Higher – Histograms

Part ofMathsStatistics

Key points about histograms

Bullet points represented by lightbulbs

  • A histogram is a type of diagram used to represent . The area of the bars represents the frequencies.

  • For a table of data where the class widths of the groups are unequal, the , which is used for the height of the bars, must be calculated.

Frequency density = Frequency ÷ Class width

  • For a table where the data has been grouped into equal class widths, the frequency density is proportional to the frequency. The frequency is often used for the height of the bars, and the graph is called a frequency diagram.
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How to construct and find frequencies from a histogram

Histogram of time taken vs frequency density
A histogram is similar to a frequency diagram but has bars of different widths.

To produce a histogram, data is required. The data should be provided in the form of a grouped .

Creating a histogram:

  1. Add two additional columns, if not provided, to the grouped frequency table. Label the first column ‘’ and the second column ‘Frequency density’.

  2. Complete the ‘Class width’ column. The class width is the difference between the upper and lower bounds of each group.

  3. Complete the ‘Frequency density’ column. The frequency density is calculated by dividing each frequency by the corresponding class width.

Histogram of time taken vs frequency density
A histogram is similar to a frequency diagram but has bars of different widths.

  1. Draw a horizontal axis. This should be a continuous number line. Look at the range of data for the class widths. Choose an appropriate scale for this axis. A may be required.

  2. Draw a vertical axis. The vertical axis is always frequency density. Choose an appropriate scale for this axis which includes the highest value in the frequency density column.

  3. Draw the bars for each row of data from the table. The width of the bar corresponds to the and of the class width. The height of the bar is the frequency density.

  4. Check you have labelled each axis correctly and give the histogram a title.

Follow the worked example below

Check your understanding

GCSE exam-style questions

Icon representing a multiple-choice question with answers A, B and C
  1. The table shows the money spent on public transport each week for 60 workers.

Calculate the frequency densities.

A two‑column table with purple headers. The left column header is ‘Cost (£)’ and the right column header is ‘Frequency’. The rows list five grouped cost intervals with their frequencies: • ‘0 < m ≤ 5’ with frequency 3 • ‘5 < m ≤ 15’ with frequency 21 • ‘15 < m ≤ 25’ with frequency 35 • ‘25 < m ≤ 40’ with frequency 12 • ‘40 < m ≤ 45’ with frequency 9

  1. The histogram shows the time taken for participants to complete a fun run.

How many participants finished the race between 40 and 45 minutes?

A histogram titled ‘A histogram to show time taken to finish a fun run’. The horizontal axis is labelled ‘Time taken (min)’ and marked at 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75 and 80. The vertical axis is labelled ‘Frequency density’ and marked at 0, 0·5, 1·0, 1·5, 2·0 and 2·5. Five blue bars show the grouped data: • 20 < t ≤ 40: bar spans from 20 to 40 with height 1·15. • 40 < t ≤ 45: bar spans from 40 to 45 with height 2·2. • 45 < t ≤ 55: tall bar from 45 to 55 with height 1·7. • 55 < t ≤ 65: bar spans from 55 to 65 with height 1·3. • 65 < t ≤ 80: wide bar from 65 to 80 with height 0·8.

  1. The histogram shows the ages of members at a gym.

How many members in total does the gym have?

A histogram titled ‘A histogram to show ages of gym club members’. The horizontal axis is labelled ‘Age (years)’ and marked at 20, 25, 30, 35, 40, 45, 50, 55 and 60. The vertical axis is labelled ‘Frequency density’ and marked at 0, 1, 2, 3 and 4. Five blue bars represent the grouped age intervals: • 20 < age ≤ 25: bar spans from 20 to 25 with a height 3·2 • 25 < age ≤ 30: bar spans from 25 to 30 with a height 3·6 • 30 < age ≤ 40: bar spans from 30 to 40 with a height 2·6. • 40 < age ≤ 60: a wide bar spanning from 40 to 60 with a height 1·2.

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How to interpret a histogram

Sometimes a histogram might be presented with an incomplete frequency density scale. In this case additional information is provided, usually the frequency of one bar.

The additional information can be used either to reconstruct the frequency density scale or to allow a to be used, to calculate the frequencies for each bar.

Follow the worked example below

GCSE exam-style questions

Icon representing a multiple-choice question with answers A, B and C
  1. The histogram shows the mass of allotment carrots.

Given there are 25 carrots with a mass between 60 and 70 grams, work out the scale for the frequency density axis.

A histogram titled ‘A histogram to show the mass of carrots’. The horizontal axis is labelled ‘Mass (g)’ and marked at 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. The vertical axis is labelled ‘Frequency density’ and marked at 0, 1, 2, 3 and 4. Six blue bars show the grouped mass intervals: • 0 < m ≤ 30: a wide bar spanning 0 to 30 with unknown height. • 30 < m ≤ 40: bar spanning 30 to 40 with unknown height. • 40 < m ≤ 60: bar spanning 40 to 60 with unknown height. • 60 < m ≤ 70: narrow bar from 60 to 70 with unknown height. • 70 < m ≤ 90: bar from 70 to 90 with unknown height. • 90 < m ≤ 100: bar from 90 to 100 with unknown height.

  1. The histogram shows the mass of allotment carrots.

A carrot is chosen at random. What is the probability it has a mass of 40 grams or less?

Give the answer as a fraction in its simplest form.

