Statistics Formulas

Statistics Formula Reference

Complete collection of statistical formulas for descriptive statistics, probability, hypothesis testing, and statistical inference

Quick ReferencePractical ApplicationsClear Explanations

Quick Formula Reference

Essential statistical formulas for quick lookup and reference

Descriptive Statistics

Mean (Average)

$x̄ = Σx / n$

Sample Variance

$s² = Σ(x - x̄)² / (n-1)$

Standard Deviation

$s = √(s²)$

Population Variance

$σ² = Σ(x - μ)² / N$

Probability Basics

Basic Probability

$P(A) = favorable outcomes / total outcomes$

Complement Rule

$P(A') = 1 - P(A)$

Addition Rule

$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$

Multiplication Rule

$P(A ∩ B) = P(A) × P(B|A)$

Normal Distribution

Z-Score

$z = (x - μ) / σ$

Standard Normal PDF

$φ(z) = (1/√2π) × e^(-z²/2)$

Normal Distribution PDF

$f(x) = (1/(σ√2π)) × e^(-(x-μ)²/2σ²)$

Central Limit Theorem

$x̄ ~ N(μ, σ²/n)$

Confidence Intervals

Mean (σ known)

$x̄ ± z(α/2) × (σ/√n)$

Mean (σ unknown)

$x̄ ± t(α/2) × (s/√n)$

Proportion

$p̂ ± z(α/2) × √(p̂(1-p̂)/n)$

Margin of Error

$E = critical value × standard error$

Probability Theory Fundamentals

Essential mathematical formulas for probability theory foundations and classical probability models

Probability Theory Formula Reference

Beginner to Intermediate

Comprehensive formulas for random experiments, events, conditional probability, independence, and Bernoulli models

Key Formulas:

$Classical Probability: P(A) = |A|/|Ω|$

$Bayes' Theorem: P(A|B) = P(B|A)P(A)/P(B)$

$Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$

$Independence: P(A ∩ B) = P(A) × P(B)$

$Binomial: P(X=k) = C(n,k)p^k(1-p)^(n-k)$

Applications:

Classical probability

Medical diagnosis

Quality control

Risk assessment

Distribution Families

Complete mathematical reference for probability distribution families and their properties

Common Distribution Families & Properties

Intermediate

Comprehensive formulas for binomial, Poisson, normal, gamma, chi-square, t-distribution, F-distribution, beta, and exponential family theory

Key Formulas:

$Binomial: P(X=k) = C(n,k) p^k (1-p)^(n-k)$

$Poisson: P(X=k) = (λ^k e^(-λ)) / k!$

$Normal: f(x) = (1/(σ√(2π))) exp(-(x-μ)²/(2σ²))$

$Gamma: f(x) = (λ^α/Γ(α)) x^(α-1) e^(-λx)$

$Chi-square: f(x) = (1/(2^(n/2)Γ(n/2))) x^(n/2-1) e^(-x/2)$

Applications:

Probability modeling

Statistical inference

Parameter estimation

Distribution theory

Mathematical Statistics

Fundamental formulas for statistical inference and mathematical statistics theory

Population vs Sample

Beginner

Key formulas distinguishing population parameters from sample statistics

Key Formulas:

$Population Mean: μ = Σx / N$

$Sample Mean: x̄ = Σx / n$

$Population Variance: σ² = Σ(x-μ)² / N$

$Sample Variance: s² = Σ(x-x̄)² / (n-1)$

$Standard Error: SE = s / √n$

Applications:

Parameter estimation

Statistical inference

Sampling theory

Data analysis

Statistical Inference

Intermediate

Formulas for making inferences about populations from sample data

Key Formulas:

$Confidence Interval: estimate ± margin of error$

$Test Statistic: (estimate - parameter) / standard error$

$P-value: P(observing result | H₀ is true)$

$Type I Error: α = P(reject H₀ | H₀ true)$

$Type II Error: β = P(fail to reject H₀ | H₁ true)$

Applications:

Hypothesis testing

Confidence intervals

Decision making

Research methodology

Point Estimation Theory

Comprehensive formulas for estimation methods, efficiency theory, and statistical inference

Point Estimation & Cramér-Rao Theory

Advanced

Complete mathematical reference for estimation methods, Fisher information, Cramér-Rao bounds, and UMVUE theory

Key Formulas:

$Fisher Information: I(θ) = E[(∂log p(X;θ)/∂θ)²]$

$Cramér-Rao Bound: Var[ĝ] ≥ [g'(θ)]²/(nI(θ))$

$MLE: θ̂ = arg max L(θ;x₁,...,xₙ)$

$MSE Decomposition: MSE[θ̂] = Var[θ̂] + Bias²[θ̂]$

$Method of Moments: θ̂ⱼ = hⱼ(a_{n,1},...,a_{n,k})$

Applications:

Optimal estimation

Efficiency analysis

UMVUE construction

Asymptotic inference

Sufficient & Complete Statistics

Essential formulas for sufficient statistics, complete statistics, and optimal estimation theory

Sufficient & Complete Statistics Theory

Advanced

Complete mathematical reference for sufficient statistics, factorization theorem, complete statistics, and UMVUE construction

Key Formulas:

$Factorization Theorem: p(x̃;θ) = g(T(x̃);θ) × h(x̃)$

$Complete Statistic: E_θ[φ(T)] = 0 ∀θ ⟹ P_θ(φ(T) = 0) = 1 ∀θ$

$Rao-Blackwell: Var(E[φ|T]) ≤ Var(φ)$

$Lehmann-Scheffé: E[φ|S] is unique UMVUE if S sufficient complete$

$Basu's Theorem: T ⊥ V if T sufficient complete, V ancillary$

Applications:

