Measures of Central Tendency
What the exam tests
Measures of central tendency locate the centre of a distribution: arithmetic mean, median, mode, geometric mean (GM) and harmonic mean (HM). Expect direct computation for raw and grouped data, weighted means, and properties.
Core rules
- Arithmetic mean: raw x̄ = Σx/n; grouped x̄ = Σfx/Σf. Step-deviation form: x̄ = A + h(Σfu/Σf), where u = (x − A)/h.
- Median: arrange raw data in order; take the middle value if n is odd and the average of the two middle values if n is even. Grouped: median = L + [(N/2 − cf)/f]h, with L the lower boundary of the median class, cf the cumulative frequency before it, f its frequency and h the width.
- Mode: the most frequent value. Grouped: mode = L + [(f1 − f0)/(2f1 − f0 − f2)]h, with f1 the modal frequency and f0, f2 the frequencies of the preceding and following classes.
- Weighted mean = Σwx/Σw.
- GM = (x1x2...xn)^(1/n); HM = n/Σ(1/x). For positive values, AM ≥ GM ≥ HM, with equality only when all values are equal.
- The median resists outliers; the mean does not. For a moderately skewed distribution, mode ≈ 3 median − 2 mean.
Worked example
For 5, 7, 8, 10, 20: mean = 50/5 = 10; the ordered median is 8; no value repeats, so the mode is ill-defined. For grouped data with N = 40, median class 20–30, cf before = 15, f = 10, L = 20, h = 10: median = 20 + [(20 − 15)/10]×10 = 25. The median class is the first class whose cumulative frequency reaches N/2 = 20.
Traps
Using N instead of N/2 in the median formula; taking cf of the median class rather than the preceding class; using class limits instead of boundaries for L; applying the empirical relation to strongly skewed data; ignoring that GM and HM require positive values.