GCSE Maths • Foundation & Higher tier
GCSE Box Plots Revision Guide
Box plots summarise a distribution using the minimum, lower quartile, median, upper quartile and maximum. This guide explains the five-number summary, range, interquartile range, accurate construction and evidence-based comparisons. You will learn what each part of a box plot shows, what it does not show and how to write clear GCSE exam answers.
Foundation
Exam practice
Worked examples
What You'll Learn
Identify the Five-Number Summary
Read the minimum, lower quartile, median, upper quartile and maximum from a scale or given information.
Calculate Range and IQR
Use the endpoints for the range and the quartiles for the interquartile range.
Draw a Box Plot
Choose a suitable scale, place all five values accurately and construct the box, median line and whiskers.
Compare Typical Values
Use the medians to compare the centres of two distributions in the context of the question.
Compare Spread
Use IQR for the middle 50% and range for the full displayed spread, supported by calculations.
Connect to Cumulative Frequency
Estimate quartiles from a cumulative frequency graph and use them in a five-number summary.
Key Rules And Formula
Five-Number Summary
Rule or formula: minimum, Q1, median (Q2), Q3, maximum Explanation: The values must be in non-decreasing order and placed on one numerical scale. Condition or reminder: Reminder: Q1 is the lower quartile and Q3 is the upper quartile.
Interquartile Range
Rule or formula: IQR = Q3 − Q1 Explanation: The IQR measures the spread of the middle 50% of the data. Condition or reminder: Interpret carefully: a smaller IQR means the central half is less spread out; it does not describe every value.
Range
Rule or formula: range = maximum − minimum Explanation: The range measures the full displayed spread from the lowest to the highest value. Condition or reminder: The range is sensitive to extreme values.
Box Meaning
Rule or formula: Q1 to Q3 contains the middle 50% Explanation: The median line divides the ordered data into lower and upper halves. Condition or reminder: Do not say the box contains 50% of the numerical scale; it represents 50% of the observations.
Compare Centre
Rule or formula: compare the medians Explanation: A higher median indicates a higher typical value for the variable being measured. Condition or reminder: Use words such as ‘better’ only when the context makes a higher or lower value desirable.
Compare Spread
Rule or formula: compare the medians Explanation: A higher median indicates a higher typical value for the variable being measured. Rule or formula: compare IQRs and, when relevant, ranges Explanation: IQR compares the central half; range compares the full displayed spread. Condition or reminder: Give both values or calculations as evidence, not only a vague visual judgement.
Lesson Modules
GCSE Maths: Comparing Box Plots
Compare medians, interquartile ranges and ranges, then write contextual conclusions supported by numerical evidence.
GCSE Maths: Drawing Box Plots
Use a five-number summary and an accurate numerical scale to draw the box, median line and whiskers.
Core Subtopics
Minimum and Maximum
Locate the smallest and largest displayed values at the ends of the whiskers.
Quartiles and Median
Recognise Q1, the median and Q3 as positions that divide ordered data into quarters.
Reading Scales
Work out the value of each interval before reading or plotting the five summary values.
Drawing the Box and Whiskers
Draw the box from Q1 to Q3, a median line inside it and whiskers to the displayed endpoints.
Comparing Distributions
Compare centre and spread separately, calculate the differences and interpret them in context.
Limitations of Box Plots
Recognise that a standard box plot does not show every observation, the mean, the mode or the detailed shape within each quarter.
Worked Examples
Example 1: Drawing and Comparing Two Box Plots
Two classes complete the same test. Class A has minimum 20, Q1 42, median 58, Q3 70 and maximum 88. Class B has minimum 25, Q1 38, median 64, Q3 82 and maximum 95. Calculate the range and IQR of each class, then compare the distributions.
Given information: Both five-number summaries are supplied, so no quartile convention is needed. Use one common scale when drawing the two box plots.
1. For Class A, calculate the range: 88 − 20 = 68.
2. For Class A, calculate the IQR: 70 − 42 = 28.
3. For Class B, calculate the range: 95 − 25 = 70.
4. For Class B, calculate the IQR: 82 − 38 = 44.
5. Compare the medians: Class B has the higher median, 64 compared with 58 for Class A.
6. Compare the spread: Class A has the smaller IQR, 28 compared with 44, so its middle 50% is less spread out. The ranges are very similar: 68 for A and 70 for B.
