Year 11 · Mathematics · Statistics
Box plots and bivariate data
Construct and compare box plots using the five-number summary, and use scatter plots to describe the relationship between two numerical variables.
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These lessons were written around Australian Curriculum strands and matched to UK year groups by age (Year 1 is the year a child turns 6 in England). They are not mapped to the National Curriculum in England or the curricula of Scotland, Wales or Northern Ireland, so check them against your own plans.
Learning goals
Learners will
- Calculate the five-number summary and interquartile range of a data set and draw a box plot.
- Compare two data sets using parallel box plots, referring to centre and spread.
- Use scatter plots to describe the relationship between two numerical variables, including direction, form and strength.
Success looks like
- I can find the minimum, lower quartile, median, upper quartile and maximum of a data set.
- I can compare two box plots using the median and interquartile range.
- I can describe a scatter plot (for example, 'strong, positive, linear') and explain why correlation does not prove causation.
The big idea
A box plot summarises a whole data set with just five numbers – the five-number summary: minimum, lower quartile Q1, median, upper quartile Q3 and maximum.
To find the quartiles, order the data and find the median. The lower quartile is the median of the lower half and the upper quartile is the median of the upper half. If there is an odd number of values, leave the median out of both halves. (Some calculators use slightly different methods, so their quartiles may differ a little.)
For example, with 2, 4, 5, 7, 8, 9, 11, 12, 15: the median is 8. The lower half 2, 4, 5, 7 has median Q1 = 4.5, and the upper half 9, 11, 12, 15 has median Q3 = 11.5. The interquartile range is IQR = 11.5 − 4.5 = 7, and the range is 15 − 2 = 13.
To draw the box plot, use a number line. Draw a box from Q1 to Q3 with a line at the median, then whiskers out to the minimum and maximum. The box shows the middle 50% of the data, and each whisker and each half of the box holds about 25%.
A common rule says a value is an outlier if it lies below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. Outliers are marked with a separate dot or cross, and the whisker stops at the most extreme value that is not an outlier.
To compare box plots drawn on the same scale, comment on centre (which median is higher?) and spread (which IQR or range is larger, meaning the data is more variable?), and use the context: 'Class B's median score was 5 marks higher, and its smaller IQR shows its results were more consistent.'
Bivariate data records two numerical variables for each individual. Plot it on a scatter plot, with the explanatory (independent) variable on the horizontal axis and the response (dependent) variable on the vertical axis.
Describe a scatter plot by its direction (positive: both increase together; negative: one increases as the other decreases), form (linear or non-linear) and strength (strong, moderate or weak, depending on how closely the points follow a pattern). Mention any outliers. A line of best fit drawn by eye can be used to make predictions, which are most reliable inside the range of the data.
Correlation does not prove causation. Ice-cream sales and sunburn cases are positively correlated, but ice-cream does not cause sunburn – both increase in hot, sunny weather.
Worked examples
Five-number summary and box plot
Find the five-number summary and IQR for these test scores: 12, 15, 9, 20, 14, 17, 11, 18, 16, 13.
- Order the data: 9, 11, 12, 13, 14, 15, 16, 17, 18, 20 (10 values).
- Median = average of the 5th and 6th values = (14 + 15) ÷ 2 = 14.5.
- Lower half 9, 11, 12, 13, 14: Q1 = 12. Upper half 15, 16, 17, 18, 20: Q3 = 17.
- Minimum = 9, maximum = 20. IQR = 17 − 12 = 5.
- Draw a box from 12 to 17 with a line at 14.5, and whiskers to 9 and 20.
Answer: Min 9, Q1 12, median 14.5, Q3 17, max 20; IQR = 5
Comparing two box plots
Two classes sat the same test. Class A: min 40, Q1 55, median 65, Q3 75, max 90. Class B: min 30, Q1 60, median 70, Q3 78, max 95. Compare the results.
- Centre: Class B's median (70) is 5 marks higher than Class A's (65).
- Spread of the middle 50%: Class A IQR = 75 − 55 = 20; Class B IQR = 78 − 60 = 18, so Class B is slightly more consistent.
- Range: Class A = 50; Class B = 65, so Class B has more extreme scores at both ends.
