Box Plots and Quartiles
You already know how averages tell you what’s “typical” in a dataset. But what if you also need to know how spread out or consistent the data is? That’s where box plots and quartiles come in — giving you a powerful visual summary of any dataset’s distribution at a glance.
Quartiles and the Five-Number Summary
Box plots display five key values from your data — together called the five-number summary.
The Five Values
- The minimum (lowest value)
- The lower quartile ($Q_1$) — 25% of data lies below this
- The median ($Q_2$) — the middle value
- The upper quartile ($Q_3$) — 75% of data lies below this
- The maximum (highest value)
Inter-Quartile Range (IQR)
The IQR is the range of the middle 50% of the data.
- Smaller IQR → data is more clustered and consistent.
- Larger IQR → data is more spread out.
- The IQR ignores extreme values, making it a more reliable measure of spread than the full range.
Worked Examples
Example 1: Finding the Five-Number Summary
Test scores for 11 students: 32, 45, 28, 38, 41, 35, 40, 29, 36, 42, 30
- Order the data:
28, 29, 30, 32, 35, 36, 38, 40, 41, 42, 45 - Min = 28, Max = 45.
- Median ($Q_2$): $(11+1)/2 = 6^{\text{th}}$ value → 36.
- $Q_1$: Lower half = 28, 29, 30, 32, 35 → $Q_1 = 30$.
- $Q_3$: Upper half = 38, 40, 41, 42, 45 → $Q_3 = 41$.
- IQR $= 41 – 30 = \mathbf{11}$.
Five-number summary: Min 28, $Q_1$ 30, Median 36, $Q_3$ 41, Max 45.
Example 2: Drawing a Box Plot
Using the five-number summary from Example 1:
Steps: Draw a number line → mark the five values → draw the box from $Q_1$ to $Q_3$ → add the median line → draw whiskers to Min and Max.
Example 3: Comparing Two Distributions
Two classes took the same test. Compare their performance using the box plots below.
Compare Averages (Median)
Class A median = 60. Class B median = 50. Class A performed better on average.
Compare Spread (IQR)
Class A IQR $= 75-40 = \mathbf{35}$. Class B IQR $= 70-30 = \mathbf{40}$. Class A was more consistent (smaller IQR).
Overall conclusion: Class A performed better on average and were more consistent with their scores.
Tutor Insights
🤔 Common Misunderstandings
- Not ordering the data first. This is the most common error. You must sort the values before finding any quartile.
- Calculating quartiles incorrectly. When the median is a single data point, exclude it from both halves when finding $Q_1$ and $Q_3$.
- Confusing Range and IQR. Range = Max − Min. IQR = $Q_3 – Q_1$. The IQR is often a better measure of spread because it ignores extreme outliers.
📝 Common Exam Mistakes
- Inaccurate drawing — a missing or incorrect scale on the number line will cost marks.
- Only comparing one measure. You must compare both average (median) and spread (IQR).
- Not stating conclusions in context — always mention what the numbers represent (e.g., “scores”, “minutes”), not just the raw values.
Practice Questions
- Five-number summary: Find the five-number summary and IQR for: 12, 15, 8, 20, 10, 18, 14, 11, 19.
- Draw a box plot: Heights (cm) of 10 plants: 25, 30, 22, 28, 35, 27, 31, 29, 26, 32.
(a) Find the five-number summary. (b) Draw the box plot. - Interpret a box plot: The box plot below shows daily maximum temperatures (°C) in June.
(a) What was the median temperature?
(b) What was the IQR?
(c) What percentage of days had a temperature below 18°C?
(d) What percentage of days had a temperature between 15°C and 22°C? - Compare two box plots: Compare journey times (mins) for students from Village A and Village B.
Write a full comparison statement using both average and spread. - Deduction: A dataset has Min = 5, $Q_1 = 12$, IQR = 10, Max = 30.
(a) Find $Q_3$. (b) If the median is 15, draw the box plot.
Show Answers
- Ordered: 8, 10, 11, 12, 14, 15, 18, 19, 20.
Min=8, $Q_1=\frac{10+11}{2}=10.5$, Median=14, $Q_3=\frac{18+19}{2}=18.5$, Max=20.
IQR $= 18.5 – 10.5 = 8$. - (a) Ordered: 22, 25, 26, 27, 28, 29, 30, 31, 32, 35.
Min=22, $Q_1=26$, Median=$\frac{28+29}{2}=28.5$, $Q_3=31$, Max=35.
(b) Draw a number line from 20–36 and place the box from 26 to 31, median line at 28.5, whiskers to 22 and 35. - (a) Median = 18°C (the line inside the box).
(b) IQR $= 22 – 15 = 7°C$.
(c) 50% (the median is the 50th percentile).
(d) 50% (between $Q_1$ and $Q_3$ is the middle 50% of the data). - Village B has a higher median (≈50 min) than Village A (≈31 min), so on average Village B students have longer journeys. Both villages have a similar IQR (≈20 min), so the consistency of journey times is comparable. However Village B has a much longer upper whisker, suggesting some students have exceptionally long journeys.
- (a) IQR $= Q_3 – Q_1 \Rightarrow 10 = Q_3 – 12 \Rightarrow \mathbf{Q_3 = 22}$.
(b) Five-number summary: Min=5, $Q_1$=12, Median=15, $Q_3$=22, Max=30. Draw a number line from 0–35 and construct the box plot accordingly.
FAQs
Q: Why use IQR instead of the full range?
A: The full range is affected by outliers (extreme values). The IQR measures only the middle 50% of the data, giving a more reliable indication of typical spread.
Q: Do I include the median when finding $Q_1$ and $Q_3$?
A: If the median is a single data point (odd-numbered dataset), exclude it from both halves. If the median is the average of two middle values (even-numbered dataset), naturally split the data into two equal halves to find $Q_1$ and $Q_3$.
Q: What makes a good comparison statement?
A: Always compare both a measure of average (the median) and a measure of spread (the IQR), and state your conclusions in the context of the question. E.g., “Class A has a higher median score (60 vs 50), suggesting better average performance. Class A also has a smaller IQR (35 vs 40), showing more consistency.”
Q: When is a box plot most useful?
A: Box plots are ideal for comparing two or more distributions side by side, and for showing skewness (whether data leans left or right). They’re less useful for small datasets where individual values matter.
Need Help with Statistics?
A qualified GCSE maths tutor can guide you through box plots, quartiles, averages, and every other statistics topic on the syllabus.
Find a GCSE Maths Tutor