Final comparison: Class B has the higher typical test score because its median is 64, compared with 58 for Class A. Class A’s central results are more consistent because its IQR is 28, compared with 44 for Class B. Their overall displayed spreads are similar because the ranges differ by only 2 marks.
Independent answer check: Class A: 20 ≤ 42 ≤ 58 ≤ 70 ≤ 88. Class B: 25 ≤ 38 ≤ 64 ≤ 82 ≤ 95. Both summaries are ordered correctly, and every range and IQR is non-negative.
Visual required: Place the two horizontal box plots on the same labelled scale from 0 to 100. Use distinct accessible colours and include the five values in the caption or accompanying text.
1. For Class A, calculate the range: 88 − 20 = 68.
2. For Class A, calculate the IQR: 70 − 42 = 28.
3. For Class B, calculate the range: 95 − 25 = 70.
4. For Class B, calculate the IQR: 82 − 38 = 44.
5. Compare the medians: Class B has the higher median, 64 compared with 58 for Class A.
6. Compare the spread: Class A has the smaller IQR, 28 compared with 44, so its middle 50% is less spread out. The ranges are very similar: 68 for A and 70 for B.
Final comparison: Class B has the higher typical test score because its median is 64, compared with 58 for Class A. Class A’s central results are more consistent because its IQR is 28, compared with 44 for Class B. Their overall displayed spreads are similar because the ranges differ by only 2 marks.
Independent answer check: Class A: 20 ≤ 42 ≤ 58 ≤ 70 ≤ 88. Class B: 25 ≤ 38 ≤ 64 ≤ 82 ≤ 95. Both summaries are ordered correctly, and every range and IQR is non-negative.
Visual required: Place the two horizontal box plots on the same labelled scale from 0 to 100. Use distinct accessible colours and include the five values in the caption or accompanying text.
- Common GCSE Mistakes
- Confusing Range and IQR
- Reading the Scale as Units of One
- Drawing the Box to the Endpoints
- Using the Mean
- Giving a Vague Comparison
- Overinterpreting the Diagram
- Assuming Every Whisker Ends at the Raw Minimum or Maximum
- Exam Tips
- Write the five summary values in order before drawing; this catches misplaced quartiles immediately.
- Calculate IQR and range beside the diagram before writing a comparison.
- When comparing two box plots, use a centre sentence and a spread sentence, each supported by numbers.
- Keep contextual meaning accurate: a higher median is not automatically ‘better’, and a smaller IQR describes only the middle 50%.
- Use a ruler and a single, evenly spaced scale. Align box plots vertically when the purpose is comparison.
- Check whether the question supplies a five-number summary, raw data or a cumulative frequency graph; the required route is different in each case.
Ready to Practice?
Printable Worksheet
Practise reading scales, calculating range and IQR, drawing box plots and comparing distributions.
Interactive Quiz
Check the five-number summary, box-plot vocabulary and short interpretation questions.
Exam-Style Questions
Answer mixed questions using supplied summaries, diagrams and contextual comparisons.
Watch the Box plot Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Frequently Asked Questions
A box plot shows a five-number summary: minimum, lower quartile, median, upper quartile and maximum, or the endpoints defined by the question’s plotting convention. It summarises the centre and spread of a distribution without showing every observation.
The box runs from Q1 to Q3 and represents the middle 50% of the ordered observations. The line inside the box marks the median.
Subtract the lower quartile from the upper quartile: IQR = Q3 − Q1. It measures the spread of the middle 50% of the data.
Compare the medians to discuss typical values, then compare the IQRs to discuss the spread of the middle 50%. Compare ranges when the full displayed spread is relevant. Support every conclusion with values and interpret it in context.
A smaller IQR means the middle 50% of the observations is less spread out. In many GCSE contexts this is described as more consistent, but it does not prove that every observation is close together.
Core box-plot interpretation, quartiles and interquartile range are relevant to both tiers; AQA places them in basic Foundation statistics content. AQA places cumulative frequency graphs in Higher-only content, so a question that obtains quartiles from such a graph is Higher Tier on AQA. Check your own exam board specification.
Not always. Different conventions can include or exclude the overall median when splitting a list, and software may use different percentile definitions. In an exam, follow the method taught for your board and any instructions given. If the five-number summary is supplied, use it directly.
A standard GCSE box plot does not show the mean, mode, individual observations or sample size. Those details require the original data or additional information.
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