IB Maths AA SL Topic 4 — Statistics Toolkit Paper 1 & 2 ~9 min read

Box & Whisker Diagrams

A box plot is a quick visual snapshot of any data set — it shows you the smallest value, the biggest value, the median, and the quartiles all in one diagram. Drawing them and reading them is straightforward once you know the 5 numbers you need.

📘 What you need to know

A box plot (or box and whisker diagram) shows 5 key values from a data set: minimum, Q₁, median, Q₃, maximum.
The box covers the middle 50% of the data — from Q₁ to Q₃.
The whiskers extend out to the smallest and largest values that are not outliers.
Outliers are marked with a cross (×) outside the whiskers.
Outliers are values below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.
Use your GDC to find the 5-number summary — enter data, run 1-Var Stats.

What does a box plot look like?

Anatomy of a box plot

The diagram shows 5 key parts:

The left whisker stretches to the smallest value that’s not an outlier.
The box goes from Q₁ (lower quartile) to Q₃ (upper quartile).
A line inside the box shows the median (Q₂).
The right whisker stretches to the largest value that’s not an outlier.
Any outliers are marked with a cross (×) past the whiskers.

The box can be drawn at any height — the height doesn’t mean anything. Only the horizontal positions matter, because they line up with the data values on the axis.

The 5-number summary

Before drawing a box plot, find these 5 values from the data:

Min

smallest

Q₁

lower quartile

Q₂

median

Q₃

upper quartile

Max

largest

📍

Use your GDC — every time

Type your data into a list and run 1-Var Stats. The calculator gives you Min, Q₁, Median, Q₃, Max all in one go. Don’t waste time finding quartiles by hand — different methods give different answers.

How to draw a box plot

5 steps to draw a box plot

Find the 5-number summary from your data (Min, Q₁, Median, Q₃, Max).
Check for outliers using Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR.
Draw a horizontal axis with an even scale that fits all your data.
Draw the box from Q₁ to Q₃, with a vertical line at the median.
Add whiskers out to the min and max (excluding outliers). Mark any outliers with a × past the whiskers.

Outlier rule

x is an outlier if x < Q₁ − 1.5 × IQR or x > Q₃ + 1.5 × IQR

🧠

Memory trick: “1.5 × IQR fences”

Imagine fences placed 1.5 × IQR away from each quartile. Anything outside those fences is an outlier and gets marked with a ×. The whisker stops at the last good data point — not at the fence itself.

What does the shape tell you?

A quick glance at a box plot tells you a lot about the data. Here’s what to look for:

Shape	What it means
Box symmetric, whiskers equal length	Data is symmetric — possibly normal-shaped.
Median closer to Q₁, longer right whisker	Data is skewed to the right (positively skewed). Most values are smaller, a few are large.
Median closer to Q₃, longer left whisker	Data is skewed to the left (negatively skewed). Most values are larger, a few are small.
Long box / wide whiskers	Data is very spread out — high variability.
Short box / short whiskers	Data is tightly bunched — low variability, very consistent.

Comparing two box plots

When comparing two data sets, draw both box plots on the same axis. Then compare:

Medians — which centre is higher?
IQRs (length of the box) — which is more spread out?
Ranges (whisker tip to whisker tip) — total spread.
Outliers — does either data set have extreme values?

Always link your comparison back to the context of the question. Don’t just say “the median of A is bigger” — say “the median wait time at A is bigger, so on average people waited longer at A”. That’s where the marks live.

Worked examples

WE 1

Find the 5-number summary and identify outliers

The distances, in metres, 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.

Use GDC’s 1-Var Stats for the quartiles.part (a) From GDC: Q₁ = 1.1 m, Q₂ = 1.3 m, Q₃ = 1.4 m Q₁ = 1.1, Q₂ = 1.3, Q₃ = 1.4 (m)part (b) IQR = Q₃ − Q₁: 1.4 − 1.1 = 0.3 IQR = 0.3 mpart (c) Lower fence: Q₁ − 1.5 × IQR = 1.1 − 1.5(0.3) = 0.65 Upper fence: Q₃ + 1.5 × IQR = 1.4 + 1.5(0.3) = 1.85 Check data: 0.5 < 0.65 → outlier! All other values are between 0.65 and 1.85. 0.5 m is an outlier always test the fence rule on the smallest and largest values

WE 2

Draw the box plot for the snail data

Using the values from WE 1, sketch a box plot for the snail distances.

