Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Centre tells you a typical value; variability tells you how much values differ.
- Mean x̄ uses every value. Median is the middle of the ordered data.
- Q1 and Q3 are the medians of the lower and upper halves; they bound the middle 50%.
- The p-th percentile has p% of the values at or below it. Q1 = 25th, Q3 = 75th percentile.
- Standard deviation s is a typical distance of the values from the mean.
Key relationships
| Statistic | How to find it | Resistant? |
|---|---|---|
| Mean | x̄ = Σxᵢ / n | No |
| Median | middle value (n odd) or mean of two middle values (n even) | Yes |
| Range | max − min | No |
| IQR | Q3 − Q1 | Yes |
| Sample standard deviation | s = √[ Σ(xᵢ − x̄)² / (n − 1) ]; variance = s² | No |
| 1.5 × IQR outlier rule | below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR | uses resistant statistics |
| 2s outlier rule | below x̄ − 2s or above x̄ + 2s | uses non-resistant statistics |
| Multiply data by k (k > 0) | centre, position, range, IQR and s all × k; variance × k² | — |
| Add c to data | centre and position + c; range, IQR and s unchanged | — |
Assumptions and conventions
- The values are measurements of one quantitative variable on the same units.
- Quartiles: split at the median; when n is odd, leave the median out of both halves (graphing-calculator method). Other methods exist, so show your halves.
- s is the sample standard deviation (n − 1). On a calculator it is Sx.
- An outlier rule only flags a potential outlier. Investigate it; do not delete it unless it is an error.
Mistakes to avoid
- Not ordering the data before finding the median or quartiles.
- IQR = Q3 − median or confusing IQR with range.
- Reporting σx (divides by n) instead of Sx.
- Forgetting the square root: s² is the variance, in squared units.
- No context: always name the variable, the individuals and the units.
- Choosing the mean for skewed data or data with outliers. Prefer the median and IQR.
- Thinking unit changes leave s unchanged: multiplying the data multiplies s.
Quick self-check
- Data (kg): 4, 7, 7, 9, 13. What is the IQR? (Q1 = 5.5, Q3 = 11, IQR = 5.5 kg)
- A data set has Q1 = 30 s and Q3 = 42 s. Is 61 s a potential outlier by the 1.5 × IQR rule? (Yes: upper fence = 42 + 18 = 60 s, and 61 > 60)
- Lengths have mean 12 cm and s = 3 cm. What are the mean and s in millimetres? (120 mm and 30 mm)
Next: practice questions.