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4MA1/1HR Jun 2023, Question 14: Drawing Cumulative Frequency Diagrams

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 6 marks

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The question, in short

From a grouped frequency table of 80 race times, complete a cumulative frequency table, draw its graph, and use it to estimate the median and interquartile range.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 1 mark

Fill in the cumulative frequency column for each time band, starting from 50 < t ≤ 60 up to 50 < t ≤ 110.

  1. 1Add the frequencies in turn.
  2. 250 < t ≤ 60: 15
  3. 350 < t ≤ 70: 15 + 16 = 31
  4. 450 < t ≤ 80: 31 + 21 = 52
  5. 550 < t ≤ 90: 52 + 14 = 66
  6. 650 < t ≤ 100: 66 + 8 = 74
  7. 750 < t ≤ 110: 74 + 6 = 80
  8. 8The cumulative frequencies are 15, 31, 52, 66, 74, 80

Answer15, 31, 52, 66, 74, 80

Part (b) · 2 marks

Plot the cumulative frequency against time on the given grid and draw the cumulative frequency curve.

  1. 1Plot each cumulative frequency at the upper end of its interval.
  2. 2The points are (60, 15), (70, 31), (80, 52), (90, 66), (100, 74) and (110, 80)
  3. 3No one took 50 minutes or less, so the curve starts at (50, 0)
  4. 4Join the points with a smooth increasing curve from (50, 0) up to (110, 80)

AnswerA smooth increasing curve from (50, 0) through (60, 15), (70, 31), (80, 52), (90, 66), (100, 74) and (110, 80)

Part (c) · 1 mark

Read from your cumulative frequency graph an estimate of the median time taken.

  1. 1The median is the time of the 80 ÷ 2 = 40th person
  2. 2Draw a line across from a cumulative frequency of 40 to the curve, then down to the time axis
  3. 3This meets the axis at about 74
  4. 4An estimate for the median time is 74 minutes

Answer74 (accept 73 to 75)

Part (d) · 2 marks

Use your graph to estimate the lower and upper quartiles, then find the interquartile range of the times.

  1. 1The lower quartile is the time of the 80 ÷ 4 = 20th person
  2. 2Reading across from a cumulative frequency of 20 and down to the axis gives a lower quartile of about 63
  3. 3The upper quartile is the time of the 3 × 80 ÷ 4 = 60th person
  4. 4Reading across from a cumulative frequency of 60 and down to the axis gives an upper quartile of about 86
  5. 5Interquartile range = 86 − 63 = 23
  6. 6An estimate for the interquartile range is 23 minutes

Answer23 (accept 21 to 24)

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