Descriptive Statistics and Measures of Location and Dispersion
Domain 1: Descriptive Statistics and Measures of Location and Dispersion
This domain tests whether you can summarise a raw or grouped frequency distribution with one representative value and one measure of spread, and justify the choice. CHSE questions usually combine a computation (grouped mean, median, quartiles, standard deviation) with a short theory part on properties or on comparing two series.
Core rules
- For grouped data use class mid-values x_i with frequencies f_i: x̄ = Σf_i x_i / Σf_i; median = l + [(N/2 − cf)/f]·h; mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]·h.
- Empirical relation for moderately skewed data: Mode ≈ 3 Median − 2 Mean.
- Dispersion: range = L − S; QD = (Q₃ − Q₁)/2; MD about median = Σf|x − M|/N; σ = √(Σf x²/N − x̄²).
- SD is independent of change of origin but is multiplied by |a| under x → ax + b.
- CV = 100σ/x̄ is unit-free, so it is the correct tool for comparing variability of two series in different units.
- Combined group: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂) and σ² = [n₁(σ₁² + d₁²) + n₂(σ₂² + d₂²)]/(n₁ + n₂), where d_i = x̄_i − x̄.
Worked example
Group A: n₁ = 40, x̄₁ = 50, σ₁ = 10; Group B: n₂ = 60, x̄₂ = 60, σ₂ = 15. x̄ = (2000 + 3600)/100 = 56, so d₁ = −6 and d₂ = +4. σ² = [40(100 + 36) + 60(225 + 16)]/100 = (5440 + 14460)/100 = 199, giving σ = 14.11. Reading the formula backwards: the +d² terms are the between-group variation, and omitting them is the commonest error.
Traps
- Using class limits instead of mid-values; forgetting to convert inclusive classes such as 10–19 by subtracting 0.5.
- Taking N/2 for the median class but forgetting the cumulative frequency of the preceding class.
- MD about the mean is always larger than MD about the median, since the median minimises it.
- Comparing standard deviations of series with different means instead of their CVs.
- Applying Sheppard's correction to open-ended, discrete or strongly skewed data.