Measures of Central Tendency and Dispersion
This domain covers averages and spread: arithmetic mean, median, mode, weighted, geometric and harmonic means, and the dispersion measures - range, quartile deviation, mean deviation, standard deviation, variance and coefficient of variation. Expect both direct computations and questions that ask which measure is appropriate or how a measure changes under transformation.
Core rules
- Grouped mean by step deviation: x̄ = A + h × (Σfu/Σf), with u = (x − A)/h.
- Median of a grouped series = l + [(N/2 − cf)/f] × h.
- Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.
- Empirical relation: Mode = 3 Median − 2 Mean (moderately skewed data).
- SD σ = √[Σ(x − x̄)²/n]; variance is its square, so SD is in the original units.
- Coefficient of variation = (σ/x̄) × 100, unit-free and used to compare consistency.
Worked example
For 4, 6, 8, 10, 12 the mean is 40/5 = 8. Deviations are −4, −2, 0, 2, 4, squared 16, 4, 0, 4, 16, total 40. Variance = 40/5 = 8, so σ = √8 ≈ 2.83. CV = (2.83/8) × 100 ≈ 35.4%. A second series with mean 8 and σ = 1.5 would have CV ≈ 18.8% and is therefore more consistent.
Traps
Applying the empirical relation to heavily skewed data; using n − 1 instead of n for a population SD as the syllabus defines it; computing CV when the mean is zero; and assuming SD changes when a constant is added to every observation - the mean shifts, the SD does not.