IB Maths AI HLStatistics ToolkitPaper 1 & 2~7 min read
Box & Whisker Diagrams
A box plot (or box and whisker diagram) is a compact picture of a dataset built from just five numbers: the minimum, lower quartile, median, upper quartile, and maximum — the five-number summary. The box covers the middle 50% of the data, the whiskers stretch out to the extremes, and any outliers are marked separately with a cross. Box plots aren’t about precision; they’re about shape and comparison. Drawn side by side, two box plots let you compare medians and spreads at a glance — which is exactly what the IB asks you to do.
📘 What you need to know
A box plot displays the five-number summary: minimum, Q1, median (Q2), Q3, maximum.
The box runs from Q1 to Q3 — it holds the middle 50% of the data. Its width is the IQR.
The line inside the box is the median.
The whiskers reach out to the smallest and largest values that are not outliers.
Outliers are plotted as a cross (×) beyond the whiskers; the whisker then stops at the most extreme value that isn’t an outlier.
A box plot is drawn against a single labelled axis with an even scale.
Box plots show the shape of a distribution (symmetry / skew) and are ideal for comparing two datasets.
Your GDC can draw a box plot from raw data — useful for checking, and it shows outliers too.
The anatomy of a box plot
Every box plot is built from the same five points. Get these and the diagram draws itself.
The five-number summary
minimum • Q1 • median (Q2) • Q3 • maximum
read all five straight off your GDC
the parts of a box and whisker diagram
The box spans Q1 to Q3 (the IQR), the median line splits it, and the whiskers reach the extremes. The full width from min to max is the range.
What a box plot does NOT show: individual data values, the mean, the mode, or the exact frequencies. It only displays the five-number summary (plus outliers).
Drawing a box plot (with outliers)
🧭 Recipe — drawing a box and whisker diagram
Find the five-number summary: min, Q1, Q2, Q3, max (use your GDC).
Check for outliers using the 1.5 × IQR rule (lower fence Q1 − 1.5×IQR, upper fence Q3 + 1.5×IQR).
Draw an even, labelled axis that comfortably covers your data range.
Draw the box from Q1 to Q3 with the median line inside.
Add whiskers to the smallest and largest values that are not outliers.
Mark any outliers with a cross (×) at their exact value.
🤔 Why does the whisker stop before the outlier?
If a whisker stretched all the way out to an outlier, the plot would look as though the data spreads evenly out to that extreme value — which is misleading. By stopping the whisker at the last “normal” value and marking the outlier separately with a cross, the diagram makes it obvious that the extreme point stands apart from the bulk of the data. So: whisker → most extreme non-outlier; cross → the outlier itself.
Reading shape from a box plot
The position of the median inside the box (and the relative whisker lengths) tells you about skew.
symmetric vs skewed distributions
If the median sits in the middle with equal whiskers, the data is roughly symmetric (possibly normal). A longer right side = positive skew; a longer left side = negative skew.
Symmetric
balanced
Median centred, whiskers similar lengths. Data could be normally distributed.
Skewed
lop-sided
Median off-centre or one whisker much longer. The long tail shows the direction of skew.
Comparing two box plots
This is the most common exam task. Two box plots are drawn on the same axis, and you compare them in context. The IB wants two comparisons: one of central tendency (medians) and one of spread (IQR or range).
🧠 Memory aid — “centre, then spread, in context”
Always make exactly two points: (1) compare the medians — “On average, A is higher/lower than B” — and (2) compare the IQRs — “A is more/less spread out (more/less consistent) than B”. Then tie each back to what it means for the real situation. Quote the actual values to back up each statement.
Worked examples
WE 1
Find the five-number summary & check for outliers
The distances (m) travelled by 15 snails in one minute are:
0.5, 0.7, 1.0, 1.1, 1.2, 1.2, 1.2, 1.3, 1.4, 1.4, 1.4, 1.4, 1.5, 1.5, 1.6
(a) Find Q1, Q2, Q3. (b) Find the IQR. (c) Identify any outliers.
(a) using GDCQ₁ = 1.1, Q₂ = 1.3, Q₃ = 1.4(b) IQRIQR = Q₃ − Q₁ = 1.4 − 1.1 = 0.3IQR = 0.3 m(c) fenceslower = 1.1 − 1.5×0.3 = 0.65upper = 1.4 + 1.5×0.3 = 1.850.5 < 0.65 → outlier; nothing above 1.850.5 m is an outlierthese five numbers (plus the outlier) are everything you need to draw the plot.
WE 2
Draw the box plot
Using the snail data from WE 1, draw a box and whisker diagram.
plot the five-number summary, treating 0.5 as an outlierbox: Q₁ = 1.1 to Q₃ = 1.4, median line at 1.3.left whisker stops at the next smallest value after the outlier = 0.7.right whisker reaches the max = 1.6. Mark 0.5 with a cross (×).
The outlier 0.5 m is a cross; the left whisker begins at 0.7 m (the next value in), not at the outlier.
WE 3
Read values off a box plot
A box plot of test scores has its left whisker at 18, box from 30 to 52 with the median line at 40, and right whisker at 70. Write down (a) the median, (b) the range, (c) the IQR.
(a) median = the line in the boxmedian = 40(b) range = max − min= 70 − 18 = 52range = 52(c) IQR = Q₃ − Q₁ = box width= 52 − 30 = 22IQR = 22box ends are the quartiles; whisker ends are the min and max.
WE 4
Comment on the shape
For the test-score box plot in WE 3 (whisker 18, box 30–52, median 40, whisker 70), describe the skew of the distribution.
compare median position & whisker lengthsmedian 40 is left of box centre (41); left whisker 12 wide, right whisker 18 widethe longer tail is on the right (higher values).positively skeweda longer right side / median nearer the left = positive (right) skew.
WE 5
Compare two distributions in context
Two surgeries’ waiting times (minutes) are shown as box plots. HealthHut: median 24, IQR 19. FitFirst: median 20, IQR 13. Compare the two distributions in context.
1. compare centres (medians)FitFirst’s median (20) is lower than HealthHut’s (24), so on average patients are seen quicker at FitFirst.2. compare spreads (IQR)FitFirst’s IQR (13) is smaller than HealthHut’s (19), so FitFirst’s waiting times are more consistent (less variable).FitFirst: lower & more consistent waitstwo comparisons, both with values, both linked to the real context.
💡 Top tips
Always label the axis with an even scale and units — marks are awarded for this.
Check for outliers first before drawing whiskers. If there’s an outlier, the whisker stops at the most extreme value that isn’t one, and the outlier gets a cross.
Box width = IQR, full width = range. You can read both straight off the diagram.
Use your GDC to draw/check a box plot from raw data — it also flags outliers automatically.
When comparing, make two points (median + spread) and always relate them to the context with actual values.
Median position shows skew: nearer the left → positive skew; nearer the right → negative skew.
⚠ Common mistakes
Drawing the whisker to the outlier. The whisker must stop at the last non-outlier; the outlier is a separate cross.
Confusing the box edges with min/max. The box edges are Q1 and Q3; the whisker ends are the min and max.
Reading the mean off a box plot. A box plot shows the median, not the mean.
Comparing only one thing. The IB wants both a centre comparison and a spread comparison.
No context in the comparison. “Median is bigger” isn’t enough — say what it means for the situation.
Uneven or unlabelled scale. Always use a consistent axis scale with units.
Next up — Cumulative Frequency Graphs. You’ve drawn box plots from raw data; but what about grouped data, where you don’t have the individual values? A cumulative frequency curve lets you estimate the median, quartiles and percentiles — and from those you can build a box plot for grouped data too.
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