Chi-squared test (or ${\displaystyle \mathrm {X} ^{2))$ test) is a statistical hypothesis test. It usually tests the hypothesis that "the experimental data does not differ from untreated data". That is a null hypothesis. The distribution of the test statistic is a chi-squared distribution when the null hypothesis is true.

The test results are regarded as 'significant' if there is only one chance in 20 that the result could be got by chance.

Groups

[change | change source]

There are three main groups of tests:

Tests for distribution check that the values follow a given probability distribution.
Tests for independence check that the values are independent; if this is the case, no value can be left out without losing information.
Tests for homogeneity: These check that all samples taken have the same probability distribution, or are from the same set of values.

Statistics

Outline
Index

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic cubic generalized/power Median Mode
Dispersion	Variance Standard deviation Average absolute deviation Coefficient of variation Percentile Range Interquartile range
Shape	Central limit theorem Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Pearson product-moment correlation
Rank correlation
- Spearman's ρ
- Kendall's τ
Partial correlation
Scatter plot

Graphics

Bar chart
Biplot
Box plot
Control chart
Correlogram
Fan chart
Forest plot
Histogram
Pie chart
Q–Q plot
Run chart
Scatter plot
Stem-and-leaf display
Radar chart
Violin plot

Data collection

Study design	Population Statistic Effect size Statistical power Optimal design Sample size determination Replication Missing data
Survey methodology	Sampling stratified cluster Standard error Opinion poll Questionnaire
Controlled experiments	Scientific control Randomized experiment Randomized controlled trial Random assignment Blocking Interaction Factorial experiment
Adaptive Designs	Adaptive clinical trial Up-and-Down Designs Stochastic approximation
Observational Studies	Cross-sectional study Cohort study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Population
Statistic
Probability distribution
Sampling distribution
- Order statistic
Empirical distribution
- Density estimation
Statistical model
- Model specification
- L^p space
Parameter
- location
- scale
- shape
Parametric family
- Likelihood (monotone)
- Location–scale family
- Exponential family
Completeness
Sufficiency
Statistical functional
- Bootstrap
- U
- V
Optimal decision
- loss function
Efficiency
Statistical distance
- divergence
Asymptotics
Robustness

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in
Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife
Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons
Parametric tests	Likelihood-ratio Score/Lagrange multiplier Wald

Specific tests

Z-test (normal) Student's t-test F-test
Goodness of fit	Chi-squared G-test Kolmogorov–Smirnov Anderson–Darling Lilliefors Jarque–Bera Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC
Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Bayesian probability
- prior
- posterior
Credible interval
Bayes factor
Bayesian estimator
- Maximum posterior estimator

Correlation	Pearson product-moment Partial correlation Confounding variable Coefficient of determination
Regression analysis	Errors and residuals Regression validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)
Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression
Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity
Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions
Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Cohen's kappa
Contingency table
Graphical model
Log-linear model
McNemar's test
Cochran-Mantel-Haenszel statistics

Multivariate

Regression
Manova
Principal components
Canonical correlation
Discriminant analysis
Cluster analysis
Classification
Structural equation model
- Factor analysis
Multivariate distributions
- Elliptical distributions
  - Normal

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality
Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey
Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)
Frequency domain	Spectral density estimation Fourier analysis Wavelet Whittle likelihood

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time
Hazard function	Nelson–Aalen estimator
Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics
Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification
Social statistics	Actuarial science Census Crime statistics Demography Econometrics Jurimetrics National accounts Official statistics Population statistics Psychometrics
Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

This short article about mathematics can be made longer. You can help Wikipedia by adding to it.