By /
GCSE Maths · Edexcel 1MA1 · Higher · Grades 5–7 · Statistics

Box Plots and Comparing Distributions

A box plot squeezes a whole data set into five numbers. The comparison question is a reliable 2–4 marks if you remember the formula: compare an average and a spread, both in context.

1Quartiles and the interquartile range

Put the data in order. For a small data set of n values:

ValuePosition in the ordered list
Lower quartile (LQ)(n + 1) ÷ 4
Median(n + 1) ÷ 2
Upper quartile (UQ)3(n + 1) ÷ 4
Interquartile rangeIQR = UQ − LQ. It measures the spread of the middle 50% of the data and isn’t affected by extreme values.
Worked example
Find the quartiles and IQR of: 12, 15, 18, 21, 24, 25, 27, 30, 33, 36, 41.
  1. n = 11, so the positions are 3, 6 and 9.
  2. LQ = 18, median = 25, UQ = 33.
  3. IQR = 33 − 18 = 15.
LQ 18, median 25, UQ 33, IQR 15

2Drawing a box plot

A box plot uses the five-number summary: minimum, LQ, median, UQ, maximum.

  1. Draw a scale.
  2. Draw a box from the LQ to the UQ.
  3. Draw a line across the box at the median.
  4. Draw whiskers out to the minimum and maximum.
1015202530354045MarkMarks
Box plot of the marks: 12, 18, 25, 33, 41

3Reading a box plot

Each section of a box plot holds about 25% of the data.

Worked example
Using the box plot above, what percentage of students scored more than 33?
  1. 33 is the upper quartile.
  2. 25% of the data lies above the UQ.
25%

4Comparing two box plots

5101520253035Time to solve puzzle (minutes)BoysGirls
Worked example
Compare the times taken by boys and girls to solve the puzzle.
  1. Average: the boys’ median (19 min) is lower than the girls’ (22 min), so on average the boys solved the puzzle faster.
  2. Spread: the boys’ IQR (11 min) is larger than the girls’ (9 min), so the girls’ times were more consistent.
Two comparisons, each with values and context
Exam tipWrite one sentence comparing the medians and one comparing the IQRs, and say what each means in the context of the question. Comparing the ranges instead of the IQRs usually loses a mark.

5Box plots from cumulative frequency graphs

For large grouped data sets, read the quartiles from a cumulative frequency graph. With n values, read across at n/4, n/2 and 3n/4. Take the minimum and maximum from the table (the ends of the first and last classes), then draw the box plot as usual.

See Cumulative Frequency Diagrams and Histograms for how to draw the graph.

Common mistakes
  • Finding quartiles from unordered data. Always sort first.
  • Comparing ranges instead of interquartile ranges.
  • Comparing the numbers without saying what they mean in context.
  • Saying a bigger box means more data. Each quarter of a box plot holds the same amount of data.

★Key Vocabulary Recap

Lower quartile
The value 25% of the way through the ordered data.
Upper quartile
The value 75% of the way through.
Interquartile range
UQ − LQ: the spread of the middle half of the data.
Five-number summary
Minimum, LQ, median, UQ, maximum.
Consistent
Having a small spread.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: S4 (Higher: box plots, quartiles, IQR))
  • S4: interpret, analyse and compare the distributions of data sets through appropriate graphical representation involving discrete, continuous and grouped data, including box plots.
  • S4: use appropriate measures of central tendency (median) and spread (range, including consideration of outliers, quartiles and inter-quartile range).
Next stepReady to practise? Try the 8 exam-style questions, then check your working against the worked solutions.
← Back to the GCSE Maths (1MA1) resource listThe Vector Academy · thevectoracademy.in

