Readings taken from a cumulative frequency curve of 200 phone call lengths give a lower quartile of 4.6 minutes and an upper quartile of 11.3 minutes. Work out the interquartile range, in minutes.
Cumulative Frequency - Edexcel GCSE Higher Maths
Typically 4-6 marks when it appears, usually in one structured question · Spec 1MA1
A cumulative frequency is a running total. You take a grouped frequency table, add each class on to everything before it, and the final figure comes out equal to the size of the whole sample. Plot each running total against the top end of its class, join the points with a smooth curve, and the graph will give you an estimate of the median, of both quartiles, and of how many items fall above or below any value on the horizontal axis. On Edexcel 1MA1 this is Higher-only content under spec refs S3 and S4.
Edexcel sets it as one structured question of three or four parts. Part (a) is a single mark for completing the cumulative frequency column. Part (b) is worth two marks for the graph itself, and the mark scheme splits them: one for points at the correct heights, one for a curve or line segments through them. The remaining parts carry the real test, and they are always interpretation: estimate the median, estimate the interquartile range, estimate how many of the 120 patients waited longer than 25 minutes. The question appears on all three papers, and Paper 1 having no calculator changes very little, because the answers come off the graph rather than out of a calculation.
Three habits protect the interpretation marks. Work out the position before you touch the graph: with 80 values the median is read across from 40, not from 50, and the quartiles come from 20 and 60. Then draw the lines across and down, because a reading a little outside the accepted range can still take a method mark when those lines are on the graph and takes nothing when they are not. And when a part asks how many were above a value, take the reading and subtract it from the total, since a cumulative frequency curve only ever tells you how many were below.
Worked example
80 people were timed while they finished a puzzle. The times, in minutes, are grouped like this: up to 10 minutes, 8 people; more than 10 and up to 20 minutes, 16 people; more than 20 and up to 30 minutes, 20 people; more than 30 and up to 40 minutes, 20 people; more than 40 and up to 50 minutes, 11 people; more than 50 and up to 60 minutes, 5 people.
(a) Work out the cumulative frequencies and write down the points that should be plotted.
(b) A cumulative frequency curve is drawn through those points. Use it to estimate the median and the interquartile range.
[5 marks]- Running totals down the column: 8, then , then 44, then 64, then 75, then 80. The last entry is 80, which matches the 80 people in the sample.B1: correct cumulative frequency column; the final total equalling the sample size is the built-in check
- Plot each total against the top of its class, not the middle: , , , , , . Join them with a smooth curve.B1: points at the upper class boundaries; plotting at midpoints scores nothing beyond a special case
- Median: half of 80 is 40, so read across from 40 on the cumulative frequency axis. That line meets the curve between and , four fifths of the way along, so the median is about 28 minutes.B1: reading across from 40 rather than from 50; median 28 minutes
- Quartiles: a quarter of 80 is 20 and three quarters is 60. Reading across from 20 gives about 17.5 minutes; reading across from 60 gives about 38 minutes.M1: both readings taken from and , with lines drawn on the graph
- Interquartile range minutes.A1: 20.5 minutes, given as one number with its unit
Practice questions
These are original questions in Edexcel 1MA1 Higher style. There are no diagrams here, so each curve is given as the list of points it passes through, and you interpolate between them exactly as you would read between two plotted points on graph paper. Any answer that would come off a curve says so in the stem and carries a reading range, so a sensible estimate near the stated value is marked correct.
No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.