Answer: Class B generally performed better (higher median) and its middle 50% was slightly more consistent (smaller IQR), although it had a larger range because of more extreme scores.
Describing a scatter plot
Hours of study and test scores for six students: (1, 52), (2, 58), (3, 61), (4, 70), (5, 74), (6, 80). Describe the relationship and predict the score for 3.5 hours of study.
- Hours studied is the explanatory variable (horizontal axis); test score is the response variable (vertical axis).
- As hours increase, scores increase, and the points lie close to a straight line: a strong, positive, linear relationship.
- Reading from a line of best fit, 3.5 hours sits between 61 and 70, so a prediction of about 65 is reasonable.
Answer: A strong, positive, linear relationship; a student studying 3.5 hours would be expected to score about 65.
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Find the median of 3, 5, 6, 9, 12, 14, 20.
9 — There are 7 values in order, so the median is the 4th value: 9.
Find the interquartile range of 3, 5, 6, 9, 12, 14, 20.
9 — Leaving out the median (9): Q1 is the median of 3, 5, 6, which is 5, and Q3 is the median of 12, 14, 20, which is 14. IQR = 14 − 5 = 9.
In a box plot, what does the box itself (from Q1 to Q3) represent?
The middle 50% of the data — The box stretches from the lower quartile to the upper quartile, covering the middle half of the data.
Which of these is NOT part of the five-number summary?
Mean — The five-number summary is minimum, Q1, median, Q3 and maximum. The mean is not included.
A data set has Q1 = 20 and Q3 = 32. Using the 1.5 × IQR rule, values above what number are outliers?
50 — IQR = 32 − 20 = 12. Upper limit = Q3 + 1.5 × 12 = 32 + 18 = 50.
As the daily temperature rises, a shop sells more cold drinks. What type of correlation is this?
Positive — Both variables increase together, which is a positive correlation.
A scatter plot of car age (horizontal axis) against value (vertical axis) shows points falling from left to right, lying very close to a straight line. How is the relationship best described?
Strong negative linear — Value decreases as age increases (negative), and the points are close to a line (strong, linear).
There is a strong positive correlation between ice-cream sales and the number of sunburn cases. Which conclusion is best?
Both are likely linked to a third factor, such as hot, sunny weather; correlation alone does not prove causation. — A correlation shows the variables move together, not that one causes the other.
Watch out for
- Thinking a longer whisker or a wider section of the box contains more data values. Each quarter holds about 25% of the data; a longer section just means the values are more spread out.
- Comparing box plots using only the maximum or minimum. Use the median for centre and the IQR for spread, which are not affected by extreme values.
- Assuming a strong correlation means one variable causes the other. A third variable or coincidence may explain the pattern.
Tips for parents
- Collect real data together – family members' reaction times, or daily temperatures for two weeks – and make a box plot from it.
- When you see a statistic in the news claiming one thing 'causes' another, ask your learner what else might explain the pattern.
- Check quartile calculations by counting: about a quarter of the data values should sit in each section of the box plot.
Go further
- Measure the height and arm span of at least eight family members or friends. Draw a scatter plot, describe the relationship and draw a line of best fit by eye.
- Record the time (in seconds) it takes to complete a puzzle with your writing hand and your other hand, ten times each. Compare the results with parallel box plots.
- Investigate how a spreadsheet calculates quartiles and compare its answers with the method in this lesson.
About 60 minutes
You'll need
- Paper, pencil and ruler
- Grid paper
- A calculator
- A tape measure (for the enrichment activity)
Key words
- Five-number summary
- The minimum, lower quartile (Q1), median, upper quartile (Q3) and maximum of a data set.
- Interquartile range (IQR)
- Q3 − Q1; the spread of the middle 50% of the data.
- Box plot
- A graph that displays the five-number summary using a box (from Q1 to Q3) and whiskers (to the minimum and maximum).
- Outlier
- A value far from the rest of the data; commonly, a value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
- Bivariate data
- Data that records two variables for each individual, such as height and arm span.
- Correlation
- A relationship between two variables, described by its direction (positive or negative) and strength.
Still tricky?
For parents
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