Measures of Central Tendency, Dispersion and Correlation
The final domain covers averages, spread and association. Expect a computation of mean by step deviation, an interpretation of standard deviation or coefficient of variation, and a conceptual question on correlation.
Core rules
- Mean can be computed by direct, assumed-mean or step-deviation methods; step deviation uses d′ = (X − A)/h and multiplies the result by h.
- Median is the positional average, best for open-end classes; mode is the most frequent value. Empirical relation: Mode = 3 Median − 2 Mean.
- Properties: Σ(X − X̄) = 0; the mean is affected by extreme values; the median is not.
- Dispersion: range, quartile deviation, mean deviation, variance and standard deviation. SD is the positive square root of variance; SD is independent of a change of origin but changes with a change of scale.
- Coefficient of variation = (SD/Mean) × 100; it compares variability across series with different means. A lower CV means greater consistency.
- Correlation: Karl Pearson r = Σxy/(nσxσy), lying between −1 and +1. Spearman rank correlation rk = 1 − 6ΣD²/(n(n² − 1)), with ranks adjusted for ties.
Worked example
For 5, 10, 15, 20, 25 the mean is 15. Deviations are −10, −5, 0, 5, 10; squares sum to 250; variance = 250/5 = 50; SD = 7.07; CV = 7.07/15 × 100 ≈ 47.1%. A second series with mean 30 and SD 7.07 has CV ≈ 23.6%, so the first series is more variable and the second more consistent despite identical SD.
Traps that cost marks
- Forgetting to multiply by h in the step-deviation method.
- Computing median from inclusive classes without conversion.
- Reporting variance as SD, or treating a high SD as high variability without checking the mean.
- Concluding causation from a high r; correlation measures association only.