BCS Written Statistics cheat sheet

Axioms of Kolmogorov: P(A) ≥ 0; P(Ω) = 1; for disjoint events P(∪Aᵢ) = ΣP(Aᵢ). Inclusion–exclusion: P(A∪B) = P(A) + P(B) − P(A∩B).

Conditional probability P(A|B) = P(A∩B)/P(B) for P(B) > 0; independence holds iff P(A∩B) = P(A)P(B). Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B).

Random variable X has expectation E(X) = Σx·P(x) or ∫x·f(x)dx; Var(X) = E(X²) − [E(X)]²; for constants a,b: E(aX+b) = aE(X)+b, Var(aX+b) = a²Var(X).

Binomial(n,p): mean np, variance np(1−p); Poisson(λ): mean and variance both λ, approximates Binomial when n large and p small with λ = np.

Normal N(μ,σ²): standardize Z = (X−μ)/σ; 68–95–99.7 rule covers approximately 68%, 95%, and 99.7% within 1, 2, 3 standard deviations.

Central Limit Theorem: for i.i.d. variables with mean μ and variance σ², the sample mean is approximately N(μ, σ²/n) as n grows, regardless of the population shape.

Exponential(λ): mean 1/λ, variance 1/λ², memoryless property P(X > s+t | X > s) = P(X > t); useful for waiting times.

Law of total probability: P(B) = ΣP(B|Aᵢ)P(Aᵢ) over a partition {Aᵢ}; joint, marginal, and conditional distributions relate by f(x,y) = f(y|x)f(x).

Estimator quality: unbiasedness E(θ̂) = θ; efficiency is lower variance among unbiased estimators; consistency means θ̂ converges in probability to θ as n→∞.

Standard error of the mean is σ/√n; when σ is unknown use s/√n with the t distribution on n−1 degrees of freedom.

Confidence interval for a mean (σ known): x̄ ± z(1−α/2)·σ/√n; for a proportion: p̂ ± z(1−α/2)·√(p̂(1−p̂)/n).

Type I error α rejects a true null; Type II error β fails to reject a false null; power = 1 − β increases with larger n, larger effect size, and larger α.

Test statistic for a mean: t = (x̄ − μ₀)/(s/√n); for two independent means use pooled variance when equal variances are assumed, otherwise Welch's approximation.

Chi-square goodness-of-fit statistic Σ(O−E)²/E with df = k−1−m, where m is the number of estimated parameters; expected counts should generally be ≥5.

p-value is the probability, under the null, of a statistic at least as extreme as observed; reject H₀ when p < α. It is not the probability that H₀ is true.

Maximum likelihood estimators maximize the likelihood (or log-likelihood); MLEs are consistent and asymptotically normal with variance given by the inverse Fisher information.

Randomization, replication, and blocking are the three core design principles; randomization protects against confounding by unmeasured factors.

One-way ANOVA partitions total sum of squares: SST = SSB + SSW, with df between = k−1 and df within = N−k; F = (SSB/(k−1))/(SSW/(N−k)).

ANOVA F-test assumes independent errors, normality within groups, and equal variances; Levene's or Bartlett's test checks homogeneity of variance.

Repeated-measures and randomized block designs remove subject or block variability, increasing power; they require sphericity assumptions for the F-test.

Factorial designs estimate main effects and interactions; a 2^k design has k factors at two levels and 2^k treatment combinations; interaction means effects are not additive.

Multiple comparisons: Tukey's HSD controls family-wise error for all pairwise means; Bonferroni divides α by the number of comparisons.

Sample size for detecting a difference in means depends on effect size, variance, α, and power; larger variance or smaller effect requires larger n.

Fixed vs random effects: fixed effects apply only to the levels studied; random effects treat levels as a sample from a population, changing the error term (EMS).

Simple random sampling (SRS): each unit has equal selection probability; sample mean is unbiased with variance (1−n/N)·S²/n, where (1−n/N) is the finite population correction.

Stratified sampling divides the population into homogeneous strata and samples within each; proportional allocation minimizes variance when stratum variances are similar; Neyman allocation is proportional to stratum standard deviation.

Cluster sampling selects groups (clusters) rather than individuals; it is cheaper but usually less precise than SRS because of intra-cluster correlation (design effect).

Systematic sampling selects every k-th unit after a random start; periodicity in the frame can bias results if the sampling interval aligns with the cycle.

Sampling and non-sampling errors: sampling error shrinks with n; non-sampling errors include nonresponse, coverage, measurement, and processing errors.

Design effect (deff) = variance under the design / variance under SRS; effective sample size = n/deff; used to adjust precision for complex designs.

Ratio and regression estimators use auxiliary information correlated with the variable of interest; they are biased in small samples but efficient when the correlation is strong.

