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Cambridge IGCSE0580

Cumulative frequency diagrams

Mathematics 0580 Chapter Notes

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Cumulative frequency diagrams
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In this chapter

  1. 1Cumulative frequency tables and diagrams
  2. 2Reading the median and quartiles
  3. 3The interquartile range
  4. 4Percentiles
  5. 5How many are above or below a value

1. Cumulative frequency tables and diagrams

A cumulative frequency is a running total of the frequencies. The last one equals the total frequency.

Plot each cumulative frequency at the upper end of its class interval, then join the points with a smooth curve. Mark each point clearly, for example with a small cross (×). Do not draw bars.

Examiner insight

“Only a few candidates made errors and these were either because they drew bars rather than a curve or they plotted the points at the mid-interval values rather than the upper end of the interval.” “The point at (20, 0) was the one most often missed out.”

Worked example 13 marks

The table gives the lengths of 40 plants.

Complete the cumulative frequencies and write down the points to plot.

length (l cm)0 < l ≤ 1010 < l ≤ 2020 < l ≤ 3030 < l ≤ 4040 < l ≤ 50
frequency491584
  1. 1

    running totals: 4, 4+9=13, 13+15=28, 28+8=36, 36+4=40 [1]

  2. 2

    upper ends: (10, 4), (20, 13), (30, 28), (40, 36), (50, 40) [1]

  3. 3

    start point at the lower end of the first interval: (0, 0) [1]

2. Reading the median and quartiles

Find the position first. For a total frequency n, the median is at n/2, the lower quartile at n/4 and the upper quartile at 3n/4.

Go across from that cumulative frequency to the curve, then down to the horizontal axis. The answer is the value on the horizontal axis, not the cumulative frequency. Check the scale of each small square before you read it.

median at n/2 LQ at n/4 UQ at 3n/4

Examiner insight

“The most common error here was candidates incorrectly reading the scale, where each small square on the horizontal axis was read as 1 second not 2 seconds, meaning often 61 was given as an answer when it should have been 62.” “Less able candidates gave the answer 40, rather than understanding that the value required was the time.”

Worked example 13 marks

The cumulative frequency curve for the journey times of 80 people passes through (8, 20), (12, 40) and (17, 60), where the times are in minutes. Estimate the median and the quartiles.

  1. 1

    median at 80/2 = 40, LQ at 80/4 = 20, UQ at (3 × 80)/4 = 60 [1]

  2. 2

    median = 12 minutes [1]

  3. 3

    lower quartile = 8 minutes, upper quartile = 17 minutes [1]

3. The interquartile range

The interquartile range is the upper quartile minus the lower quartile. Read both quartiles from the horizontal axis first, then subtract.

To compare two sets of data, say which median is greater and which interquartile range is smaller. Stating the values alone is not a comparison.

IQR = UQ - LQ

Examiner insight

“Some less able candidates subtracted the frequencies at the upper and lower quartiles to obtain a frequency of 80 and read off the median value again.” “The inter-quartile range was well answered, although a common error, made by a number of candidates, was to calculate upper quartile – lower quartile = 300 – 100 = 200 and then effectively find the median again by reading the value at 200.”

Worked example 13 marks

The cumulative frequency curve for the marks of 120 students passes through (35, 30), (52, 60) and (70, 90). Estimate the interquartile range.

  1. 1

    quartile positions: 120/4 = 30 and (3 × 120)/4 = 90 [1]

  2. 2

    lower quartile = 35, upper quartile = 70 [1]

  3. 3

    IQR = 70 - 35 = 35 [1]

4. Percentiles

The pth percentile is the value that p% of the data lie below. Work out p% of the total frequency first.

Then read across from that cumulative frequency to the curve and down to the horizontal axis. The lower quartile is the 25th percentile and the upper quartile is the 75th.

position of the pth percentile = p/100 × n

Examiner insight

“A significant number of candidates did not pick up on the 80th percentile, instead finding the time corresponding to a cumulative frequency of 80 leading to the very common error of 60.” “Two common errors included calculating the cumulative frequency for the 40th percentile and giving this as their answer or simply reading off the mass for a cumulative frequency of 40.”