Practice questions

GCSE Maths · Edexcel 1MA1 · Higher · Grades 5–7 · Statistics

Box Plots and Comparing Distributions — Practice Questions

8 exam-style questions (17 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Higher · Grade 5
  1. (2 marks)
    Grade 5
    Find the median and the interquartile range of: 3, 7, 8, 12, 14, 15, 20.
  2. (3 marks)
    Grade 5
    The masses (kg) of 11 dogs are: 41, 35, 52, 47, 38, 60, 44, 50, 39, 55, 42. Find the five-number summary.
  3. (2 marks)
    Grade 5
    Draw a box plot for the dogs’ masses in Question 2.
Higher · Grade 6
  1. (1 mark)
    Grade 6
    120 students took a test. The upper quartile was 68 marks. About how many students scored more than 68?
  2. (2 marks)
    Grade 6
    Using the box plots for boys and girls in the notes, find the median and IQR for each group.
  3. (2 marks)
    Grade 6
    Compare the distributions of the boys’ and girls’ puzzle times.
Higher · Grade 7
  1. (2 marks)
    Grade 7
    Ayesha says: “The girls’ box is narrower, so fewer girls took the test.” Explain why she is wrong.
  2. (3 marks)
    Grade 7
    The cumulative frequency graph for the times of 80 runners gives these readings: CF = 20 at 31 minutes, CF = 40 at 36 minutes, CF = 60 at 42 minutes. The fastest time was 22 minutes and the slowest 58 minutes. Draw a box plot and find the IQR.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
← Back to the GCSE Maths (1MA1) resource listThe Vector Academy · thevectoracademy.in

Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Higher · Grades 5–7 · Statistics

Box Plots and Comparing Distributions — Worked Solutions

Full worked solutions to all 8 questions (17 marks), set out the way an Edexcel examiner wants to see them.

Higher · Grade 5
  1. (2 marks)
    Grade 5
    Find the median and the interquartile range of: 3, 7, 8, 12, 14, 15, 20.
    Solution
    1. n = 7: positions 2, 4, 6.
    2. LQ = 7, median = 12, UQ = 15.
    3. IQR = 15 − 7 = 8.
    Median 12, IQR 8
  2. (3 marks)
    Grade 5
    The masses (kg) of 11 dogs are: 41, 35, 52, 47, 38, 60, 44, 50, 39, 55, 42. Find the five-number summary.
    Solution
    1. Order: 35, 38, 39, 41, 42, 44, 47, 50, 52, 55, 60.
    2. Positions 3, 6, 9 for LQ, median, UQ.
    35, 39, 44, 52, 60
  3. (2 marks)
    Grade 5
    Draw a box plot for the dogs’ masses in Question 2.
    Solution
    3035404550556065Mass (kg)Mass
    Box 39 to 52, median 44
Higher · Grade 6
  1. (1 mark)
    Grade 6
    120 students took a test. The upper quartile was 68 marks. About how many students scored more than 68?
    Solution
    1. 25% of the data lies above the UQ: 0.25 × 120.
    30 students
  2. (2 marks)
    Grade 6
    Using the box plots for boys and girls in the notes, find the median and IQR for each group.
    Solution
    1. Boys: median 19, IQR 25 − 14 = 11.
    2. Girls: median 22, IQR 26 − 17 = 9.
    Boys 19, 11; girls 22, 9
  3. (2 marks)
    Grade 6
    Compare the distributions of the boys’ and girls’ puzzle times.
    Solution
    1. Median: boys 19 < girls 22, so the boys were faster on average.
    2. IQR: boys 11 > girls 9, so the girls were more consistent.
    Boys faster on average; girls more consistent
Higher · Grade 7
  1. (2 marks)
    Grade 7
    Ayesha says: “The girls’ box is narrower, so fewer girls took the test.” Explain why she is wrong.
    Solution
    1. A box plot shows how spread out the data is, not how many values there are.
    2. Each part of each box plot holds 25% of that group, whatever the group’s size.
    Width shows spread, not number of people
  2. (3 marks)
    Grade 7
    The cumulative frequency graph for the times of 80 runners gives these readings: CF = 20 at 31 minutes, CF = 40 at 36 minutes, CF = 60 at 42 minutes. The fastest time was 22 minutes and the slowest 58 minutes. Draw a box plot and find the IQR.
    Solution
    1. n = 80: LQ at 20 → 31 min; median at 40 → 36 min; UQ at 60 → 42 min.
    2. IQR = 42 − 31 = 11 minutes.
    202530354045505560Time (minutes)Runners
    IQR = 11 minutes
← Back to the GCSE Maths (1MA1) resource listThe Vector Academy · thevectoracademy.in

Book a free trial class

Tell us a little about your requirement.