IB Maths AI HLStatistics ToolkitPaper 1 & 2~6 min read
Outliers
An outlier is a data value that sits far away from the rest — a 29-year-old at a children’s birthday party, or a £200,000 salary in a list where everyone else earns £30,000. The IB gives you one precise rule to decide whether a value counts as an outlier: it must lie more than 1.5 × IQR beyond the nearest quartile. Learn the two boundary formulas, compare each value against them, and you can flag every outlier in seconds. The harder skill — also examined — is deciding whether a flagged outlier should be removed: that depends entirely on context, not on the maths.
📘 What you need to know
An outlier is an extreme value that doesn’t fit with the rest of the data — much bigger or much smaller than everything else.
The 1.5 × IQR rule: a value x is an outlier if it lies more than 1.5 × IQR belowQ1 or more than 1.5 × IQR aboveQ3.
Lower boundary (fence): Q1 − 1.5 × IQR. Any value below this is an outlier.
Upper boundary (fence): Q3 + 1.5 × IQR. Any value above this is an outlier.
Recall: IQR = Q3 − Q1 (in the formula booklet).
The outlier rule itself is NOT in the formula booklet — you must memorise the 1.5 × IQR boundaries.
Removing outliers depends on context: remove if it’s an error; keep if it’s a valid data value.
Outliers strongly affect the mean, range and standard deviation, but not the median and IQR.
The 1.5 × IQR rule
To test for outliers you first need the quartiles Q1 and Q3 (use your GDC), then the IQR. The rule sets up two “fences” — one below the data and one above it. Anything outside the fences is an outlier.
Outlier boundaries (the 1.5 × IQR rule)x is an outlier if x < Q1 − 1.5 × IQR or x > Q3 + 1.5 × IQR
NOT in the formula booklet — memorise this ✗
where the outlier fences sit
The box holds the middle 50% of the data. The fences sit 1.5 × IQR out from each quartile. Anything beyond a fence (the red zones) is flagged as an outlier.
🧭 Recipe — testing for outliers
Find Q1 and Q3 using your GDC (statistics mode).
Calculate the IQR: IQR = Q3 − Q1.
Lower fence: Q1 − 1.5 × IQR.
Upper fence: Q3 + 1.5 × IQR.
Compare each value: anything below the lower fence or above the upper fence is an outlier.
Should you remove an outlier?
Finding an outlier is the maths. Deciding what to do with it is the judgement — and the IB tests this. The answer always depends on context, never on the number alone.
Remove it
if it’s an error
The value was recorded wrong or doesn’t belong. E.g. 17 typed as 71, or an adult’s age in children’s data.
Keep it
if it’s valid
The value is genuine, just extreme. E.g. a company director’s salary, or a very tall student. It’s real data.
🤔 Why not just delete every outlier?
An outlier isn’t automatically “bad data”. A director earning ten times more than every other employee is a real, correct value — deleting it would hide a true feature of the dataset. Only remove an outlier when you have a reason to believe it’s a mistake (a recording error, a value that can’t be valid in context). When in doubt, the data should be checked rather than silently dropped. Always justify your decision in words.
Which measures do outliers affect?
This comes up constantly in exam interpretation questions. Outliers drag some statistics around badly and leave others untouched — knowing which is which tells you when to trust the mean versus the median.
Measure
Affected by outliers?
Why
Mean
Yes ✗
uses every value, so one extreme value shifts it
Range
Yes ✗
uses the max and min directly
Standard deviation
Yes ✗
measures spread about the mean — extremes inflate it
Median
No ✓
only the middle position matters, not the extremes
IQR
No ✓
uses the central 50% only, ignoring the tails
🧠 Memory aid — “middle measures are safe”
The statistics based on position in the middle (median, IQR) shrug off outliers, because an extreme value at the end doesn’t change what’s in the centre. The statistics that use every value (mean, range, standard deviation) get pulled around. So if a dataset has outliers, prefer the median and IQR to describe it.
Worked examples
WE 1
Identify the outliers
The ages, in years, of children at a birthday party are given below.
