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Cambridge IGCSE0580

Histograms

Mathematics 0580 Chapter Notes

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Histograms
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Histograms notes

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In this chapter

  1. 1Frequency density
  2. 2Drawing a histogram
  3. 3Reading frequencies from a histogram
  4. 4Bar heights from one given bar

1. Frequency density

In a histogram the bars can have different widths. The area of a bar then shows the frequency, so the height cannot be the frequency. The height is the frequency density.

To find it, divide the frequency by the class width. The class width is the upper boundary minus the lower boundary.

frequency density = frequency/(class width)

frequency = frequency density × class width

Examiner insight

“A common error resulted from incorrectly calculating the frequency density by dividing the interval widths by the frequencies.” “Some divided all the frequencies by the smallest class width, some divided the frequencies by the upper bounds of the intervals, and some multiplied the midpoint values by the total frequency.”

Worked example 13 marks

Do not use a calculator.

A class 20 < h ≤ 35 has frequency 27. Work out its frequency density.

  1. 1

    class width = 35 - 20 = 15 [1]

  2. 2

    frequency density = frequency/(class width) = 27/15 [1]

  3. 3

    = 1.8 [1]

2. Drawing a histogram

Work out the frequency density of every class first. Write the values down before you draw.

Each bar covers its class width on the horizontal axis, with no gaps between bars. Its height is its frequency density. Label the vertical axis "Frequency density".

Examiner insight

“Others lost marks for incorrect widths of bars, usually the first which was often drawn over the interval 0 to 60.” “Candidates are advised to use a ruler when drawing histograms.”

Worked example 14 marks

The table shows the times, t minutes, taken by 51 students to finish a test.

Give the position and height of each bar of a histogram for this information.

time (t minutes)0 < t ≤ 1010 < t ≤ 3030 < t ≤ 40
frequency12309
  1. 1

    bars run from 0 to 10, 10 to 30 and 30 to 40 [1]

  2. 2

    0 < t ≤ 10: 12 ÷ 10 = 1.2 [1]

  3. 3

    10 < t ≤ 30: 30 ÷ 20 = 1.5 [1]

  4. 4

    30 < t ≤ 40: 9 ÷ 10 = 0.9 [1]

  5. 5

    draw the bars touching, with the vertical axis labelled Frequency density

3. Reading frequencies from a histogram

To interpret a histogram, turn each height back into a frequency. Multiply the frequency density by the class width.

The vertical axis is not the frequency. To find a total, work out each bar's frequency, then add them.

Examiner insight

“A significant number gave an answer of 65, 67 or 96, thinking that the vertical axis represented frequency and not frequency density.” “Errors seen using frequency density were multiplying it by the upper bound of the group or used with a group width of 10 for both.”

Worked example 14 marks

Do not use a calculator.

In a histogram, the bar for 10 < x ≤ 25 has height 2.4 on the frequency density axis. The bar for 25 < x ≤ 45 has height 1.1.

Work out the total frequency for 10 < x ≤ 45.

  1. 1

    frequency = frequency density × class width [1]

  2. 2

    10 < x ≤ 25: 2.4 × 15 = 36 [1]

  3. 3

    25 < x ≤ 45: 1.1 × 20 = 22 [1]

  4. 4

    total = 36 + 22 = 58 [1]

4. Bar heights from one given bar

Sometimes the height of one bar is given in centimetres. Then the heights are not frequency densities. They are a fixed multiple of them.

Find the frequency density of the given bar. Divide its height by that to get the scale. Multiply every other frequency density by the scale.

Examiner insight

“Many calculated the scale factor from the frequencies leading to a scale factor of 17.2 ÷ 43 = 0.4.” “Many correct calculations for the frequency densities were completed but it was common to give these as the final answers instead of going on to find the heights.”

Worked example 13 marks

The table shows some information about the lengths of 145 pieces of wire. Stanley wants to draw a histogram to represent the information. Complete the table.

length (L cm)70 < L ≤ 9090 < L ≤ 100100 < L ≤ 115115 < L ≤ 120
frequency40354525
height of bar in histogram (cm)4
  1. 1

    first bar: 40 ÷ 20 = 2, height 4, so the scale is 2 cm for each unit

  2. 2

    90 < L ≤ 100: 35 ÷ 10 = 3.5, 3.5 × 2 = 7 [1]

  3. 3

    100 < L ≤ 115: 45 ÷ 15 = 3, 3 × 2 = 6 [1]

  4. 4

    115 < L ≤ 120: 25 ÷ 5 = 5, 5 × 2 = 10 [1]

Recap

  • Frequency density = frequency ÷ class width.
  • The height of a bar is its frequency density. Its area is the frequency.
  • Bars cover their class widths with no gaps. The vertical axis is labelled Frequency density.
  • Frequency = frequency density × class width.
  • Given one bar's height: find its scale, then multiply every other frequency density by it.
  • Show each frequency density before you draw or scale.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. State the formula for frequency density.1 mark
  2. Do not use a calculator. The class 20 < x ≤ 30 has frequency 45. Work out its frequency density.2 marks
  3. Do not use a calculator. In a histogram, the bar for 10 < t ≤ 25 has height 3.2 on the frequency density axis. Work out the frequency of this class.2 marks
  4. Do not use a calculator. The table gives the ages, a years, of 80 visitors. Work out the frequency density of each class. | age (a years) | 0 < a ≤ 10 | 10 < a ≤ 30 | 30 < a ≤ 40 | 40 < a ≤ 80 | | --- | --- | --- | --- | --- | | frequency | 12 | 30 | 18 | 20 |3 marks
  5. A histogram has a bar for the class 20 < m ≤ 50, which has frequency 60. State where the bar starts and ends on the horizontal axis, and the height of the bar.3 marks
  6. In a histogram, 4 cm of height stands for 1 unit of frequency density. The class 8 < t ≤ 20 has frequency 17. Calculate the height of its bar, correct to 3 significant figures.3 marks
  7. Do not use a calculator. A histogram is drawn for 60 lengths, l cm. The bar for 0 < l ≤ 10 has frequency 20 and height 5 cm. The bar for 10 < l ≤ 30 has frequency 24. Work out the height of this bar.4 marks
  8. Do not use a calculator. In a histogram, the bar for 5 < t ≤ 15 stands for 36 people and has height 9 cm. The bar for 15 < t ≤ 45 has height 4 cm. How many people does it stand for?4 marks
  9. Do not use a calculator. A histogram shows the speeds, v km/h, of 100 cars. It has three bars. The bar for 30 < v ≤ 40 has height 2.6 and the bar for 40 < v ≤ 60 has height 2.2. The third bar starts at 60 and has height 1.5. All three heights are on the frequency density axis. Work out the speed at which the third bar ends.4 marks
  10. A courier records the mass, m kg, of each of 120 parcels and draws a histogram with three bars. The bar for 0 < m ≤ 2 stands for 24 parcels and has height 6 cm. The bar for 2 < m ≤ 5 has height 8 cm. The third bar is for 5 < m ≤ 10. Work out the height of the third bar.5 marks

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