Key Idea: This topic is about summarising a data set with a few numbers — an average (centre) and a spread. It runs through both papers: by-hand work on Paper 1 and 1-Var Stats on the GDC for Paper 2.
📊 The three averages
| Average | What it is | Example: 4, 7, 7, 9, 13 |
|---|---|---|
| Mean x̄ | Sum of values ÷ how many (n). | (4+7+7+9+13) ÷ 5 = 8 |
| Median | Middle value once the data is ordered. | 7 (the 3rd of 5) |
| Mode | Value that occurs most often. | 7 (appears twice) |
- each value (or class **midpoint** for grouped data)
- the frequency of that value
- the total frequency = how many data items
Multiply each value by its frequency, add up Σfx, then divide by the total frequency Σf — not by the number of rows. The mean also gives the total: sum = mean × n, which is how you find a missing value.
📦 Spread: range, IQR & standard deviation
- lower quartile — median of the lower half
- upper quartile — median of the upper half
- standard deviation — typical distance from the mean (variance = σ²)
| Measure | Uses | Outlier? |
|---|---|---|
| Range = max − min | The two extremes. | Sensitive |
| IQR = Q₃ − Q₁ | Middle 50% of the data. | Resists outliers |
| Std deviation σ | Every value, spread about the mean. | Pulled by extremes |
Add c to every value → mean + c, σ unchanged (the spread doesn't move). Multiply by k → mean and σ both × |k|. On the GDC read σx (population SD — the IB syllabus value), never Sx. If one value is far from the rest, quote the median, which resists outliers.
✏️ IB-style worked examples
IB-style question — mean from a frequency table (4.3.1)
The number of pets owned by 30 students is recorded: 0 → 11, 1 → 9, 2 → 7, 3 → 3. Find the mean number of pets.
Step by step:
Form Σfx — multiply each value by its frequency and add.
Divide by the total frequency Σf = 30.
Mean ≈ 1.07 pets.
IB-style question — estimated mean of grouped data (4.3.2)
Journey times (minutes) for 40 commuters are grouped: 0–10 → 6, 10–20 → 14, 20–30 → 12, 30–40 → 8. Estimate the mean and state the modal class.
Step by step:
Use class midpoints 5, 15, 25, 35 as the values, then form Σfx.
Divide by Σf = 40 for the estimated mean.
The modal class has the greatest frequency (14).
Estimated mean ≈ 20.5 min; modal class 10 ≤ t < 20.
IB-style question — median, quartiles & IQR (4.3.3)
Find the median, Q₁, Q₃ and the IQR of 6, 9, 9, 11, 14, 17, 21.
Step by step:
n = 7 (already ordered); the median is the 4th value.
Leave the median out: lower half 6, 9, 9 and upper half 14, 17, 21.
IQR is the spread of the middle 50%.
Median = 11, Q₁ = 9, Q₃ = 17, IQR = 8.
IB-style question — shift and scale the data (4.3.4)
A data set has mean 30 and standard deviation 6. Every value is decreased by 4, then the result is multiplied by 3. Find the new mean and standard deviation.
Step by step:
Subtract 4: the mean drops by 4, the spread (σ) is unchanged.
Multiply by 3: both the mean and σ are multiplied by |3|.
New mean = 78, new standard deviation = 18.
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Important: For Σfx ÷ Σf, the denominator is the total frequency (add up the f-column), not the number of different values. In the pets example divide by 30, not 4. The same trap hits grouped data — divide by Σf, and remember the answer is only an estimate because you used midpoints.
Tap each card to reveal the answer.
Median of 4, 8, 8, 10, 12, 15 9 — even n, so average the two middle values (8 and 10).
Five numbers have mean 12; four are 9, 11, 13, 15. Find the fifth. 12 — total = 12 × 5 = 60, then 60 − (9+11+13+15) = 12.
Q₁ = 9, Q₃ = 17 — what is the IQR? 8 — IQR = Q₃ − Q₁ = 17 − 9.
Which is unaffected by an outlier: range or IQR? IQR — it uses only the middle 50%, so an extreme value barely moves it.
Mean 20, σ = 4; add 5 to every value. New mean and σ? Mean = 25, σ = 4 — adding shifts the mean but leaves the spread unchanged.
Which 1-Var Stats output is the IB standard deviation? σx — the population standard deviation, not Sx.
Exam Tips
- Order the data before finding the median or quartiles; for odd n, leave the median out of both halves.
- Mean from a table = Σfx ÷ Σf — divide by the TOTAL frequency, not the number of rows.
- Grouped data: use class midpoints; the mean is an ESTIMATE because exact values are lost.
- Adding c leaves σ unchanged; multiplying by k multiplies the mean and σ by |k|.
- On Paper 2, one 1-Var Stats run gives x̄, σx and Q₁/Med/Q₃ — read σx, not Sx.