O-Level E-Maths Cumulative Frequency: Read the Curve in the Right Direction

Decide what the answer should measure

An SGExams student asked what happens if a cumulative frequency reading is slightly off. The most useful response is to make the estimate reproducible. Check the axis scales, draw guides and record what the curve supports. There is no need to assume a universal marking tolerance from a forum reply.

The horizontal axis normally shows a measured quantity such as mass, and the vertical axis shows a running count. Start on the horizontal axis when the question gives a measurement and asks how many. Start on the vertical axis when it gives a count or percentile and asks for a measurement. Write the unit you expect before drawing any lines.

Work from a count to a measurement

Imagine a curve for 80 parcels. The median is estimated by locating cumulative frequency 40, moving horizontally to the curve, then dropping vertically to the mass axis. If the crossing is at 185 g, the estimated median is 185 g. The answer is a mass, so 40 is an intermediate reading, even if it is the easiest number to spot.

For the lower and upper quartiles use 20 and 60 in this example. If they give 150 g and 220 g, the interquartile range is 70 g. Write the two estimated quartiles before subtracting so you can check the order. Follow the question if it specifies how to obtain or present a value.

Work from a measurement to a count

Suppose the curve at 200 g shows a cumulative count of about 50. In this example, that gives about 30 parcels above 200 g out of 80. If selecting one parcel at random, the estimated probability for that event is 30/80. Read the axis label and the question wording before deciding whether a complement is needed.

For a count between two boundaries, take the difference between the corresponding cumulative counts. This avoids adding a group that already includes the lower readings. When a question asks about exact endpoints, check the stated grouping and what the data allow you to infer.

Make a defensible estimate

Find two numbered grid marks and count the intervals between them. If a 20-unit gap has ten equal intervals, each represents two units. Use a ruler to trace to and from the curve and avoid claiming more precision than the drawing supports.

The official 2026 Mathematics 4052 syllabus includes cumulative frequency diagrams, quartiles and percentiles. Practise one question in each direction, then compare your traced lines against a worked answer. If the result differs, check whether the cause is scale reading, the wrong axis or the wrong event.

Use the units to catch a reversal

Try a second example: a running-time curve has 60 runners and shows 45 at 28 minutes. What percentage finished within 28 minutes? Divide 45 by 60 to get 75%. For the upper quartile, start from cumulative frequency 45 and read back to 28 minutes. The same intersection gives two different answers because the questions ask for different quantities. If you write 45 minutes for the quartile, your units have exposed the reversal.

Choose a suitable practice question from our Secondary Maths papers.

Sources and further reading

• Some clarifications for EMath (22 October 2024).
• Official syllabus reference. Checked on 5 November 2026.