Statistics and Probability
The statistics part tests central tendency and dispersion of grouped data; the probability part tests classical probability, the addition rule, and independence.
Grouped data
- Mean by step deviation: x̄ = a + h(Σfᵢuᵢ/Σfᵢ), where uᵢ = (xᵢ − a)/h.
- Median: l + [(N/2 − cf)/f]·h, with l the lower limit of the median class, cf the cumulative frequency before it, f its frequency, h the class width. The median class is the first class whose cumulative frequency reaches or exceeds N/2.
- Mode: l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]·h.
- Standard deviation: σ = √(Σfᵢxᵢ²/N − x̄²); variance is σ².
Probability
P(A) = n(A)/n(S). P(A∪B) = P(A) + P(B) − P(A∩B); for mutually exclusive events the last term vanishes. P(A′) = 1 − P(A). For independent events P(A∩B) = P(A)P(B). Conditional probability P(A|B) = P(A∩B)/P(B).
Worked example
Given P(A) = 0.5, P(B) = 0.3 and P(A∩B) = 0.2: P(A∪B) = 0.5 + 0.3 − 0.2 = 0.6, so P(A′∩B′) = P((A∪B)′) = 1 − 0.6 = 0.4. The events are not independent because 0.5 × 0.3 = 0.15 ≠ 0.2.
Traps that cost marks
- Picking the median class by looking at frequencies rather than cumulative frequency against N/2.
- Adding P(A) and P(B) without subtracting the overlap when the events are not mutually exclusive.
- Treating "independent" and "mutually exclusive" as the same idea.
- Using N instead of N/2, or the wrong h, in the median formula.