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How do you draw and read a cumulative frequency curve, find the quartiles and interquartile range, and use box plots to compare distributions?

Construct and interpret a cumulative frequency curve to estimate the median, quartiles and interquartile range, and draw and compare box plots from five-number summaries (Higher tier).

A focused answer to the WJEC GCSE Mathematics statistics content on cumulative frequency and box plots, covering constructing and reading cumulative frequency curves, estimating the median and quartiles, finding the interquartile range, and drawing and comparing box plots at Higher tier.

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  1. What this dot point is asking
  2. Building a cumulative frequency curve
  3. Median, quartiles and the IQR
  4. Box plots
  5. Comparing distributions
  6. Why this matters

What this dot point is asking

This is the main Higher-tier statistics topic. WJEC asks you to construct and read a cumulative frequency curve, to estimate the median and the quartiles from it, to calculate the interquartile range as a measure of spread, and to draw and compare box plots built from a five-number summary. The recurring extended question gives two box plots (or two curves) and asks you to compare the distributions using a median and the interquartile range, in context. It is examined on Unit 2 and rewards accurate reading and clear, comparative reasoning.

Building a cumulative frequency curve

Cumulative frequency is a running total of the frequencies.

Plotting against the upper boundary (not the midpoint) is essential, because the running total counts everything up to and including that value.

Median, quartiles and the IQR

The curve estimates the values that split the data into quarters.

So for n=80n = 80, read at 4040, 2020 and 6060. The IQR is a better measure of spread than the range because it ignores the extreme values at each end.

Box plots

A box plot displays the five-number summary visually.

You can draw a box plot directly from a five-number summary, or read one off a cumulative frequency curve.

Comparing distributions

Comparing two box plots is the highest-value skill here.

Why this matters

Cumulative frequency and box plots are the signature Higher-tier statistics topic and a reliable source of extended marks on Unit 2. The skills (drawing the curve against upper boundaries, reading the median and quartiles, computing the IQR, and constructing and comparing box plots) are examined together, and the comparative "compare the distributions" question rewards AO2 and AO3 reasoning. The IQR's resistance to extreme values makes it the preferred measure of spread here, and the habit of comparing both an average and a spread, in context, secures the final marks.

Exam-style practice questions

Practice questions written in the style of WJEC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

WJEC 20193 marksA cumulative frequency curve is drawn for the times of 8080 runners. Explain how to estimate the median and the interquartile range from the curve. (Higher, Unit 2, calculator.)
Show worked answer →

The median is at the value where the cumulative frequency is half the total: 80÷2=4080 \div 2 = 40. Read across from 4040 on the cumulative axis to the curve and down to the time axis.

The lower quartile is at 14×80=20\tfrac{1}{4}\times 80 = 20 and the upper quartile at 34×80=60\tfrac{3}{4}\times 80 = 60; read these off the same way.

The interquartile range is upper quartile minus lower quartile.

Markers award marks for the positions (4040, 2020 and 6060 on the cumulative axis) and for the interquartile range as the difference of the quartiles. Using n+12\tfrac{n+1}{2} positions (for a list, not a curve) is a common confusion.

WJEC 20224 marksTwo box plots compare boys' and girls' scores. The boys have median 5252 and IQR 2424; the girls have median 5858 and IQR 1212. Compare the two distributions. (Higher, Unit 2, calculator.)
Show worked answer →

Compare the medians: the girls' median (5858) is higher than the boys' (5252), so on average the girls scored more highly.

Compare the spread using the interquartile range: the girls' IQR (1212) is smaller than the boys' (2424), so the girls' scores were more consistent (less spread out).

Markers give marks for comparing the medians in context and for comparing the IQRs in context. A full comparison needs both an average (the median) and a measure of spread (the IQR), worded for the situation. Comparing only one loses marks.

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