UMVUE construction

Optimal estimation

Information theory

Statistical sufficiency

Confidence Intervals & Interval Estimation

Essential formulas for confidence interval construction and interval estimation theory

Confidence Intervals Theory & Applications

Intermediate

Complete mathematical reference for interval estimation, pivotal quantities, and confidence interval construction methods

Key Formulas:

$Coverage Probability: P_θ{θ ∈ [θ̂_L, θ̂_U]}$

$Normal Mean (σ known): x̄ ± u_{α/2} × σ/√n$

$Normal Mean (σ unknown): x̄ ± t_{α/2}(n-1) × s/√n$

$Normal Variance: [(n-1)s²/χ²_{α/2}(n-1), (n-1)s²/χ²_{1-α/2}(n-1)]$

$Pivotal Quantity: G(X̃,θ) with known distribution$

Applications:

Parameter estimation

Hypothesis testing

Statistical inference

Quality control

Hypothesis Testing & Statistical Inference

Complete formulas for hypothesis testing, test statistics, decision rules, and GLRT methods

Hypothesis Testing Theory & Applications

Intermediate

Comprehensive formulas for statistical hypothesis testing, error analysis, and test construction methods

Key Formulas:

$Type I Error: α(θ) = P_θ(X̃ ∈ D | θ ∈ Θ₀)$

$Type II Error: β(θ) = P_θ(X̃ ∈ D̄ | θ ∈ Θ₁)$

$Power Function: g(θ) = P_θ(X̃ ∈ D) = 1 - β(θ)$

$U-Test: U = (X̄ - μ₀)/(σ/√n) ~ N(0,1)$

$T-Test: T = (X̄ - μ₀)/(S/√n) ~ t(n-1)$

$Chi-square Test: χ² = (n-1)S²/σ₀² ~ χ²(n-1)$

$GLRT: λ(x̃) = sup_Θ L(θ)/sup_Θ₀ L(θ)$

Applications:

Statistical testing

Quality control

A/B testing

Medical research

Nonparametric Hypothesis Testing

Complete formulas for distribution-free statistical tests including sign tests, rank-based methods, and goodness-of-fit procedures

Nonparametric Testing Theory & Applications

Intermediate

Comprehensive formulas for distribution-free hypothesis testing, sign tests, rank-based methods, and independence analysis

Key Formulas:

$Sign Test: N^+ = \sum I(X_i > t_0) \sim B(n, \theta)$

$Wilcoxon Rank Sum: W = \sum R_i, E[W] = \frac{n(m+n+1)}{2}$

$Chi-square Test: \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

$K-S Test: D_n = \sup_x |F_n(x) - F_0(x)|$

$Independence: E_{ij} = \frac{n_{i\cdot} n_{\cdot j}}{n}$

$Run Test: E[R] = \frac{2n_1 n_2}{n_1 + n_2} + 1$

Applications:

Robust testing

Small samples

Ordinal data

Distribution-free analysis

Bayesian Statistics & Inference

Complete formulas for Bayesian statistical inference including Bayes' theorem, conjugate priors, and posterior analysis

Bayesian Inference Theory & Applications

Intermediate to Advanced

Comprehensive formulas for Bayesian statistical analysis, prior-posterior relationships, and credible intervals

Key Formulas:

$Bayes' Theorem: π(θ|x̃) = p(x̃|θ)π(θ) / p(x̃)$

$Beta-Binomial: Beta(a,b) + Binomial(n,x) → Beta(a+x, b+n-x)$

$Gamma-Poisson: Gamma(α,β) + Poisson(Σx) → Gamma(α+Σx, β+m)$

$Normal-Normal: τₙ⁻² = τ⁻² + nσ⁻², μₙ = (τ⁻²μ₀ + nσ⁻²x̄)τₙ²$

$Credible Interval: P(θ ∈ [θ_L, θ_U] | data) = 1-α$

$Posterior Predictive: p(z|data) = ∫ p(z|θ)π(θ|data)dθ$

Applications:

Parameter estimation

Uncertainty quantification

Decision making

Machine learning

Coming Soon

Additional statistical formula collections are being developed

More Formulas Coming Soon

Advanced Probability Distributions

Intermediate

Coming Soon

Formulas for binomial, Poisson, exponential, and other key distributions

Key Formulas:

$Binomial: P(X=k) = C(n,k) × p^k × (1-p)^(n-k)$

$Poisson: P(X=k) = (λ^k × e^(-λ)) / k!$

$Exponential: f(x) = λe^(-λx), x ≥ 0$

$Chi-square: χ² = Σ((O-E)²/E)$

Applications:

Probability modeling

Statistical testing

Quality control

Risk analysis

Regression Analysis Formulas

Advanced

Coming Soon

Formulas for linear regression, correlation, and model diagnostics

Key Formulas:

$Linear Regression: ŷ = a + bx$

$Slope: b = Σ((x-x̄)(y-ȳ)) / Σ(x-x̄)²$

$Correlation: r = Σ((x-x̄)(y-ȳ)) / √(Σ(x-x̄)²Σ(y-ȳ)²)$

$R-squared: R² = SSR / SST$

Applications:

Predictive modeling

Relationship analysis

Forecasting

Data science

📊 How to Use Statistical Formulas Effectively

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Understand Your Data

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Interpret Results

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