Min = 0.5, Q₁ = 1.1, Q₂ = 1.3, Q₃ = 1.4, Max = 1.6 0.5 is an outlier — left whisker ends at next-smallest value (0.7). Step 1: Draw an axis from 0.4 to 1.7 m. Step 2: Draw box from 1.1 to 1.4, with a line at 1.3. Step 3: Right whisker from 1.4 to 1.6 (max). Step 4: Left whisker from 1.1 to 0.7 (next smallest after outlier). Step 5: Mark × at 0.5. Box plot drawn ✓ whisker ends at next “good” value (0.7), not at the outlier (0.5)

WE 3

Read information from a box plot

The box plot below shows test scores for a class of 30 students.

(a) Find the median. (b) Find the IQR. (c) Estimate how many students scored above 80.

Read the 5 key positions off the diagram.part (a) Median = vertical line in the box: Median = 65part (b) Box edges: Q₁ = 50, Q₃ = 80. IQR = 80 − 50 = 30 IQR = 30part (c) Q₃ = 80 means 75% of data is below 80. So 25% scored above 80: 25% × 30 = 7.5 About 7 or 8 students scored above 80 remember each section of the box plot holds 25% of the data

WE 4

Compare two doctor surgeries (SME canonical)

The box plots below show waiting times in minutes at HealthHut and FitFirst surgeries. Compare the distributions in context.

Always compare a measure of centre + a measure of spread, in context.centre — median HealthHut median = 20 min, FitFirst median = 24 min. HealthHut’s median is smaller (20 < 24). On average patients are seen quicker at HealthHutspread — iqr HealthHut IQR = 30 − 15 = 15 min FitFirst IQR = 31 − 18 = 13 min FitFirst’s IQR is smaller (13 < 15). Wait times at FitFirst are more consistent always link the comparison back to context — that’s where the marks are!

WE 5

Check whether a value is an outlier

A data set has Q₁ = 14, Q₃ = 22. Determine whether 35 is an outlier.

Q₁ = 14, Q₃ = 22 Find the IQR, then check the upper fence. IQR: 22 − 14 = 8 Upper fence: Q₃ + 1.5 × IQR = 22 + 1.5(8) = 34 Compare 35 with the fence: 35 > 34 ✓ Yes — 35 is an outlier if the value is bigger than Q₃ + 1.5×IQR, it’s a high outlier

💡 Top tips

Always use your GDC for the 5-number summary. Enter data, run 1-Var Stats, read off Min, Q₁, Med, Q₃, Max.
Check for outliers before drawing. Find Q₁ − 1.5 × IQR and Q₃ + 1.5 × IQR — anything outside is an outlier.
If there’s an outlier, the whisker ends at the next value (not the outlier). Mark the outlier with a × past the whisker.
Each section of the box plot holds 25% of the data. Use this to estimate how many values fall in any range.
Always include units when stating Q₁, Q₃, IQR, etc.
Comparing two box plots: always quote a measure of centre (median) and a measure of spread (IQR), and link both back to context.
Box plots can be drawn vertically too — same idea, just rotated 90°. Same axis rules apply.

⚠ Common mistakes

Drawing the whisker out to the outlier. The whisker ends at the smallest/largest non-outlier value. Outliers are crosses, not whisker tips.
Forgetting to test for outliers. Always calculate the fences before drawing — an outlier you missed costs marks.
Reading the wrong line as the median. The median is the line inside the box, not the edges of the box (those are Q₁ and Q₃).
Saying “the median of A is bigger” without context. You need to relate it to what the data represents — “wait times at A were on average longer”.
Comparing only one statistic. Examiners want both centre and spread. One alone misses the bigger picture.
Uneven scales. The horizontal axis must have evenly-spaced tick marks, with units clearly labelled.
Forgetting to mark the outlier. If the outlier check picks one out, mark it with × past the whisker — don’t just ignore it.
Confusing box width with frequency. The box height/width doesn’t represent anything — only horizontal position matters.

Box plots are exam gold — they’re easy to draw, easy to read, and worth solid marks. The next note covers cumulative frequency graphs, which let you build a box plot from grouped data too.

Need help with Box & Whisker Diagrams?

Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.

Book Free Session →