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Cambridge IGCSE0607

Cumulative frequency diagrams

International Mathematics 0607 Chapter Notes

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Cumulative frequency diagrams
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In this chapter

  1. 1Cumulative frequency tables
  2. 2Drawing a cumulative frequency diagram
  3. 3Median, quartiles and interquartile range
  4. 4Percentiles and counting above a value

1. Cumulative frequency tables

Cumulative frequency is a running total. For each class, add its frequency to the total of all the classes before it.

The cumulative frequency of a class is the number of values up to and including its upper end. The last cumulative frequency is the total number of values, so use it as a check.

cumulative frequency of a class = its frequency + the previous cumulative frequency

Examiner insight

“Some just transferred the frequency values, others put in mid-interval values and a few put in the lower interval value plus the frequency or the upper interval value minus the frequency.” “The fact that the final cumulative frequency was not 120 should have alerted them.”

Worked example 11 mark

Each of 100 students recorded the time spent doing homework one evening.

Complete the cumulative frequency table.

time (t minutes)0 < t ≤ 6060 < t ≤ 9090 < t ≤ 120120 < t ≤ 180
frequency7314517
  1. 1

    the first class is t ≤ 60: its cumulative frequency is the first frequency, 7

  2. 2

    add the next frequency: 7 + 31 = 38

  3. 3

    add again: 38 + 45 = 83, so the missing values are 38, 83 [1]

  4. 4

    check: 83 + 17 = 100, the total number of students

2. Drawing a cumulative frequency diagram

Plot each cumulative frequency against the upper end of its class, never the middle. Mark each point with a small cross (×).

Join the points with a smooth curve, not straight lines and not blocks. The curve always rises. It can be extended back to the origin.

A cumulative frequency diagram is not a frequency diagram: it plots running totals, so it never goes down.

Examiner insight

“In questions like this, the x co-ordinate is plotted at the top end of the interval so this first point was (20, 18).” “A number, however, drew a frequency diagram rather than a cumulative frequency curve and this meant they were ineligible for credit in parts (iii)(a) and (b).”

Worked example 12 marks

The table shows some information about the heights of 120 trees.

Draw a cumulative frequency diagram to show this information.

height, h cmh ≤ 50h ≤ 100h ≤ 150h ≤ 200h ≤ 250h ≤ 300
cumulative frequency6164482108120
  1. 1

    plot the six points at the upper end of each class: (50, 6), (100, 16), (150, 44), (200, 82), (250, 108), (300, 120) [1]

  2. 2

    join them with a smooth curve that rises through every point [1]

3. Median, quartiles and interquartile range

Read these from the vertical axis first. For n values, the median is at 1/2n, the lower quartile is at 1/4n and the upper quartile is at 3/4n.

Go across from that cumulative frequency to the curve, then down to the horizontal axis and read the value there.

The interquartile range is the upper quartile minus the lower quartile. Both are values from the horizontal axis. A smaller interquartile range means the values are more consistent.

IQR = upper quartile - lower quartile

Examiner insight

“To find the median from a cumulative frequency curve, candidates must look to half-way up the frequency axis, go across to the curve and read the value from the horizontal axis.” “The curve has to be read at a quarter the way from the top (at 75) and from the bottom (at 25) then the values subtracted.”

Worked example 12 marks

The cumulative frequency diagram shows the heights of 120 students.

Use the cumulative frequency diagram to find an estimate for the interquartile range.

  1. 1

    1/4 × 120 = 30 and 3/4 × 120 = 90: read across at each, then down: lower quartile ≈ 120 cm, upper quartile ≈ 139 cm [1]

  2. 2

    interquartile range = 139 - 120 = 19 cm [1]

4. Percentiles and counting above a value

A percentile is read the same way. For the pth percentile, go across from p/100 × n on the vertical axis, to the curve, then down.

To find how many values are more than a given value, read the cumulative frequency at that value. That is the number up to it, so subtract it from the total.

Take care with the scale: check what one small square is worth on each axis before you read.

number above x = n - (cumulative frequency at x)

Examiner insight

“A number of candidates omitted the subtraction from 120.” “A number of candidates misread the scale and read off at for example 215 or 240.”

Worked example 12 marks

The cumulative frequency diagram shows the heights of 120 students.

Use the cumulative frequency diagram to find an estimate for the number of students that have a height of more than 145 cm.

  1. 1

    read up from 145 cm to the curve, then across: cumulative frequency ≈ 106 [1]

  2. 2

    number above = 120 - 106 = 14 [1]

Recap

  • Cumulative frequency is a running total. The last value is the total number of values.
  • Plot each cumulative frequency at the upper end of its class and join the points with a smooth curve.
  • Mark points with small crosses. The curve never goes down.
  • Median at 1/2n, lower quartile at 1/4n, upper quartile at 3/4n, all on the vertical axis.
  • Interquartile range = upper quartile − lower quartile, read from the horizontal axis.
  • A smaller interquartile range means more consistent data.
  • The pth percentile is read at p/100 × n.
  • Number more than x = total − cumulative frequency at x.

End-of-chapter exercise

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

  1. Do not use a calculator. A cumulative frequency table is drawn up for 80 values. State the last cumulative frequency in the table.1 mark
  2. Do not use a calculator. The table gives the masses of 40 parcels. Complete the cumulative frequency row. | mass (m kg) | 0 < m ≤ 10 | 10 < m ≤ 20 | 20 < m ≤ 30 | 30 < m ≤ 40 | 40 < m ≤ 50 | | --- | --- | --- | --- | --- | --- | | frequency | 4 | 9 | 15 | 8 | 4 | | cumulative frequency | | | | | |2 marks
  3. Do not use a calculator. A cumulative frequency table gives 13 for the class 10 < m ≤ 20. Write down the coordinates of the point to plot for this class.1 mark
  4. Do not use a calculator. A cumulative frequency diagram is drawn for 120 values. State the cumulative frequency at which you read the lower quartile, the median and the upper quartile.3 marks
  5. The cumulative frequency diagram shows the times 40 students took to solve a puzzle. (a) Use it to estimate the median. (b) Use it to estimate the interquartile range.3 marks
  6. The cumulative frequency diagram shows the times 40 students took to solve a puzzle. (a) Estimate the number of students who took more than 34 minutes. (b) Estimate the 90th percentile.3 marks
  7. Class A has median 62 and interquartile range 14 in a test. Class B has median 58 and interquartile range 9. Compare the two classes.2 marks
  8. Do not use a calculator. A cumulative frequency table for the delivery times of 50 parcels gives these values. Find (a) the frequency of the class 20 < t ≤ 30 and (b) the number of parcels with t > 30. | time (t minutes) | t ≤ 10 | t ≤ 20 | t ≤ 30 | t ≤ 40 | t ≤ 50 | | --- | --- | --- | --- | --- | --- | | cumulative frequency | 4 | 14 | 38 | 48 | 50 |2 marks
  9. The cumulative frequency diagram shows the times 160 runners took to finish a hill race. (a) The fastest 20% of the runners win a medal. Estimate the longest time that wins a medal. (b) The slowest 15% of the runners took more than T minutes. Estimate T. (c) Estimate the percentage of the runners who took more than 48 minutes.6 marks
  10. The cumulative frequency diagram shows how many minutes late each of 80 trains on a railway line was. (a) Estimate the median and the interquartile range. (b) Estimate the number of trains that were more than 24 minutes late but not more than 40 minutes late. (c) Ten more trains are added to the data. Each of them was more than 50 minutes late. Estimate the median for all 90 trains.7 marks

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