Sheet № 67 · Higher only · AQA · Edexcel · OCR
Cumulative Frequency and Box Plots –
Cumulative frequency diagrams and box plots are Higher-tier statistics topics that appear on AQA, Edexcel and OCR GCSE papers almost every year. They allow you to estimate the median, quartiles and interquartile range from grouped data — all things you cannot find exactly from a grouped frequency table alone. Being able to draw and interp
§Key definitions
Cumulative frequency
is a running total of frequencies. For each class interval, you add the frequency of that class to the total of all previous classes. The result tells you how many data items are less than or equal to the upper boundary of that class.
Question:
The table shows the distances (km) 80 people travel to work.
Comparing averages:
Year 11 has a higher median (62 vs 55), suggesting Year 11 performed better on average.
Comparing spread:
Year 11 IQR = 74 − 48 = 26. Year 10 IQR = 68 − 40 = 28. The IQR is slightly smaller for Year 11, meaning their middle 50% of marks were more consistent.
Comparing range:
Year 10 range = 90 − 22 = 68. Year 11 range = 95 − 30 = 65. The ranges are similar.
§Formulas to memorise
IQR = Q3 − Q1
Minimum — value
Lower quartile (Q1) — 25% of the data lies below this
Median (Q2) — 50% of the data lies below this
Upper quartile (Q3) — 75% of the data lies below this
Maximum — value
Calculate the IQR = Q3 − Q1.
Worked example
The table shows the distances (km) 80 people travel to work. | Distance (km) | 0 ≤ d < 5 | 5 ≤ d < 10 | 10 ≤ d < 15 | 15 ≤ d < 20 | 20 ≤ d < 30 | |---|---|---|---|---|---| | Frequency | 8 | 20 | 28 | 16 | 8 | (a) Draw a cumulative frequency
Working:
⚠ Common mistakes
- ✗Plotting at midpoints instead of upper class boundaries — cumulative frequency points must be plotted at the upper boundary.
- ✗Using n ÷ 2 + 1 instead of n ÷ 2 — for grouped data and cumulative frequency curves, use n ÷ 2 (the +1 is for listed data only).
- ✗Drawing a jagged line instead of a smooth curve — the examiner expects a smooth S-shaped curve through the points.
- ✗Forgetting the (0, 0) point — start the curve at the lower boundary of the first class with a cumulative frequency of zero.
- ✗Confusing IQR with range — the IQR is Q3 − Q1, not max − min.
✦ Exam tips
- →Draw accurately — use the full width of the graph paper and plot points with small crosses, not dots.
- →Read values carefully — when estimating from the curve, draw horizontal and vertical guide lines with a ruler and pencil. The examiner will check your reading to within half a square.
- →Comparison questions — always make two comparative statements: one about an average (median) and one about spread (IQR). Use the context of the question (e.g. "Year 11 performed better on average because…").
- →Box plots can appear alongside histograms — draw them on the same horizontal scale if required. For histogram basics, see histograms.
- →Revisit key formulas on our GCSE Maths formulas page.