Nonresponse bias occurs when respondents differ systematically from nonrespondents; weighting adjustments and follow-up attempts can reduce it.

Simple linear regression: ŷ = b₀ + b₁x, where b₁ = Sxy/Sxx and b₀ = ȳ − b₁x̄; residuals e = y − ŷ have mean zero and are orthogonal to fitted values.

Coefficient of determination R² = SSR/SST is the proportion of variance explained; adjusted R² penalizes adding predictors: 1 − (1−R²)(n−1)/(n−k−1).

OLS assumptions: linearity, independent errors, homoscedasticity, normality of errors, and no perfect multicollinearity; violations require transformation, robust SEs, or weighted least squares.

Multicollinearity inflates standard errors and makes coefficients unstable; variance inflation factor VIF = 1/(1−R²ⱼ); values above about 10 signal serious problems.

Logistic regression models log-odds: log(p/(1−p)) = β₀ + β₁x; coefficients are interpreted as changes in log-odds, and exp(β) gives odds ratios.

Principal component analysis finds orthogonal linear combinations that maximize variance; components are ordered by eigenvalue, and the first component explains the most variance.

Factor analysis models observed variables as linear combinations of latent factors plus error; rotation (varimax) aids interpretation of factor loadings.

Discriminant analysis and MANOVA handle multiple dependent variables; Hotelling's T² is the multivariate analogue of the t statistic.

Time series components: trend (long-term direction), seasonality (repeating pattern), cyclical (longer irregular swings), and irregular (random) variation; additive vs multiplicative models.

Stationarity means constant mean, variance, and autocovariance over time; the Augmented Dickey–Fuller test checks for a unit root (nonstationarity).

Differencing removes trend (first difference) and seasonality (seasonal difference at lag s); ARIMA(p,d,q) combines autoregressive, integrated (d differences), and moving average terms.

Autocorrelation function (ACF) and partial autocorrelation function (PACF) guide model identification: AR(p) cuts off at lag p in the PACF; MA(q) cuts off at lag q in the ACF.

Moving average smoothing with window m averages the last m observations; exponential smoothing weights recent observations more heavily with smoothing constant α (0 < α < 1).

Seasonal decomposition: seasonal index = observed value / moving average (multiplicative) or observed minus moving average (additive); seasonal adjustment removes the seasonal component.

Forecast accuracy measures: MAE, MSE, RMSE, and MAPE; MAPE is undefined when actual values are zero and is asymmetric.

Ljung–Box test checks whether autocorrelations up to a given lag are jointly zero, testing for remaining structure in residuals.

Crude birth rate = (live births / mid-year population) × 1000; crude death rate = (deaths / mid-year population) × 1000; both are affected by age structure.

Age-specific fertility rate (ASFR) = births to women aged a / women aged a × 1000; total fertility rate (TFR) = sum of ASFRs × age-interval width / 1000.

Life expectancy at birth e₀ is the average years lived from birth under current mortality; life tables use age-specific death rates (qₓ) and survivorship (lₓ).

Infant mortality rate = deaths under age 1 / live births × 1000; under-5 mortality extends the measure to age five.

Population growth: natural increase = births − deaths; net migration adds or subtracts; exponential growth P(t) = P₀e^{rt} where r is the growth rate.

Standardization: direct standardization applies a study population's age-specific rates to a standard age structure; indirect standardization uses a standard set of rates (SMR).

Census and vital registration are the main official sources; intercensal estimates use cohort-component or ratio methods; demographic dividend refers to a favorable age structure.

Dependency ratio = (population aged 0–14 + 65+) / population aged 15–64 × 100; sex ratio = males/females × 100.

Reproducibility: set random seeds for simulations; document software versions; separate data cleaning, analysis, and reporting scripts.

Data types matter: numeric, factor (categorical), and character; factors must be converted before regression, and missing values (NA) require explicit handling.

Common R/Python functions: mean, sd, var, t.test, lm, aov, chisq.test; Python equivalents include numpy.mean, scipy.stats.ttest_ind, statsmodels.OLS.

Resampling: the bootstrap estimates standard errors and confidence intervals by resampling with replacement; permutation tests assess significance without distributional assumptions.

Cross-validation splits data into training and test folds (k-fold, leave-one-out) to estimate out-of-sample error and guard against overfitting.

Simulation: Monte Carlo methods generate random samples to approximate integrals and distributions; variance of the estimate decreases roughly as 1/√n.

Numerical stability: avoid subtracting nearly equal numbers; scale variables before iterative algorithms; check convergence of optimization routines.

Data visualization and diagnostics: residual plots, Q–Q plots, boxplots, and histograms reveal outliers, skewness, and assumption violations before formal testing.