IB Maths AI SL Topic 4 — Statistics Toolkit Paper 1 & 2 1.5 × IQR rule ~6 min read

Outliers

An outlier is a data value far outside the rest. The IB uses a strict definition: a value is an outlier if it lies more than 1.5 × IQR from the nearest quartile. Whether to keep or remove an outlier depends on whether it is a genuine extreme value or just a recording error.

📘 What you need to know

Definition: a value x is an outlier if x < Q₁ − 1.5 IQR OR x > Q₃ + 1.5 IQR.
Fences: the two boundary values Q₁ − 1.5 IQR (lower) and Q₃ + 1.5 IQR (upper). Anything beyond a fence is an outlier.
Boundary values are NOT outliers: the rule says more than 1.5 IQR, not “equal or more”. A value exactly on a fence is kept.
Remove if it is an error: a typo, double-counted entry, or wrong unit — remove and recompute.
Keep if it is genuine: a real extreme value (mansion in a row of houses, prodigy in a class) — keep but flag it.
On a box plot: outliers are drawn as separate × marks, and the whisker stops at the most extreme non-outlier.

The 1.5 × IQR rule visualised

Picture the box plot. The box covers the middle 50% of the data (Q₁ to Q₃). Then extend invisible “fences” 1.5 IQRs further out on each side. Anything that escapes those fences is an outlier.

The box covers Q₁ to Q₃ (IQR = 12). Orange fences sit 1.5 IQR beyond each quartile: 40 − 18 = 22 and 52 + 18 = 70. The whiskers end at 35 and 55 (the most extreme non-outliers), and the values 12 and 120 fall beyond the fences — both flagged as outliers (red ×).

The outlier rule x is an outlier ⇔ x < Q₁ − 1.5 IQR or x > Q₃ + 1.5 IQR

where IQR = Q₃ − Q₁

🧭 Recipe — identify outliers

Order the data and find Q₁ and Q₃. Use the GDC on Paper 2.
Compute IQR = Q₃ − Q₁.
Compute the two fences: lower = Q₁ − 1.5 IQR · upper = Q₃ + 1.5 IQR.
List any data values below the lower fence or above the upper fence.
Decide: keep (genuine extreme) or remove (clear error / typo). Justify with the context.

Strict inequality: the rule says more than 1.5 IQR. A value exactly on the fence (e.g. x = Q₁ − 1.5 IQR) is NOT an outlier — it’s the borderline case.

Worked examples

WE 1

Identify outliers from given quartiles

A data set has Q₁ = 12 and Q₃ = 24. Determine which of the following values are outliers:

3, 15, 18, 28, 50

Step 1 — IQR IQR = 24 − 12 = 12 Step 2 — fences lower = 12 − 1.5 × 12 = 12 − 18 = −6 upper = 24 + 1.5 × 12 = 24 + 18 = 42 Step 3 — check each value 3 ∈ [−6, 42] → not outlier 15, 18, 28 → all inside → not outliers 50 > 42 → outlier ✓ only 50 is an outlier don’t be fooled by 3 — it looks small, but it’s still above the lower fence of −6. Compare to the FENCE, not to the median.

WE 2

Full workflow from raw data

The 5 km running times (min) of 11 club athletes are:

35, 38, 40, 42, 44, 45, 47, 48, 50, 52, 95

Identify any outliers.

Step 1 — quartiles (n = 11) median = 6th value = 45 lower 5: 35, 38, 40, 42, 44 → Q₁ = 40 upper 5: 47, 48, 50, 52, 95 → Q₃ = 50 Step 2 — IQR IQR = 50 − 40 = 10 Step 3 — fences lower = 40 − 15 = 25 upper = 50 + 15 = 65 Step 4 — flag values outside [25, 65] 95 > 65 → outlier outlier: 95 min 95 minutes for 5 km is extremely slow — likely a recording error or a walker. Decide context before removing it.

WE 3

Two outliers (low AND high)

The annual incomes of 10 employees (in $1000s) are:

12, 35, 40, 42, 45, 47, 50, 52, 55, 120

Identify any outliers.

