ConceptProbability & Statistics

Key formulas

Classical probability: P(A) = n(A)/n(S), 0 ≤ P(A) ≤ 1; P(certain)=1, P(impossible)=0
Combinations C(n,k)=n!/(k!(n−k)!); permutations A(n,k)=n!/(n−k)!
Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B)
Complement: P(A^c) = 1 − P(A); at-least-one: P(≥1) = 1 − P(none)
Conditional: P(A|B) = P(A∩B)/P(B); Multiplication: P(A∩B)=P(A)P(B|A); Independence: P(A∩B)=P(A)P(B)
Mean: x̄ = (Σ x_i)/n; weighted/combined mean weights by group sizes
Variance: s² = (1/n) Σ (x_i − x̄)²; standard deviation s = √s²; range = x_max − x_min
Linear transform: E(aX+b)=aE(X)+b; D(aX+b)=a² D(X)
Pooled variance of two groups: weight within-group variances by size AND add between-group (mean-shift) term
Normal PDF: f(x) = 1/(σ√2π) · e^−(x−μ²/(2σ²)); curve symmetric about μ; total area = 1
Empirical rule: ≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ
Standard normal: Z = (x−μ)/σ; Φ(x)+Φ(−x)=1
Estimated count in an interval = P(interval) × population size
Least-squares slope: b̂ = (Σ x_i y_i − n x̄ ȳ)/(Σ x_i² − n x̄²); intercept â = ȳ − b̂ x̄
Normal approximation to binomial (De Moivre-Laplace): X~B(n,p) ≈ N(np, np(1−p))

If the concept underneath still feels loose, patch it here before returning to CSCA questions.

Combining strata/groups

pooled mean and pooled variance (between + within), size-ratio weighting

No misconceptions are logged for this concept yet, so there is nothing named here to trip you up. Practice it and anything that does catch you will be added.

Practice this concept →