Comparing Data Distributions
Theory
To compare two or more data sets, compare their centre, spread and shape. This Year 10 maths page shows how to read parallel box plots drawn on the same scale, compare medians, IQRs and ranges, decide which group is more consistent, describe each shape, and make statements about percentages and counts of each group. In the Victorian Curriculum this is Level 10 content; in the Australian Curriculum it is Year 10 content. Curriculum: VC2M10ST01 / AC9M10ST02.
Parallel box plots (side-by-side box plots) show the five-number summaries of two or more data sets, one above the other, on one common scale.
Compare three things. Centre: the group with the higher median generally has higher scores. Spread: a smaller IQR (shorter box) or range means the scores are closer together, so the group is more consistent. Shape: symmetrical or skewed, and any outliers.
Comparing a quartile of one group with a quartile of another gives percentage statements. If group A's \(Q_3\) is below group B's median, then at least \(75\%\) of A is below at least half of B.
A good comparison names the statistic and the numbers: "Class B's median (\(72\)) is \(10\) marks higher than Class A's (\(62\))."
| To compare | Use | Larger value means |
|---|---|---|
| Centre | median | generally higher scores |
| Spread of the middle half | IQR \(= Q_3 - Q_1\) | less consistent |
| Overall spread | range \(= \max - \min\) | more spread out (affected by outliers) |
How to compare parallel box plots
- Read the five-number summary of each group from the common scale.
- Compare the medians: which is higher, and by how much?
- Compare the IQRs and ranges: which group is more consistent?
- Describe the shape of each and note any outliers.
- Conclude in context, quoting the numbers.
Statements about percentages
- Find where the given value sits on that group's plot: min, \(Q_1\), median, \(Q_3\) or max.
- Count quarters: below \(Q_1\) is \(25\%\), below the median \(50\%\), below \(Q_3\) \(75\%\).
- Multiply by that group's total for a count.
(i) How much higher is 10Q's median?
(ii) Find each IQR and range.
(iii) Which class is more consistent?
Read each five-number summary: 10P \(38, 52, 61, 70, 88\); 10Q \(45, 58, 66, 72, 84\).
| \(\text{(i) } 66 - 61\) | \(=\) | \(\mathbf{5}\text{ marks}\) |
| \(\text{IQR}_P\) | \(=\) | \(70 - 52 = \mathbf{18},\) \(\text{range}_P = 88 - 38 = \mathbf{50}\) |
| \(\text{IQR}_Q\) | \(=\) | \(72 - 58 = \mathbf{14},\) \(\text{range}_Q = 84 - 45 = \mathbf{39}\) |
(iii) 10Q has the smaller IQR and range, so its marks are closer together: 10Q is more consistent. It also has the higher median.
(i) About how many of Shop A's deliveries took \(25\) minutes or less?
(ii) What percentage of Shop B's took longer than \(22\) minutes?
(iii) True or false: Shop A's median is lower than Shop B's.
| \(\text{(i) } 25 = \text{median}_A\) | \(\Rightarrow\) | \(50\% \text{ of } 60 = \mathbf{30}\) |
| \(\text{(ii) } 22 = Q_1 \text{ of } B\) | \(\Rightarrow\) | \(\text{above } Q_1 = \mathbf{75\%}\) |
| \(\text{(iii) } 25\) | \(<\) | \(28 \Rightarrow \textbf{true}\) |
First decide which of the five values the number is on that group's plot, then use quarters of that group's total.
\(11\) scores each: the median is the 6th; the halves are the 5 scores either side.
| \(\text{Ana}\) | \(:\) | \(8,\ 11,\ 14,\ 16,\ 21\) |
| \(\text{Ben}\) | \(:\) | \(5,\ 9,\ 16,\ 22,\ 30\) |
Centre: Ben's median (\(16\) km) is higher than Ana's (\(14\) km). Spread: Ben's IQR is \(13\) km and range \(25\) km; Ana's are \(5\) km and \(13\) km. Ben usually rides further but varies much more; Ana is more consistent.
(i) Describe the shape of each group.
(ii) Is the maximum of \(Y\) an outlier?
(iii) For which group is the mean likely to be greater than the median?
(i) \(X\): every section is \(5\) or \(10\) long and balanced about the median: symmetrical. \(Y\): short on the left (\(4, 3\)), long on the right (\(9, 24\)): positively skewed.
| \(\text{(ii) IQR}_Y\) | \(=\) | \(36 - 24 = 12\) |
| \(\text{upper limit}\) | \(=\) | \(36 + 1.5 \times 12 = 54\) |
| \(60\) | \(>\) | \(54 \Rightarrow \textbf{outlier}\) |
(iii) Group \(Y\): its long right tail (and the outlier) pull the mean above the median.
Common pitfalls
Frequently asked questions
What are parallel box plots used for?
They compare two or more data sets by drawing their box plots on the same scale, so you can compare the medians, the spreads and the shapes at a glance.
How do you compare two box plots?
Compare the medians for the centre, the interquartile ranges and ranges for the spread, and the shapes, including any outliers. Quote the numbers in your answer.
Which box plot is more consistent?
The one with the smaller interquartile range, because its middle half of scores are closer together. A smaller range also suggests consistency, but the range is affected by outliers.
What does it mean if one median is above the other box?
If one group's median is above the other group's upper quartile, at least half of the first group scored more than at least three quarters of the second. It is a strong sign the first group scored higher.
Why must parallel box plots use the same scale?
Only a common scale lets you compare positions and lengths directly. Two box plots on different scales can look alike even when their values are very different.