IB Maths AI SL Topic 4 — Statistics Toolkit Paper 1 & 2 Range, 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.
Variance σ² = average of squared deviations from the mean. Units: data units².
Standard deviation σ = √variance. Same units as the data — the headline measure of spread.
GDC: 1-Var Stats gives x̄, σ_x, 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 (x_i − 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 σ_x; 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 — order 36, 37, 38, 39, 40, 41, 42, 43, 44, 45 Range range = 45 − 36 = 9 Quartiles: n = 10 → split into two halves of 5 lower half: 36, 37, 38, 39, 40 → Q₁ = 38 upper half: 41, 42, 43, 44, 45 → Q₃ = 43 IQR IQR = 43 − 38 = 5 range = 9 · IQR = 5 for 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 — mean x̄ = (12+15+18+14+16) / 5 = 75/5 = 15 Step 2 — deviations and squares 12 → −3 → 9 15 → 0 → 0 18 → 3 → 9 14 → −1 → 1 16 → 1 → 1 Step 3 — variance σ² = (9+0+9+1+1) / 5 = 20/5 = 4 Step 4 — std dev σ = √4 = 2 variance = 4 cm² · std dev = 2 cm variance 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.

Range 92 − 56 = 36 Median: n = 11 (odd) → 6th value median = 78 Quartiles: exclude median, each half has 5 lower half: 56, 62, 68, 71, 75 → Q₁ = 68 (3rd) upper half: 80, 83, 85, 88, 92 → Q₃ = 85 (3rd) IQR IQR = 85 − 68 = 17 range = 36 · median = 78 · IQR = 17 odd-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.

Mean x̄ = 15/5 = 3 Squared deviations (1−3)² = 4 (3−3)² = 0 (2−3)² = 1 (4−3)² = 1 (5−3)² = 4 Variance σ² = 10/5 = 2 Std dev σ = √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 each A: 100/5 = 20, B: 100/5 = 20 ✓ Player A — squared deviations from 20 4, 4, 0, 1, 1 → sum = 10 σ²_A = 10/5 = 2 → σ_A = √2 ≈ 1.41 Player B — squared deviations from 20 64, 64, 25, 25, 0 → sum = 178 σ²_B = 178/5 = 35.6 → σ_B ≈ 5.97 Compare σ_A < σ_B → A is more consistent Player 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 outlier 280, 290, 295, 305, 310, 320, 850 range = 850 − 280 = 570 median = 305 (4th) lower half: 280, 290, 295 → Q₁ = 290 upper half: 310, 320, 850 → Q₃ = 320 IQR = 320 − 290 = 30 (b) Remove the outlier 280, 290, 295, 305, 310, 320 range = 320 − 280 = 40 Q₁ = 290, Q₃ = 310 IQR = 310 − 290 = 20 Comment range collapsed 570 → 40 (massive change) IQR barely moved 30 → 20 IQR is far more robust to outliers than the range this 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 σ_x (population), not s_x (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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