Boxplots
A box plot shows the five-number summary of a data set: a box from \(Q_1\) to \(Q_3\) with a line at the median, and whiskers out to the minimum and maximum. Each section holds about a quarter of the scores. In NSW Year 10 Advanced maths you draw box plots, read them, and use them to make statements about the data (NESA outcome MA5‑DAT‑C‑01).
You will draw box plots from lists, dot plots, column graphs and tables, and read percentages and counts from them, such as how many units rent for between two prices. You will explain why a box plot can have no lower whisker and match a data set to a box plot. Harder questions decide which statements must, could or cannot be true, and rebuild all the scores from a summary and a mean. It is part of Year 10 Advanced Single Variable and Bivariant Statistics.
Every question with a fully worked solution.
- Boxplots - Video - The Five Number Summary, Boxplots, and Outliers Watch
Theory
A box plot (box-and-whisker plot) draws the five-number summary to scale: the box runs from \(Q_1\) to \(Q_3\) with a line at the median, and whiskers reach the minimum and maximum. This Year 10 Advanced maths page covers drawing box plots from any data display, marking outliers, reading percentages and counts, and working backwards from a summary to the data.
A box plot is drawn above a number line. The box spans the middle \(50\%\) of the data, from the lower quartile \(Q_1\) to the upper quartile \(Q_3\); its length is the IQR. A vertical line inside it marks the median. The whiskers join the box to the minimum and maximum.
Each of the four sections (whisker, half-box, half-box, whisker) contains about \(25\%\) of the scores. A longer section means those scores are more spread out, not that there are more of them.
An outlier may be drawn as a separate point; the whisker then ends at the most extreme score that is not an outlier. A common test: a score more than \(1.5 \times \text{IQR}\) below \(Q_1\) or above \(Q_3\).
A box plot does not show the mean, the mode or the number of scores.
| Part of the box plot | Fraction of the scores (about) |
|---|---|
| below \(Q_1\), or above \(Q_3\) | \(25\%\) each |
| between \(Q_1\) and \(Q_3\) (the box) | \(50\%\) |
| above \(Q_1\), or below \(Q_3\) | \(75\%\) |
| above or below the median | \(50\%\) |
Drawing a box plot
- Order the data (or use running totals for a table or graph) and find the five-number summary.
- Draw a number line with an even scale that covers the minimum to the maximum.
- Draw the box from \(Q_1\) to \(Q_3\) and a line at the median.
- Add the whiskers to the minimum and maximum, or to the last non-outlier, marking any outlier as a dot.
Reading a box plot
- Read the five values from the scale.
- Percentages: use \(25\%\), \(50\%\), \(75\%\); multiply by the number of scores for 'about how many'.
- Spread and shape: IQR \(= Q_3 - Q_1\); compare the two halves of the box and the two whiskers.
- Special cases: no whisker means the minimum (or maximum) equals the quartile; a median on the edge of the box means it equals a quartile.
- Must, could or cannot: only the five values are certain. Statements about the mean or exact individual scores can only be 'could'.
- Working backwards: place the known values at their positions, then use the mean (for a total) and the order of the scores.
(i) Find the five-number summary.
(ii) Use the test "more than \(1.5 \times \text{IQR}\) beyond a quartile" to show that \(52\) is an outlier.
(iii) Draw the box plot and describe the shape.
(i) The data are in order. Median: the \(6\)th score, \(22\). Lower half \(12, \ldots, 21\): \(Q_1 = 18\). Upper half \(24, \ldots, 52\): \(Q_3 = 27\). Summary: \(\mathbf{12,\ 18,\ 22,\ 27,\ 52}\).
(ii) The IQR is \(27 - 18 = 9\).
| \(Q_3 + 1.5 \times \text{IQR}\) | \(=\) | \(27 + 13.5 = 40.5\) |
\(52 > 40.5\), so \(52\) is an outlier. (Nothing is below \(18 - 13.5 = 4.5\).)
(iii) Draw \(52\) as a separate point and end the upper whisker at \(30\), the largest time that is not an outlier.
The median is nearer \(Q_1\) and there is a long tail on the right: positively skewed.
Read the five values: \(14,\ 19,\ 23,\ 34,\ 61\). Each section holds about a quarter of the people.
| \(\text{(i)}\ \ Q_1 \text{ to } Q_3\) | \(=\) | \(\dfrac{1}{2} \times 160 \approx \mathbf{80}\) |
| \(\text{(ii)}\ \ \text{above } Q_1\) | \(=\) | \(\dfrac{3}{4} \times 160 \approx \mathbf{120}\) |
| \(\text{(iii)}\ \ \text{IQR}\) | \(=\) | \(34 - 19 = \mathbf{15}~\text{years}\) |
(iv) The top quarter runs from \(34\) to \(61\), a spread of \(27\) years, the longest section.
(v) False. Each of those sections holds about \(25\%\) of the people. A longer section means the ages are more spread out, not that there are more people.
| People in the car | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
|---|---|---|---|---|---|---|
| Frequency | \(3\) | \(5\) | \(12\) | \(3\) | \(1\) | \(1\) |
Running totals: \(3,\ 8,\ 20,\ 23,\ 24,\ 25\). The median is the \(13\)th car; each half has \(12\) cars.
| \(\text{median}\) | \(=\) | \(13\text{th} = 3 \quad (8 < 13 \le 20)\) |
| \(Q_1\) | \(=\) | \(\text{average of 6th and 7th} = 2\) |
| \(Q_3\) | \(=\) | \(\text{average of 19th and 20th} = 3\) |
Summary: \(\mathbf{1,\ 2,\ 3,\ 3,\ 6}\).
The median equals \(Q_3\), so the median line sits on the right edge of the box. So many cars hold \(3\) people that the \(13\)th to the \(20\)th cars are all \(3\).
(i) Show that the 3rd and 5th scores add to \(15\).
(ii) List every possible set of scores.
(iii) Which set is it if the mode is \(4\)?
(i) With \(7\) scores, \(Q_1\) is the \(2\)nd score, the median the \(4\)th and \(Q_3\) the \(6\)th. So the scores are \(2,\ 4,\ s_3,\ 9,\ s_5,\ 13,\ 20\).
| \(2 + 4 + s_3 + 9 + s_5 + 13 + 20\) | \(=\) | \(7 \times 9 = 63\) |
| \(s_3 + s_5\) | \(=\) | \(15\) |
(ii) The order needs \(4 \le s_3 \le 9\) and \(9 \le s_5 \le 13\). Pairs adding to \(15\): \((4, 11),\ (5, 10),\ (6, 9)\). The sets are
(iii) Only the first set has \(4\) as its mode: \(\mathbf{2,\ 4,\ 4,\ 9,\ 11,\ 13,\ 20}\).
Common pitfalls
Frequently asked questions
How do you draw a box plot?
Find the five-number summary: minimum, lower quartile, median, upper quartile and maximum. Draw a scaled number line, a box from the lower to the upper quartile with a line at the median, and whiskers out to the minimum and maximum.
What percentage of the data is in the box of a box plot?
About 50 percent. The box runs from the lower quartile to the upper quartile, so it holds the middle half of the scores.
How are outliers shown on a box plot?
An outlier can be drawn as a separate dot or cross. The whisker then ends at the most extreme score that is not an outlier.
Why does a box plot sometimes have no whisker?
If the minimum equals the lower quartile, or the maximum equals the upper quartile, that whisker has length zero, so it does not appear.
Can you find the mean from a box plot?
No. A box plot shows only the five-number summary. The mean needs every score, or at least their total.