GCSE Maths Cumulative Frequency & Box Plots Notes
Cumulative frequency adds running totals to reveal the median and quartiles. Box plots then summarise the spread of a data set at a glance.
Key concepts
- Cumulative frequency
- A running total of frequencies. Plot it against the upper class boundary and join with a smooth curve.
- Median (Q2)
- Read across at n/2 on the cumulative frequency axis, then down to the value.
- Quartiles (Q1, Q3)
- Read across at n/4 for the lower quartile and 3n/4 for the upper quartile.
- Box plot
- Shows minimum, Q1, median, Q3 and maximum — a five-figure summary of the data.
Key formulas
Interquartile range
IQR = Q3 − Q1
A measure of spread that ignores extreme values.
Worked example
60 values are plotted. Read the median and IQR when Q1 = 22, Q2 = 30, Q3 = 41.
- 1The median is Q2, read at n/2 = 30 on the cumulative axis: 30.
- 2Q1 = 22 (at 15) and Q3 = 41 (at 45).
- 3IQR = Q3 − Q1 = 41 − 22.
Answer: Median = 30, IQR = 19
Common mistakes
✗ Plotting cumulative frequency at the midpoint of a class.
✓ Plot at the upper class boundary, not the midpoint.
✗ Reading quartiles at the wrong fractions.
✓ Use n/4, n/2 and 3n/4 of the total frequency on the vertical axis.
✗ Confusing range and interquartile range.
✓ Range = max − min; IQR = Q3 − Q1 (the middle 50%).
Quick check
Try these, then reveal the answers.
If Q1 = 12 and Q3 = 28, find the IQR.Reveal answer
16
For 80 values, at what cumulative frequency do you read the median?Reveal answer
40
What five values make a box plot?Reveal answer
Min, Q1, median, Q3, max
Exam tips
- Join cumulative frequency points with a smooth curve, not straight lines, for full marks.
- Draw feint lines across and down when reading quartiles so the examiner sees your method.
- Compare two box plots by contrasting both an average (median) and a spread (IQR).
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Frequently asked questions
Why use the interquartile range instead of the range?+
The IQR ignores extreme outliers, so it describes the spread of the central data more reliably than the range.
What does a longer box on a box plot show?+
A larger interquartile range, meaning the middle half of the data is more spread out.
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