Measures of Dispersion
What the domain covers
This domain tests absolute and relative measures of dispersion: range, quartile deviation, mean deviation, standard deviation, variance and coefficient of variation. The exam expects correct formulas, interpretation and comparison of consistency between data sets.
Core rules
- Range = L − S; coefficient of range = (L − S)/(L + S).
- Quartile deviation QD = (Q3 − Q1)/2; coefficient = (Q3 − Q1)/(Q3 + Q1).
- Mean deviation about A = Σf|x − A|/Σf; it can be about mean, median or mode.
- Variance σ² = Σf(x − x̄)²/Σf; standard deviation σ = √variance. Shortcut: σ² = Σfx²/Σf − (x̄)².
- Coefficient of variation CV = (σ/x̄) × 100%; it is unit-free and used to compare consistency. Smaller CV means more consistent.
Worked example
For 4, 6, 8, 10, 12: mean = 8. Squared deviations are 16, 4, 0, 4, 16, total 40. Variance = 40/5 = 8, so σ = 2.828. CV = (2.828/8) × 100 = 35.35%. A second series with mean 50 and σ 10 has CV 20%, so it is relatively more consistent despite a larger absolute σ.
Traps
Never report a negative SD; variance is in squared units while SD is in original units. For grouped data use class midpoints with frequencies, not raw class limits. CV is a percentage, so do not compare it directly with σ. QD ignores extreme values and is not the same as standard deviation.