Measures of dispersion — A/L Combined Mathematics

Measures of dispersion quantify the extent of spread, scatter, or variability of data values around their central mean: (1) Range: Difference between maximum and minimum values (X_max - X_min); (2) Mean Absolute Deviation: (∑ f_i |x_i - x̄|) / N; (3) Variance: Mean squared deviation from arithmetic mean: σ² = [∑ f_i (x_i - x̄)²] / N = [ (∑ f_i x_i²) / N ] - x̄²; (4) Standard Deviation: Positive square root of variance: σ = √Variance; (5) Coefficient of Variation (relative dispersion for comparing datasets with different units or means): CV = (σ / x̄) × 100%. Under linear transformation y = ax + b: ȳ = a x̄ + b, Variance(y) = a² Variance(x), and σ_y = |a| σ_x (location shifts b do not alter dispersion).

Check before you answer (A/L Applied Maths Paper 2)

1. Alternative Variance Formula: Applying σ² = (∑ f_i x_i² / N) - x̄² to compute variance efficiently without individual deviations.
2. Linear Coding Properties: Explaining why adding a constant b leaves variance unchanged, while multiplying by a scales variance by a² and standard deviation by |a|.
3. Consistency Comparison using CV: Comparing two groups (e.g., student marks or factory outputs) and concluding that the smaller CV exhibits greater consistency/uniformity.