IB Maths AI SLTopic 4 — Statistics ToolkitPaper 1 & 2Range, IQR, std dev~7 min read
Measures of Dispersion
Two data sets can share the same mean and still tell totally different stories — one tightly grouped, the other wildly spread out. Dispersion measures fill that gap: range, interquartile range (IQR), variance, and standard deviation. Each captures spread in its own way.
📘 What you need to know
Range = max − min. Simple, but very sensitive to outliers.
Quartiles: Q1 (lower 25% cut-off), Q2 (median), Q3 (upper 25% cut-off). Find them by splitting the ordered data at the median.
Interquartile range IQR = Q3 − Q1. Captures the middle 50% of the data — barely affected by outliers.
Varianceσ2 = average of squared deviations from the mean. Units: data units².
Standard deviationσ = √variance. Same units as the data — the headline measure of spread.
GDC: 1-Var Stats gives x̄, σx, Q1, median, Q3 in one go. Use it on Paper 2.
Range, quartiles & IQR
To find quartiles by hand, order the data, then split at the median:
• If n is even: lower half is the first n/2 values; upper half is the last n/2.
• If n is odd: exclude the median itself; lower half is everything below it, upper half everything above.
Q1 is the median of the lower half. Q3 is the median of the upper half.
Variance & standard deviation
Range and IQR only use a few values. Standard deviation uses every value, measuring the average distance from the mean.
The recipe: subtract the mean from each value (the “deviation”), square it, average those squares (that’s variance), then take the square root.
Both data sets have mean = 20, but Data A is tightly grouped (small σ) while Data B is widely spread (large σ). The shaded bands span ±1σ from the mean — the wider the band, the more dispersed the data.
The dispersion formulae
range = max − min · IQR = Q3 − Q1
variance σ2 = Σ (xi − x̄)2n · std dev σ = √σ2
🧭 Recipe — tackle any dispersion question
Order the data from smallest to largest. Required before any quartile work.
Range: subtract smallest from largest — one line.
Quartiles: split at the median; Q1 = median of lower half, Q3 = median of upper half.
IQR: Q3 − Q1.
Std dev / variance: enter the list in the GDC, run 1-Var Stats, read off σx; square it for variance.
Outlier-robust? Range and standard deviation are pulled hard by outliers. The IQR ignores them by construction — it only uses the middle 50%. That’s why box plots use IQR.
Worked examples
WE 1
Range and IQR from raw data
A coffee shop records its daily sales (in hundreds) over 10 days:
38, 42, 36, 45, 41, 39, 44, 37, 43, 40
Find the range and the IQR.
Step 1 — order36, 37, 38, 39, 40, 41, 42, 43, 44, 45Rangerange = 45 − 36 = 9Quartiles: n = 10 → split into two halves of 5lower half: 36, 37, 38, 39, 40 → Q₁ = 38upper half: 41, 42, 43, 44, 45 → Q₃ = 43IQRIQR = 43 − 38 = 5range = 9 · IQR = 5for even n, each half has n/2 values. Q₁ and Q₃ are their medians.
WE 2
Variance and standard deviation by hand
The heights (cm) of 5 seedlings are:
12, 15, 18, 14, 16
Find the mean, the variance, and the standard deviation.
Step 1 — meanx̄ = (12+15+18+14+16) / 5 = 75/5 = 15Step 2 — deviations and squares12 → −3 → 915 → 0 → 018 → 3 → 914 → −1 → 116 → 1 → 1Step 3 — varianceσ² = (9+0+9+1+1) / 5 = 20/5 = 4Step 4 — std devσ = √4 = 2variance = 4 cm² · std dev = 2 cmvariance has units cm² (squared!). Std dev is in the same units as the data — more interpretable.
WE 3
Range, median & IQR from 11 values
The test scores of 11 students (already in order) are:
56, 62, 68, 71, 75, 78, 80, 83, 85, 88, 92
Find the range, median, and IQR.