2, 7, 5, 4, 8, 4, 6, 5, 5, 29, 2, 5, 13
Using your GDC, Q1 = 4 and Q3 = 7. Identify any outliers.
Step 1: IQRIQR = Q₃ − Q₁ = 7 − 4 = 3Step 2: lower fenceQ₁ − 1.5×IQR = 4 − 1.5×3 = 4 − 4.5 = −0.5Step 3: upper fenceQ₃ + 1.5×IQR = 7 + 1.5×3 = 7 + 4.5 = 11.5Step 4: compare valuesnothing below −0.5; values above 11.5 are 13 and 29outliers: 13 and 29find both fences, then scan the data for anything outside them.
WE 2
Decide which outlier to remove
From WE 1, the outliers are 13 and 29. The data records the ages of children at a party. Suggest which value(s) should be removed, justifying your answer.
consider each in context13 is a valid age for a child → keep it (it is a real data value, just on the high side).29 is the age of an adult, not a child → remove it (it does not belong in data about children).remove 29; keep 13always justify with the context — “child vs adult” — not just the maths.
WE 3
Lower outlier — distances
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
Given Q1 = 1.1 and Q3 = 1.4, identify any outliers.
Step 1: IQRIQR = 1.4 − 1.1 = 0.3Step 2: lower fence1.1 − 1.5×0.3 = 1.1 − 0.45 = 0.65Step 3: upper fence1.4 + 1.5×0.3 = 1.4 + 0.45 = 1.85Step 4: compare0.5 < 0.65 → outlier; nothing above 1.850.5 m is an outlieroutliers can be on the low side too — always check both fences.
WE 4
No outliers present
A data set has Q1 = 22 and Q3 = 34. The smallest value is 15 and the largest is 40. Determine whether either extreme value is an outlier.
Step 1: IQRIQR = 34 − 22 = 12Step 2: fenceslower = 22 − 1.5×12 = 22 − 18 = 4upper = 34 + 1.5×12 = 34 + 18 = 52Step 3: compare15 > 4 ✓ and 40 < 52 ✓ — both inside the fencesno outliersa value being the max or min does NOT make it an outlier — it must cross a fence.
WE 5
Effect on the mean and median
A small company lists five annual salaries (£000s): 28, 30, 31, 33, 150. The £150,000 value is the director’s salary and is confirmed correct. Comment on which average best represents a typical employee.
mean(28+30+31+33+150)/5 = 272/5 = 54.4medianmiddle of ordered data = 31the mean (£54.4k) is pulled up by the director’s salary and sits above four of the five employees.the median (£31k) is not affected by the outlier and better reflects a typical employee.use the medianthe outlier is valid (don’t remove it), but the median describes the data more fairly.
💡 Top tips
Memorise the boundaries: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. The outlier rule is not in the formula booklet (only IQR = Q3 − Q1 is).
Always check both fences. Outliers can be unusually small as well as unusually large.
Use your GDC for the quartiles — type in the data and read off Q1 and Q3.
Justify removal in words. “Remove because it’s an adult’s age in children’s data” earns the mark; just writing “remove 29” usually doesn’t.
Link to averages. If a dataset has outliers, the median and IQR are the safer summary statistics.
On a box plot, outliers are drawn as a cross, and the whisker stops at the most extreme value that is not an outlier.
⚠ Common mistakes
Forgetting the 1.5 factor. The fence is Q1 − 1.5 × IQR, not Q1 − IQR.
Assuming the max or min is automatically an outlier. It only counts if it lies beyond a fence.
Adding instead of subtracting at the lower fence. Subtract for the lower boundary, add for the upper.
Removing every outlier. Valid extreme values (like a director’s salary) should be kept.
Not justifying the remove/keep decision. The IB wants a context-based reason, not just the value.
Thinking outliers affect the median or IQR. They don’t — only the mean, range and SD.
Next up — Box & Whisker Diagrams. Now that you can find quartiles and flag outliers, you’ll learn to display them: the box shows the middle 50% (Q1 to Q3), the whiskers reach the most extreme non-outlier values, and any outliers are marked with a cross. You’ll also see how to compare two datasets side by side using their medians and IQRs.
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