Measures of dispersion — A/L Combined Mathematics

What must I remember about Measures of dispersion?

• Measures of Dispersion (Variation): Statistical metrics quantifying the degree of spread, scatter, or variability of data values around their central location.
• Range: Difference between maximum and minimum values: Range = X_max - X_min (simple but sensitive to extreme outliers).
• Mean Absolute Deviation (MAD): Average distance of observations from mean: MAD = (∑ f_i * |x_i - x̄|) / N.
• Variance (σ² or s²): The arithmetic mean of squared deviations from the arithmetic mean: σ² = (1/N) ∑ f_i (x_i - x̄)² = [(1/N) ∑ f_i * x_i²] - (x̄)² (computational formula).
• Standard Deviation (σ or s): Positive square root of variance: σ = √[(1/N) ∑ f_i * x_i² - (x̄)²]; expressed in the SAME physical units as the original observations.

Test yourself (1 minute)

1. Effect of Linear Transformations

   If y_i = a * x_i + b: New mean ȳ = a * x̄ + b; New variance σ_y² = a² * σ_x² (constant b has zero effect on spread); New standard deviation σ_y = |a| * σ_x.

2. Coefficient of Variation (CV)

   Relative dispersion metric independent of measurement units: CV = (σ / x̄) * 100%; lower CV indicates greater consistency / reliability.

3. Combined Variance of two samples

   For sample 1 (size n1, mean x̄1, variance s1²) and sample 2 (size n2, mean x̄2, variance s2²): Combined variance s² = [n1(s1² + d1²) + n2(s2² + d2²)] / (n1 + n2) where d1 = x̄1 - x̄_c, d2 = x̄2 - x̄_c.

Practise Measures of dispersion questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 8

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Where does Measures of dispersion fit in the syllabus?

Year 2 · Term 3