A histogram titled ‘A histogram to show the mass of carrots’. The horizontal axis is labelled ‘Mass (g)’ and marked at 0, 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100. The vertical axis is labelled ‘Frequency density’ and goes from 0 to 2.5. Six blue bars show the grouped mass intervals: • 0 < m ≤ 30: a wide bar spanning 0 to 30 with height of 0·5. • 30 < m ≤ 40: bar spanning 30 to 40 with height of 1·5. • 40 < m ≤ 60: bar spanning 40 to 60 with height of 2·0. • 60 < m ≤ 70: narrow bar from 60 to 70 with height of 2·5. • 70 < m ≤ 90: bar from 70 to 90 with height of 1·25. • 90 < m ≤ 100: bar from 90 to 100 with height of 0·5.

  1. The histogram shows the length of telephone calls to a call centre.

There were 36 calls that lasted less than 10 minutes.

How many calls had a duration of 30 minutes and over?

A histogram titled ‘A histogram to show the length of telephone calls’. The horizontal axis is labelled ‘Time (min)’ and marked at 0, 10, 20, 30, 40, 50 and 60. The vertical axis is labelled ‘Frequency density’ and is blank. Six blue bars represent grouped call lengths: • 0 < t ≤ 10: bar spanning 0 to 10 with unknown height. • 10 < t ≤ 15: bar spanning 10 to 15 with unknown height. • 15 < t ≤ 20: tall narrow bar from 15 to 20 with unknown height. • 20 < t ≤ 30: bar spanning 20 to 30 with unknown height. • 30 < t ≤ 40: bar spanning 30 to 40 with unknown height. • 40 < t ≤ 60: wide bar spanning 40 to 60 with unknown height.

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How to find the median on a histogram

To estimate a in a histogram, find a vertical line that chops the total area of the bars in half.

The line should have half of the area (or frequency) to the left of it and half the area to the right.

The value where this line intersects the 𝑥-axis is the median.

If a table with the frequencies is not provided, calculate these first.

Follow the worked example below

GCSE exam-style questions

Icon representing a multiple-choice question with answers A, B and C
  1. The histogram shows the length of 60 bags.

The frequency for each bar is provided.

Use the histogram to estimate the median length.

A histogram titled ‘A histogram to show the length of bags’. The horizontal axis is labelled ‘Length (cm)’ and marked at 40, 50, 60, 70, 80 and 90. The vertical axis is labelled ‘Frequency density’ and marked at 0, 0·5, 1·0, 1·5, 2·0 and 2·5. Five blue bars represent grouped bag‑length intervals. Each bar contains its frequency written in black text. • 40 < l ≤ 45: bar spans 40 to 45, height 1·6, with the number ‘8’ inside. • 45 < l ≤ 55: narrow bar from 45 to 55, height 2·2, with the number ‘22’ inside. • 55 < l ≤ 70: wide bar from 55 to 70, height 1·0, with the number ‘15’ inside. • 70 < l ≤ 80: bar from 70 to 80, height around 0·8, with the number ‘9’ inside. • 80 < l ≤ 85: bar from 80 to 85, height around 1·2, with the number ‘6’ inside.

  1. The histogram shows the mark in a music theory assessment for 800 people.

The frequency for each bar is provided.

Use the histogram to estimate the median mark.

A histogram titled ‘A histogram to show music theory marks’. The horizontal axis is labelled ‘Mark’ and marked at 30, 40, 50, 60, 70, 80, 90 and 100. The vertical axis is labelled ‘Frequency density’ and marked at 0, 5, 10, 15, 20 and 25. Six blue bars represent grouped marks, and each bar contains its frequency in black text: • 35 < mark ≤ 40: bar from 35 to 40 with height 10; the number ‘50’ is written inside. • 40 < mark ≤ 50: bar from 40 to 50 with height 12; the number ‘120’ is written inside. • 50 < mark ≤ 60: bar from 50 to 60 with height 15.5; the number ‘180’ is written inside. • 60 < mark ≤ 65: narrow bar from 60 to 65 with height 20; the number ‘100’ is written inside. • 65 < mark ≤ 95: very wide bar from 65 to 95 with height 11.5; the number ‘330’ is written inside. • 95 < mark ≤ 100: narrow bar from 95 to 100 with height 4; the number ‘20’ is written inside.

  1. The histogram shows the height of 80 trees in a park.

Use the histogram to estimate the median height.

A histogram titled ‘A histogram to show the height of trees’. The horizontal axis is labelled ‘Height (m)’ and marked at 0, 2, 4, 6, 8, 10, 12, 14, 16 and 18. The vertical axis is labelled ‘Frequency density’ and marked at 0, 1, 2, 3, 4, 5, 6, 7 and 8. Six blue bars represent grouped height intervals: • 0 < h ≤ 2: bar spanning 0 to 2, height about 1. • 2 < h ≤ 6: bar spanning 2 to 6, height about 6. • 6 < h ≤ 10: bar spanning 6 to 10, height about 7 (the tallest bar). • 10 < h ≤ 12: bar spanning 10 to 12, height about 6. • 12 < h ≤ 14: bar spanning 12 to 14, height about 4. • 14 < h ≤ 18: bar spanning 14 to 18, height about 2.

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Quiz – Higher – Histograms

Practise what you've learned about histograms with this quiz.

Now you've revised histograms, why not look at Higher – Solving problems using Venn diagrams?

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