Parallel Boxplots
Parallel box plots draw two or more box plots on the same scale so that data sets can be compared at a glance. In NSW Year 10 Advanced maths you compare their medians, their spreads (range and IQR) and their shapes, and use the quartiles to estimate how many scores lie above or below a value (NESA outcome MA5‑DAT‑C‑01).
You will compare sprint times of boys and girls, push-ups at two schools, waiting times at two restaurants, ages at a theme park and a museum, and hours worked by Year 10 and Year 12 students. You will draw parallel box plots from dot plots, tables and stem-and-leaf plots, decide which statement about two data sets is not correct, and check a clinic against a target for waiting times. Harder questions find unknown scores so that two sets match. It is part of Year 10 Advanced Single Variable and Bivariant Statistics.
Every question with a fully worked solution.
- Parallel Boxplots - Video - Parallel Box Plots Watch
Theory
Parallel box plots draw two or more box plots on the same scale so that groups can be compared at a glance: centre (median), spread (IQR and range) and shape. This Year 10 Advanced maths page covers drawing them from data, reading percentages and counts for groups of different sizes, testing statements, and harder problems using quartile positions and unknowns.
Parallel box plots are box plots for different groups drawn one above the other over one number line. Because the scale is shared, values can be compared directly.
To compare groups, describe: the centre (which median is higher, and by how much), the spread (IQR and range: a smaller IQR means more consistent data), and the shape (symmetrical or skewed), plus any outliers.
A single value can be a different statistic for each group: \(30\) might be the median of one group and the lower quartile of another. About \(50\%\) of a group lies above its median and \(75\%\) above its \(Q_1\).
Box plots hide the group sizes. Percentages come from the plot; counts need the number in each group.
| Compare | Use | Say |
|---|---|---|
| centre | medians | "B's median is \(8\) higher, so B scored higher in general" |
| spread | IQRs (and ranges) | "A's smaller IQR means its middle half is more consistent" |
| shape | position of median, whisker lengths | "A is positively skewed; B is symmetrical" |
| overlap | one group's quartile against the other's | "75% of B scored above A's median" |
Drawing parallel box plots
- Find the five-number summary of each group (from lists, dot plots, stem-and-leaf plots or tables).
- Draw one scale that covers every group, and label each box plot.
- Draw each box plot above the scale, with outliers marked if needed.
Comparing and answering questions
- Centre, spread, shape: compare medians, IQRs and ranges, and skewness. Quote the numbers.
- Counts: for each group find what the value is (min, \(Q_1\), median, \(Q_3\), max), convert to a fraction of that group, then multiply by its size. Add the groups.
- Statements: test each one against the five values. "Every" needs the minimum or maximum; "at least half" needs a median; "at least \(75\%\)" needs a quartile.
- Targets: check every condition (for example median at most \(20\) and IQR at most \(15\)) for each group.
- Unknowns: write the IQR or median of each group, set the given condition as an equation and solve; check the five values are in order.
Route A: \(2,\ 3,\ 4,\ 4,\ 5,\ 6,\ 7,\ 8,\ 12\)
Route B: \(0,\ 1,\ 2,\ 2,\ 3,\ 6,\ 9,\ 11,\ 14\)
(i) Find each five-number summary and IQR.
(ii) Draw parallel box plots.
(iii) Which route is more reliable?
(i) Each set has \(9\) scores, so the median is the \(5\)th and each half has \(4\).
| \(\text{A}:\ \) | \(\) | \(2,\ 3.5,\ 5,\ 7.5,\ 12 \qquad \text{IQR} = 7.5 - 3.5 = \mathbf{4}\) |
| \(\text{B}:\ \) | \(\) | \(0,\ 1.5,\ 3,\ 10,\ 14 \qquad \text{IQR} = 10 - 1.5 = \mathbf{8.5}\) |
(ii) Use one scale for both.
(iii) Route B has the lower median (\(3\) min against \(5\) min), so its typical delay is shorter. But its IQR is more than twice as large and its upper quarter reaches \(14\) min. Route A is more reliable (more consistent): its middle half of delays lies within \(4\) minutes.
(i) \(40\) is the Year \(9\) median but the Year \(10\) lower quartile.
| \(\text{Year 9}:\ \dfrac{1}{2} \times 60\) | \(=\) | \(30\) |
| \(\text{Year 10}:\ \dfrac{3}{4} \times 100\) | \(=\) | \(75\) |
| \(\text{total}\) | \(=\) | \(\mathbf{105}\) |
| \(\dfrac{105}{160} \times 100\%\) | \(=\) | \(65.625\% \approx \mathbf{65.6\%}\) |
(ii) True. The Year \(9\) median (\(40\)) equals the Year \(10\) \(Q_1\), and about \(75\%\) of a group is at or above its \(Q_1\).
(iii) False. The Year \(10\) minimum is \(20\) minutes, below \(40\).
Count each group with its own size; do not average the percentages.
(i) Read P: \(\text{IQR} = 34 - 20 = 14\). Set Q's IQR equal to it.
| \((3m - 4) - (2m - 6)\) | \(=\) | \(14\) |
| \(m + 2\) | \(=\) | \(14\) |
| \(m\) | \(=\) | \(\mathbf{12}\) |
| \(Q_1\) | \(=\) | \(2(12) - 6 = \mathbf{18}\) |
| \(Q_3\) | \(=\) | \(3(12) - 4 = \mathbf{32}\) |
Check the order: \(8 < 18 < 30 < 32 < 50\ \checkmark\)
(ii) Q has the higher median (\(30\) against \(26\)). The IQRs are equal (\(14\)), so the middle halves are equally spread, but Q has the larger range (\(42\) against \(33\)). In Q the median is close to \(Q_3\): Q is negatively skewed, while P is close to symmetrical (if anything, slightly positively skewed).
Use the positions. With \(24\) scores in order, the median is the average of the \(12\)th and \(13\)th, and \(Q_1\) is the average of the \(6\)th and \(7\)th.
Class A: if the average of the \(6\)th and \(7\)th is \(65\), the \(7\)th score is at least \(65\) (the larger of two numbers is at least their average). So the \(7\)th to \(24\)th scores are all at least \(65\): \(24 - 6 = 18\) students.
Class B: the \(13\)th score is at least \(65\), so the \(13\)th to \(24\)th are: \(12\) students.
| \(18 + 12\) | \(=\) | \(\mathbf{30}\) |
At least \(30\) students scored \(65\) or more. (More are possible if the \(6\)th or \(12\)th score is also \(65\).)
Common pitfalls
Frequently asked questions
What are parallel box plots?
They are two or more box plots drawn on the same scale, one above the other, so different groups can be compared directly.
How do you compare two box plots?
Compare the medians for the centre, the interquartile ranges and ranges for the spread, and the shapes. Quote the actual values and say what they mean in context.
Which group is more consistent?
The group with the smaller interquartile range, because its middle half of scores is closer together.
How do you find how many people are above a value in parallel box plots?
For each group, work out which statistic the value is, such as the median or a quartile, turn it into a percentage of that group, multiply by the group size, then add the groups.
Can you tell how many scores are in each group from a box plot?
No. A box plot shows the five-number summary only, so the group sizes must be given.