Comparing Data Distributions
Theory
To compare two or more data sets, compare their centre, spread and shape. This Year 10A 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, using back-to-back stem-and-leaf plots, histograms, dot plots and the standard deviation as well. It is Level 10 content in the Year 10A course (Victorian Curriculum 10A) and Year 10 content in the Australian Curriculum. 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 (\(62\))."
Other displays compare groups too. A back-to-back stem-and-leaf plot shares one stem between two groups; the left-hand leaves are read outwards from the stem. Histograms and dot plots drawn on the same scale show the modal class, the shape and any clusters or outliers.
The standard deviation \(\sigma\) measures how far the scores are spread from the mean (use the \(\sigma_x\) key in statistics mode). For roughly symmetrical data with no outliers compare means and standard deviations; for skewed data or data with outliers compare medians and IQRs.
| 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) |
| Display | Read off |
|---|---|
| back-to-back stem-and-leaf | ordered scores of both groups: medians, quartiles, ranges |
| dot plots on one scale | every score: medians, modes, shape; mean and \(\sigma\) with a calculator |
| histograms on one scale | the modal class (tallest column), the shape, the class holding the median |
| parallel box plots | five-number summaries: medians, IQRs, ranges, skewness |
| summary table | mean against median for skew; \(\sigma\) against IQR for the effect of 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.
| Class P | Stem | Class Q |
|---|---|---|
| \(\) | \(3\) | \(5\) |
| \(8\ \ 2\) | \(4\) | \(6\) |
| \(7\ \ 5\ \ 1\) | \(5\) | \(8\) |
| \(7\ \ 3\ \ 2\ \ 0\) | \(6\) | \(4\ \ 6\ \ 8\) |
| \(1\) | \(7\) | \(0\ \ 3\ \ 5\ \ 9\) |
| \(4\) | \(8\) | \(2\) |
Key: \(2\,|\,4\,|\,6\) means \(42\) for class P and \(46\) for class Q
(i) Find the median, IQR and range of each class.
(ii) Compare the classes.
(i) With \(11\) marks, the median is the \(6\)th, \(Q_1\) the \(3\)rd and \(Q_3\) the \(9\)th. Read class P's leaves from the stem outwards (right to left).
Five-number summaries:
| \(\text{P}\) | \(:\) | \(42,\ 51,\ 60,\ 67,\ 84\) |
| \(\text{Q}\) | \(:\) | \(35,\ 58,\ 68,\ 75,\ 82\) |
| \(\text{IQR}_P\) | \(=\) | \(67 - 51 = \mathbf{16}\) |
| \(\text{range}_P\) | \(=\) | \(84 - 42 = \mathbf{42}\) |
| \(\text{IQR}_Q\) | \(=\) | \(75 - 58 = \mathbf{17}\) |
| \(\text{range}_Q\) | \(=\) | \(82 - 35 = \mathbf{47}\) |
Medians: P \(\mathbf{60}\), Q \(\mathbf{68}\).
(ii) Centre: Q's median is \(8\) marks higher, so Q generally did better. Spread: the IQRs are almost equal (\(16\) and \(17\)); Q's larger range comes from one low mark, \(35\). Shape: Q is negatively skewed (tail of low marks); P is fairly symmetrical with one high mark, \(84\).
(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.
When should I compare standard deviations instead of IQRs?
When both data sets are roughly symmetrical with no outliers. If either set is skewed or has an outlier, the IQR is the fairer measure of spread.