Question 1 Warm-up [2 marks] Question 2 Warm-up [2 marks] A cumulative frequency curve is drawn for the times, in minutes, that 120 patients waited to be seen at a surgery. It passes through these points: up to 5 minutes, 18; up to 10 minutes, 42; up to 15 minutes, 72; up to 20 minutes, 102; up to 25 minutes, 116; up to 30 minutes, 120. Use the curve to estimate how many patients waited 15 minutes or less. Since the figure is read from a graph, an answer within 2 patients is accepted.Question 3 Exam pace [2 marks] A cumulative frequency curve is drawn for the times, in minutes, that 120 patients waited to be seen at a surgery. It passes through these points: up to 5 minutes, 18; up to 10 minutes, 42; up to 15 minutes, 72; up to 20 minutes, 102; up to 25 minutes, 116; up to 30 minutes, 120. Estimate the upper quartile of the waiting times, in minutes. The value is read off the curve, so an answer within 1 minute is accepted.Question 4 Stretch [3 marks] The heights of 300 sunflowers were measured. A cumulative frequency curve of the heights, in centimetres, passes through these points: up to 100 cm, 12; up to 120 cm, 50; up to 140 cm, 150; up to 160 cm, 250; up to 180 cm, 290; up to 200 cm, 300. Estimate the lower quartile and the upper quartile of the heights, both in centimetres. Each value is read off the curve, so an answer within 3 cm is accepted for each.Question 5 Stretch [3 marks] 200 people took part in a 5 km fun run. A cumulative frequency curve of their finishing times, in minutes, passes through these points: up to 25 minutes, 30; up to 30 minutes, 70; up to 35 minutes, 120; up to 40 minutes, 170; up to 45 minutes, 192; up to 50 minutes, 200. The slowest 20 per cent of the runners all took longer than a certain time. Estimate that time, in minutes. The value is read off the curve, so an answer within 1 minute is accepted.Question 6 Stretch [4 marks] The heights of 300 sunflowers were measured. A cumulative frequency curve of the heights, in centimetres, passes through these points: up to 100 cm, 12; up to 120 cm, 50; up to 140 cm, 150; up to 160 cm, 250; up to 180 cm, 290; up to 200 cm, 300. Estimate how many of the sunflowers had a height greater than 130 cm and less than or equal to 150 cm. Two readings come off the curve, so an answer within 8 plants is accepted.
Common mistakes examiners see
Plotting each cumulative total against the midpoint of its class, so a table with classes up to 10, 20, 30 and 40 produces points at 5, 15, 25 and 35 and a curve squashed into the left of the grid.
Plot at the upper class boundary. The running total up to 20 minutes is only known at 20 minutes, so it belongs above the 20, and the curve should finish at the top right corner of the data.
Reading the median across from 50 on the cumulative frequency axis because the median is 'the halfway point', when the sample is 80 and the read should be at 40.
Write down before you go near the graph. With 80 values the median is read from 40; with 120 values it is read from 60. The number 50 only ever appears if the sample size happens to be 100.
Taking the quartiles at 25 and 75 on the cumulative frequency axis, or at a quarter and three quarters of the way along the horizontal axis instead.
Use and on the vertical axis every time. For 300 plants that is 75 and 225, and both readings come back down to the horizontal axis to give the two quartile values.
Answering 'how many took more than 25 minutes' with the cumulative frequency read at 25 minutes, giving 116 out of 120 rather than 4.
The curve counts items below a value, so subtract the reading from the total. Say the sentence to yourself as you write it: 116 were at or below 25 minutes, so were above.
Giving the interquartile range as 'from 17.5 to 38', or subtracting the wrong way round to get a negative number.
The interquartile range is one number, upper quartile minus lower quartile, and it is never negative. Write on its own line so the examiner can see both readings and the subtraction.
Frequently asked questions
- Is cumulative frequency on the Foundation paper?
- No. Cumulative frequency graphs are Higher-only content on Edexcel 1MA1, along with box plots and histograms with unequal class widths. Foundation candidates meet grouped frequency tables and the estimated mean, but not the curve or the quartiles read from it.
- Do you use n over 2 or n plus 1 over 2 for the median on a cumulative frequency curve?
- Use . Adding one is the rule for finding the median of a list of raw values; on a cumulative frequency graph the accepted method is to read across from half the total frequency. For the quartiles the same applies, so read from and .
- Where do the points go when you draw the curve?
- At the upper boundary of each class, plotted against the running total up to that point. A class of more than 20 and up to 30 minutes with a cumulative frequency of 44 is plotted at 30 on the horizontal axis, not at 25. Points at midpoints are the single most common reason for losing the plotting marks.
- How many marks is a cumulative frequency question worth on Edexcel Higher?
- Usually 4-6 marks in one structured question: one mark for the table, two for the graph, and the rest for the readings. It is often followed by a box plot part, which reuses the quartiles you have already found, so a good curve can be worth more than the question itself.