Measures of Dispersion
What the exam tests
Dispersion measures spread: range, quartile deviation, mean deviation, variance and standard deviation, plus relative measures such as coefficient of variation. Expect computation for raw or grouped data, comparison of two series, and combined variance.
Core rules
- Range = max − min. Coefficient of range = (max − min)/(max + min).
- Quartile deviation QD = (Q3 − Q1)/2; coefficient = (Q3 − Q1)/(Q3 + Q1). Q1 = L + [(N/4 − cf)/f]h; Q3 = L + [(3N/4 − cf)/f]h.
- Mean deviation about A: MD = Σf|x − A|/N. It is least when A is the median.
- Variance σ² = Σf(x − x̄)²/N = Σfx²/N − x̄². Standard deviation σ = √variance.
- Coefficient of variation CV = (σ/x̄)×100%; it compares variability of series with different units or means.
- Combined variance of two groups: σ² = [n1(σ1² + d1²) + n2(σ2² + d2²)]/(n1 + n2), where d1 = x̄1 − x̄, d2 = x̄2 − x̄ and x̄ is the combined mean.
- Adding a constant leaves SD unchanged; multiplying by k multiplies SD by |k|.
Worked example
Series A: mean 50, SD 5, so CV = 10%. Series B: mean 80, SD 6, so CV = 7.5%. B is relatively more consistent although its SD is larger. For 2, 4, 6, 8, 10: mean = 6; squared deviations 16, 4, 0, 4, 16 sum to 40; σ² = 40/5 = 8; σ = 2.83.
Traps
Dividing by n − 1 when the paper expects the population formula with N; using mean deviation about the mean when the question specifies median; comparing SDs alone for series with different means; forgetting the positive root; using the grouped quartile formula when raw-data ranking is intended.