Step 1 — quartiles (n = 10) median = (45+47)/2 = 46 lower 5: 12, 35, 40, 42, 45 → Q₁ = 40 upper 5: 47, 50, 52, 55, 120 → Q₃ = 52 Step 2 — IQR & fences IQR = 12 lower = 40 − 18 = 22 upper = 52 + 18 = 70 Step 3 — values outside [22, 70] 12 < 22 → outlier 120 > 70 → outlier outliers: 12 and 120 an outlier set can be one-sided, two-sided, or empty. Always check BOTH fences — students often forget to test for low outliers.

WE 4

Keep or remove? — justify the decision

Seven recent house sale prices on a quiet street are recorded (in $000s):

240, 260, 275, 290, 305, 320, 750

(a) Show that 750 is an outlier. (b) State whether 750 should be removed, with justification.

(a) Quartiles (n = 7, odd) median = 4th value = 290 lower 3: 240, 260, 275 → Q₁ = 260 upper 3: 305, 320, 750 → Q₃ = 320 IQR = 60 upper fence = 320 + 90 = 410 750 > 410 → outlier ✓ (b) Decision 750k is a plausible mansion price — not an error keep 750 — it is a genuine extreme value, not a typo “Keep / remove” decisions need a CONTEXT-based reason. “It’s plausibly real” → keep. “Clearly a typo / impossible value” → remove.

WE 5

Find the smallest integer that is an outlier

A data set has Q₁ = 18 and Q₃ = 26. Find the smallest integer value that would be classified as an upper outlier.

Step 1 — IQR & upper fence IQR = 26 − 18 = 8 upper fence = 26 + 1.5 × 8 = 26 + 12 = 38 Step 2 — outlier means STRICTLY greater than 38 38 itself: NOT an outlier (on the boundary) 39: 39 > 38 ✓ smallest integer outlier = 39 trap: the value AT the fence is not an outlier. The smallest integer above 38 is 39.

WE 6

Comprehensive — full Paper-2 style question

A commuter records her daily journey times (min) over 11 working days:

12, 17, 18, 19, 21, 22, 24, 25, 26, 28, 50

(a) Find Q₁, Q₃ and the IQR. (b) Identify any outliers. (c) Suggest whether the outlier should be removed.

(a) Quartiles (n = 11, odd) median = 6th value = 22 lower 5: 12, 17, 18, 19, 21 → Q₁ = 18 upper 5: 24, 25, 26, 28, 50 → Q₃ = 26 IQR = 26 − 18 = 8 Q₁ = 18 · Q₃ = 26 · IQR = 8 (b) Fences lower = 18 − 12 = 6 upper = 26 + 12 = 38 12 not < 6 → not outlier 50 > 38 → outlier ✓ outlier: 50 min (c) Decision 50 min is plausible — a bad-traffic day keep 50; it is a genuine value commuting times can spike for legitimate reasons (accidents, delays). Unless told it was a recording error, this outlier represents real-world variability and should be kept.

💡 Top tips

Compute the fences ONCE; then it’s just a “less than / greater than” check for every data value.
Check BOTH ends — outliers can be low, high, or both.
Boundary values are NOT outliers: the rule uses strict inequality.
“Keep or remove” needs CONTEXT: typo/error → remove. Genuine extreme value → keep.
Box plot drawing: whisker stops at the smallest/largest non-outlier; outliers become × marks beyond.

⚠ Common mistakes

Using 1.5 × range instead of 1.5 × IQR. It’s the interquartile range, not the full range.
Forgetting to halve… wait, just forgetting Q₁: students often compute only the upper fence and miss low outliers.
Treating fence values as outliers: a value at the fence is fine; it’s only an outlier if it goes past the fence.
Removing every outlier automatically: only remove genuine errors. A legitimate extreme value should be kept and discussed.
Drawing the box plot whisker to the outlier: the whisker must STOP at the most extreme non-outlier; the outlier is a separate × mark.

Next up: Box & Whisker Diagrams. You’ve already seen the box plot in action; now you’ll draw and read them formally — from the five-number summary (min, Q₁, median, Q₃, max), with outliers shown separately. Box plots are the fastest way to compare two distributions at a glance.

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