Range92 − 56 = 36Median: n = 11 (odd) → 6th valuemedian = 78Quartiles: exclude median, each half has 5lower half: 56, 62, 68, 71, 75 → Q₁ = 68 (3rd)upper half: 80, 83, 85, 88, 92 → Q₃ = 85 (3rd)IQRIQR = 85 − 68 = 17range = 36 · median = 78 · IQR = 17odd-n case: drop the median from both halves before finding Q₁ and Q₃. The middle 50% lies between 68 and 85.
WE 4
Standard deviation involving a surd
A football team scores the following goals in 5 matches:
1, 3, 2, 4, 5
Find the variance and standard deviation, giving the std dev to 3 sf.
Meanx̄ = 15/5 = 3Squared deviations(1−3)² = 4(3−3)² = 0(2−3)² = 1(4−3)² = 1(5−3)² = 4Varianceσ² = 10/5 = 2Std devσ = √2 ≈ 1.414…variance = 2 · σ ≈ 1.41 goals (3 sf)when σ isn’t a whole number, exams ask for 3 sf. Keep √2 exact through the calculation and round only at the end.
WE 5
Compare two data sets — which is more consistent?
Two basketball players’ last five game scores:
Player A: 18, 22, 20, 19, 21
Player B: 12, 28, 15, 25, 20
Both means equal 20. Calculate each std dev and decide who is more consistent.
Mean of eachA: 100/5 = 20, B: 100/5 = 20 ✓Player A — squared deviations from 204, 4, 0, 1, 1 → sum = 10σ²_A = 10/5 = 2 → σ_A = √2 ≈ 1.41Player B — squared deviations from 2064, 64, 25, 25, 0 → sum = 178σ²_B = 178/5 = 35.6 → σ_B ≈ 5.97Compareσ_A < σ_B → A is more consistentPlayer A more consistent (σ ≈ 1.41 vs 5.97)same mean ≠ same data. Std dev tells the rest of the story: smaller σ → tighter, more reliable scores.
WE 6
Outlier sensitivity: range vs IQR
Seven recent house sale prices in a street (in $000s):
280, 310, 295, 320, 305, 290, 850
(a) Find the range and IQR. (b) The 850 is a mansion — an outlier. Recompute with it removed, and comment.
(a) Order with the outlier280, 290, 295, 305, 310, 320, 850range = 850 − 280 = 570median = 305 (4th)lower half: 280, 290, 295 → Q₁ = 290upper half: 310, 320, 850 → Q₃ = 320IQR = 320 − 290 = 30(b) Remove the outlier280, 290, 295, 305, 310, 320range = 320 − 280 = 40Q₁ = 290, Q₃ = 310IQR = 310 − 290 = 20Commentrange collapsed 570 → 40 (massive change)IQR barely moved 30 → 20IQR is far more robust to outliers than the rangethis is exactly why IQR is preferred for skewed or outlier-heavy data. Range is dramatic but misleading.
💡 Top tips
Always order before quartiles. Forgetting this is the most common mark-loser.
Use the GDC’s 1-Var Stats for Paper 2 — mean, std dev, Q₁, median, Q₃ in seconds.
Std dev units = data units; variance units = (data units)². State the units.
For odd n, exclude the median value when finding Q₁ and Q₃.
Same mean, different spread: always report std dev or IQR alongside the mean — the mean alone is half the story.
⚠ Common mistakes
Range = max + min: it’s max − min. Don’t add.
Forgetting to square root the variance when asked for std dev. σ² = 25 → σ = 5, not 25.
Wrong std dev on the GDC: pick σx (population), not sx (sample). IB uses the population formula.
Computing IQR from the whole list instead of from the quartiles. IQR is always Q3 − Q1.
Reporting variance with wrong units: variance is in squared units — write “cm²” not “cm”.
Next up: Frequency Tables. When data is repeated, we don’t list every value — we group them with their frequencies. The mean / median / mode / std dev formulae still apply, but adapted to use frequencies. The GDC has a “List with Frequency” mode that does the heavy lifting.
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