Tamil Nadu HSE First Year Statistics: Measures of Dispersion

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 = Σ|x−A|/n; it uses absolute deviations.
• Variance σ² = Σ(x−x̄)²/n; SD σ = √variance. For grouped data multiply squared deviations by frequencies.
• Coefficient of variation CV = (σ/x̄)×100, which is unit-free.
• SD is independent of change of origin but changes with scale; variance is in squared units.

Worked example

For 4, 6, 8, 10, 12: mean = 8. Deviations are −4, −2, 0, 2, 4; squares sum to 16+4+0+4+16 = 40. Variance = 40/5 = 8, SD = √8 ≈ 2.83. CV = 2.83/8×100 ≈ 35.4%. The series has moderate relative variability.

Traps

• Dividing by n−1 in this syllabus unless told; HSE typically uses n.
• Comparing SD alone across series with different means; use CV.
• Ignoring that range uses only two values.
• Forgetting QD covers the middle 50%, not all data.
• Reversing the CV formula or omitting the percentage.