes. (6 marks)"** **Step 1: find every class width.** 0 ≤ t < 20 has width 20. 20 ≤ t < 30 has width 10. 30 ≤ t < 50 has width 20. Write the widths down before doing anything else. **Step 2: divide.** With frequencies of 40, 45 and 35, the frequency densities are 40 ÷ 20 = 2, 45 ÷ 10 = 4.5, and 35 ÷ 20 = 1.75. **Step 3: plot frequency density up the vertical axis**, label that axis, and draw the bars touching, each spanning its own class. **Step 4: the estimate.** Everyone under 30 minutes is already counted: 40 + 45 = 85. The remaining part runs from 30 to 35, which is 5 of the third class's width of 20, so a quarter of its 35 values, giving 8.75, which is about 9. The estimate is roughly 94 people. **Step 5: say the assumption.** "This assumes the times are evenly spread within the 30 to 50 class." **The two places marks are lost**: writing the class widths down wrong at step 1, which poisons everything after it, and forgetting to label the vertical axis as frequency density rather than frequency.

Reading a histogram

A histogram is printed in the exam and you are asked to take information from it. Work through the decisions.

• The question asks which class contains the most data. What do you do?
• The vertical axis is unlabelled. What does that change?
• You are asked to estimate how many values are below a figure that falls in the middle of a class. How do you write the answer?

Your turn: build and read

A set of continuous data is grouped into classes of 0 ≤ x < 10 (frequency 25), 10 ≤ x < 30 (frequency 60) and 30 ≤ x < 35 (frequency 20). Work with it.

• Give the class width and the frequency density for each of the three classes, showing your working
• State which class contains the most data, and explain why that is not the same as which bar is tallest
• Estimate how many values are less than 20, and state the assumption your method makes
• Explain in one sentence why the vertical axis must be labelled frequency density rather than frequency