Statistics
Statistics
The Statistics section tests descriptive summary, probability rules, standard distributions, correlation and regression, and the interpretation of survey or index-number data. ACET favours conceptual checks: can you pick the right summary, the right probability rule, and the right interpretation?
Core rules
- Centre and spread: the mean is pulled by outliers, the median is not. For a sample, variance uses n − 1; for a population it uses n.
- Coefficient of variation: CV = (standard deviation ÷ mean) × 100. It is unit-free, so it compares variability across datasets with different scales.
- Probability: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for independent events P(A ∩ B) = P(A)P(B); conditional probability is P(A | B) = P(A ∩ B) ÷ P(B).
- Distributions: binomial has mean np and variance np(1 − p); Poisson has mean = variance = λ; normal data follow roughly 68–95–99.7 within 1, 2 and 3 standard deviations.
- Association: Pearson r lies in [−1, 1] and measures linear association only; the least-squares line passes through (x̄, ȳ) with slope b = r × (s_y ÷ s_x).
Worked example
A random variable has E[X] = 4 and Var(X) = 9. Find E[2X + 3] and Var(2X + 3).
Expectation is linear: E[2X + 3] = 2(4) + 3 = 11. For variance, an additive constant shifts nothing: Var(2X + 3) = 2² × Var(X) = 4 × 9 = 36. The common error is writing 2 × 9 = 18, forgetting that variance scales by the square of the multiplier.
Traps
Mixing n and n − 1; reading r = 0 as no relationship at all; extrapolating a regression line far beyond the data; using the mean on clearly skewed data; and confusing conditional with joint probability.