Sheet № 192 · Higher only · AQA · Edexcel · OCR
Box Plots and Comparing Distributions –
Box plots (also called box-and-whisker diagrams) are a Higher-tier topic that features prominently on AQA, Edexcel and OCR GCSE Maths papers. They provide a visual summary of data using five key values and are especially useful for comparing two distributions. You need to draw box plots, read values from them, calculate the interquartile
§Key definitions
Question:
Draw a box plot for this data set: 4, 7, 9, 12, 15, 18, 21, 24, 28.
Answer:
Five-number summary: 4, 8, 15, 22.5, 28.
(a)
IQR = Q3 − Q1 = 42 − 20 = 22.
(b)
Range = 50 − 12 = 38.
Q1 (Foundation):
A box plot has Q1 = 15 and Q3 = 35. What is the IQR?
§Formulas to memorise
Interquartile Range (IQR) = Q3 − Q1
Range = Maximum − Minimum
Minimum — the smallest value.
Lower quartile (Q1) — the value one quarter of the way through the ordered data.
Median (Q2) — the middle value.
Upper quartile (Q3) — the value three quarters of the way through the ordered data.
Maximum — the largest value.
Positive skew — the median is closer to Q1, and the right whisker is longer.
Negative skew — the median is closer to Q3, and the left whisker is longer.
Symmetrical — the median is roughly central in the box.
Worked example
Draw a box plot for this data set: 4, 7, 9, 12, 15, 18, 21, 24, 28.
Working:
⚠ Common mistakes
- ✗Confusing Q1 and Q3 with the minimum and maximum. The box shows Q1 to Q3; the whiskers extend to the min and max.
- ✗Not ordering data before finding quartiles. Quartiles only work on ordered data.
- ✗Using range instead of IQR for comparison. The IQR is usually more useful because it ignores outliers and focuses on the central 50%.
- ✗Only comparing one measure. When comparing distributions, always comment on both an average (median) and a measure of spread (IQR or range).
✦ Exam tips
- →When comparing box plots, always make two comparison points: one about the median (average) and one about the IQR or range (spread). Use context — for example, "Class B scored higher on average and their marks were more consistent."
- →Read box plot values carefully from the scale — misreading by one division is a common error.
- →If asked about skewness, look at where the median sits within the box and compare whisker lengths.
- →For related cumulative frequency, see cumulative frequency and box plots. For key formulas, visit our GCSE Maths formulas page.