Cumulative frequency and box plot questions are some of the most predictable marks in O-Level E Math (4048). They appear almost every year in Paper 2, they are usually worth 8 to 12 marks in total, and the method barely changes from paper to paper. Yet students still drop marks on them, usually for small mechanical reasons rather than conceptual ones.
Here is exactly how to attack these questions, the way we drill them at Dojo.
Step 1: Build the Cumulative Frequency Table Correctly
You will usually be given a grouped frequency table, for example masses of 80 students grouped as 40 < m ≤ 45, 45 < m ≤ 50, and so on.
The single most common error is plotting at the class midpoint. Cumulative frequency is plotted at the upper class boundary, because the running total tells you how many values are less than or equal to that boundary.
So for the group 45 < m ≤ 50 with a running total of 23, your point is (50, 23), not (47.5, 23).
Also remember the starting point. If no student has mass 40 or below, plot (40, 0). Questions often award a mark for this point alone, and it anchors the left tail of your curve.
Step 2: Draw a Curve That Earns the Marks
Graph marks in E Math are given for scale, plotting, and shape. Protect all three.
- Use the scale stated in the question. If it says 2 cm to 5 units, use it. A different scale can cost you the whole plotting mark.
- Label both axes with the quantity and units, for example "Mass (kg)" and "Cumulative frequency".
- Plot points with a sharp pencil and a small cross or dot.
- Join with a smooth curve, not straight line segments, unless the question specifically asks for a polygon.
A curve that dips or wiggles backwards is a sign of a misplotted point. Cumulative frequency never decreases.
Step 3: Read Off Median, Quartiles and Percentiles
For a cumulative frequency curve with n values:
- Median: read across at n/2
- Lower quartile Q1: read across at n/4
- Upper quartile Q3: read across at 3n/4
- pth percentile: read across at p/100 × n
With 80 students, you read across at 40, 20 and 60. Do not use (n+1)/2 here. That formula is for an ordered list of individual data values, not for a continuous curve.
Always draw dashed construction lines from the vertical axis across to the curve, then down to the horizontal axis. Examiners look for these lines as evidence of method, and they save you if your final read-off is slightly off. Accept a small tolerance, usually half a small square.
Interquartile Range and "How Many" Questions
Interquartile range = Q3 minus Q1. Write it as a single number with units, and state what it measures if asked: it is a measure of the spread of the middle 50% of the data.
For a question like "estimate the number of students with mass greater than 62 kg", read up from 62, across to the cumulative frequency, then subtract from the total. Students who forget the subtraction lose an easy mark. If the question asks for a probability, divide by the total and leave it as a fraction or 3 significant figures.
Step 4: Draw the Box Plot Properly
A box-and-whisker plot needs the five figure summary: minimum, Q1, median, Q3, maximum.
When you draw it:
- Use the same horizontal scale as the axis given, and use a ruler.
- Draw the box from Q1 to Q3 with a clear vertical line at the median.
- Extend whiskers to the minimum and maximum values, not to the axis ends.
- Keep the box a sensible height, and align it with the given axis.
If the question gives you the box plot and asks you to extract values, read carefully. The median line is often not in the centre of the box, and that asymmetry is usually the point of the question.
Step 5: Comparison Questions and Mark-Scheme Phrasing
When you are asked to compare two distributions, the mark scheme almost always wants two comparisons in context: one about average and one about spread.
A full-mark answer looks like this:
"The median mass of Class A (58 kg) is higher than that of Class B (52 kg), so students in Class A are generally heavier. The interquartile range of Class A (9 kg) is smaller than that of Class B (14 kg), so the masses in Class A are more consistent."
Notice what is happening there. Each statement quotes a value, names the measure, and interprets it in context. Answers like "Class A is better" or "Class B is more spread out" without figures usually score zero.
Use median for average and IQR for spread. Do not compare means from a box plot, because a box plot does not show the mean.
Common Mistakes Checklist
- Plotting at midpoints instead of upper boundaries
- Using (n+1)/2 for the median on a curve
- Missing the (lowest boundary, 0) starting point
- No dashed read-off lines
- Forgetting to subtract from the total in "more than" questions
- Comparing only the average and not the spread
- Comparing spread without quoting IQR values
Work through three past year Paper 2 statistics questions with this checklist beside you, and these marks become almost automatic.
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