Measures of Central Tendency and Dispersion
What the exam tests
Questions ask you to compute mean, median and mode for raw or grouped data, and to measure spread with range, quartile deviation, mean deviation or standard deviation. Interpretation through the coefficient of variation is also tested.
Core rules
- Raw mean: x̄ = Σx/n. Grouped mean: x̄ = Σfx/Σf, where x is the class midpoint.
- Median (grouped): M = L + [(N/2 − cf)/f] × h, with L the lower limit of the median class.
- Mode (grouped): Mo = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h.
- Variance = Σ(x − x̄)²/n; standard deviation σ = √variance.
- Coefficient of variation CV = (σ/x̄) × 100%; the series with the larger CV is more variable.
Worked example
For 4, 6, 8, 10, 12: x̄ = 40/5 = 8. Deviations: −4, −2, 0, 2, 4; squared: 16, 4, 0, 4, 16; sum = 40. Variance = 40/5 = 8; σ = √8 ≈ 2.828. CV = (2.828/8) × 100 ≈ 35.36%. Median = 8 (middle value); mode is not unique here because every value occurs once.
Traps
- Using class limits instead of midpoints for grouped means.
- Forgetting the cumulative frequency in the median formula, or choosing the wrong median class.
- Dividing by n − 1 when the syllabus expects the population variance with n.
- Comparing standard deviations of series with very different means instead of using CV.
- Using the modal class's lower limit incorrectly when classes are inclusive; convert to exclusive classes first.