Overview
This chapter has two halves: measures of dispersion (how spread out data is) and probability (chance of events). Both are scoring if your tables and substitutions are neat.
1. Measures of Dispersion
Range = Largest value − Smallest value
Variance (σ²) = (Σx²)/n − [(Σx)/n]² = mean of squares − square of mean
Standard deviation σ = √variance
Coefficient of variation CV = (σ / mean) × 100%
A smaller standard deviation (or CV) means the data is more consistent.
2. Probability
P(E) = favourable outcomes / total outcomes, 0 ≤ P(E) ≤ 1
P(impossible) = 0, P(certain) = 1
P(not E) = 1 − P(E)
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually exclusive: P(A ∩ B) = 0
Worked Examples
Example 1 — Standard deviation
Find σ for the data 2, 4, 6, 8, 10.
n = 5, Σx = 30 → mean = 6
Σx² = 4 + 16 + 36 + 64 + 100 = 220
variance = 220/5 − 6² = 44 − 36 = 8
σ = √8 ≈ 2.83
Example 2 — Coefficient of variation
Mean = 50, σ = 5. Find CV.
CV = (5/50) × 100 = 10%
Example 3 — Comparing consistency
A: mean 40, σ 4. B: mean 50, σ 4. Which is more consistent?
CV(A) = 4/40 × 100 = 10% ; CV(B) = 4/50 × 100 = 8%
B has the smaller CV → B is more consistent.
Example 4 — Simple probability
A die is thrown. Find P(even number).
Favourable = {2,4,6} = 3, total = 6
P(even) = 3/6 = 1/2
Example 5 — Addition rule
A card is drawn from 52. Find P(king or heart).
P(king) = 4/52, P(heart) = 13/52, P(king ∩ heart) = 1/52
P = 4/52 + 13/52 − 1/52 = 16/52 = 4/13
Common Mistakes
- Forgetting to subtract (mean)² in the variance formula.
- Leaving variance as the answer when standard deviation is asked.
- Adding probabilities of overlapping events without subtracting the intersection.
- Writing a probability greater than 1.
Exam Tips & Revision Checklist
- σ = √variance; CV compares consistency (smaller = better).
- For "or" probability, check whether events overlap before adding.
- Practised: range, variance, σ, CV, simple probability, addition rule.
Frequently Asked Questions
Standard deviation (σ) measures how far the values spread from the mean. A larger σ means more spread; σ = √variance.