Worked example 12 marks

The cumulative frequency curve for the heights of 150 plants passes through (45, 60) and (62, 120). Estimate the 40th percentile.

  1. 1

    40/100 × 150 = 60 [1]

  2. 2

    read across from 60 and down: 45 [1]

5. How many are above or below a value

To find how many values are below a number, read the cumulative frequency at that number.

To find how many are above it, subtract that reading from the total frequency. To find how many lie between two values, read both cumulative frequencies and subtract.

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

Examiner insight

“Instead of reading off the cumulative frequencies for both of the masses 250 and 400, several read off for only one of these values and subtracted the reading from 120.” “Some gave a correct reading from their graph but then did not subtract this from 200.”

Worked example 12 marks

The cumulative frequency curve for the lifetimes of 200 bulbs passes through (900, 52), where the lifetime is in hours. Estimate how many bulbs last more than 900 hours.

  1. 1

    cumulative frequency at 900 hours = 52 [1]

  2. 2

    200 - 52 = 148 bulbs [1]

Recap

  • Extended: cumulative frequency is a running total, plotted at the upper end of each interval.
  • Extended: join the plotted points with a smooth curve, not bars.
  • Extended: median at n/2, lower quartile at n/4, upper quartile at 3n/4.
  • Extended: read across to the curve and down. The answer is on the horizontal axis.
  • Extended: interquartile range = upper quartile − lower quartile.
  • Extended: the pth percentile is at p/100 × n.
  • Extended: number above a value = total frequency − cumulative frequency at that value.

End-of-chapter exercise

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

  1. Do not use a calculator. The frequencies for four time intervals are 5, 12, 20 and 8. Write down the cumulative frequencies.2 marks
  2. Do not use a calculator. The times of 20 runners are grouped as 20 < t ≤ 30 (frequency 6), 30 < t ≤ 40 (frequency 10) and 40 < t ≤ 50 (frequency 4). Write down the four points to plot on a cumulative frequency diagram.3 marks
  3. Do not use a calculator. The cumulative frequency curve for 60 students passes through (14, 15), (22, 30) and (31, 45), where the values are times in minutes. Write down the lower quartile, the median and the upper quartile.3 marks
  4. Do not use a calculator. The cumulative frequency curve for the times of 200 runners passes through (24, 50), (31, 100) and (39, 150), where the times are in seconds. Estimate the interquartile range.3 marks
  5. Do not use a calculator. The cumulative frequency curve for the times of 90 students passes through (57, 54), where the times are in minutes. Estimate the 60th percentile.2 marks
  6. The cumulative frequency curve for the masses of 120 parcels passes through (2.5, 78), where the masses are in kilograms. Estimate how many parcels have a mass greater than 2.5 kg.2 marks
  7. The cumulative frequency curve for the heights of 300 trees passes through (6, 84) and (9, 246), where the heights are in metres. Estimate how many trees have a height between 6 m and 9 m.2 marks
  8. Class A and class B sit the same test. For class A the median mark is 62 and the interquartile range is 18. For class B the median mark is 55 and the interquartile range is 9. Make two comparisons between the marks of the two classes.2 marks
  9. The cumulative frequency curve for the times, t minutes, that 150 runners take to finish a race passes through (30, 0), (40, 18), (45, 51), (50, 96), (55, 129), (60, 144) and (70, 150). (a) Find the number of runners in each of the intervals 30 < t ≤ 40, 40 < t ≤ 50, 50 < t ≤ 60 and 60 < t ≤ 70. (b) The fastest 34% of the runners win a medal. Find the longest time that wins a medal. (c) The slowest 14% of the runners take longer than k minutes. Find k.6 marks
  10. The cumulative frequency curve for the heights of some plants in field P passes through (12, 45), (15, 90), (19, 135) and (21, 144), where the heights are in centimetres. The upper quartile is 19 cm. (a) Show that there are 180 plants in field P. (b) Find the interquartile range. (c) 21 cm is the kth percentile. Find k. (d) The plants in field Q have a median height of 17 cm and an interquartile range of 4 cm. Make two comparisons between the heights in the two fields.8 marks

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