Measures of Central Tendency
What the domain covers
The exam tests computation and interpretation of arithmetic mean, median, mode, weighted mean, geometric mean and harmonic mean, including grouped data. You should know which average suits skewed data and how the empirical relation links mean, median and mode.
Core rules
- Direct mean for a frequency distribution: x̄ = Σfx/Σf. Step-deviation: x̄ = A + h(Σfd/Σf).
- Median (grouped): M = L + [(N/2 − cf)/f] × h, where L is the lower boundary of the median class, cf the cumulative frequency before it, f its frequency and h its width.
- Mode (grouped): Z = L + [(f1 − f0)/(2f1 − f0 − f2)] × h.
- Weighted mean uses Σwx/Σw; geometric mean is the nth root of the product; harmonic mean is n/Σ(1/x).
- For a moderately skewed distribution, Mode ≈ 3 Median − 2 Mean.
Worked example
Classes 0–10, 10–20, 20–30, 30–40 have frequencies 5, 8, 12, 5; N = 30, so N/2 = 15. Cumulative frequencies are 5 and 13, so the median class is 10–20. M = 10 + [(15 − 5)/8] × 10 = 10 + 12.5 = 22.5. Mean from midpoints is (5×5 + 15×8 + 25×12 + 35×5)/30 = 650/30 ≈ 21.67.
Traps
Use N/2, not N, in the median formula, and use the cumulative frequency before the median class. Do not use class marks as boundaries. The empirical relation is an approximation, not an identity. A mean can be pulled by outliers, so report median for skewed income data.