Topic 4 covers data analysis and probability theory. SL: statistics, probability rules, binomial and normal distributions. HL extends to Bayes\'s theorem, continuous random variables, probability density functions, and the Poisson distribution.
Measures of centre: mean, median, mode. Measures of spread: range, IQR, variance σ², standard deviation σ. For grouped data: estimated mean = Σfx / Σf. Cumulative frequency curves give median and quartiles. Box plots display the five-number summary.
P(A) = n(A)/n(S) for equally likely outcomes. Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B). Conditional probability: P(A|B) = P(A∩B)/P(B). Independent events: P(A∩B) = P(A)·P(B). Tree diagrams and Venn diagrams for visualisation.
X ~ B(n, p): n independent trials, each with success probability p. P(X = r) = C(n,r)·pʳ·(1−p)ⁿ⁻ʳ. Mean E(X) = np; Var(X) = np(1−p). Used when counting successes in fixed number of independent trials with constant probability.
X ~ N(μ, σ²): continuous, symmetric, bell-shaped. 68–95–99.7 rule. Standardisation: Z = (X − μ)/σ ~ N(0,1). Use GDC for P(X < a), inverse normal. Applications: modelling heights, test scores, measurement errors. HL: linear combinations of normal variables.
Use binomial when: counting discrete successes in a fixed number of independent trials with constant probability. Use normal when: modelling continuous data that is symmetric and bell-shaped (heights, errors, aggregated scores). The normal can also approximate the binomial when n is large and p is not too extreme (np > 5 and n(1−p) > 5).
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