""" .. _statsrefmanual: ========================================== Statistical functions (:mod:`scipy.stats`) ========================================== .. currentmodule:: scipy.stats This module contains a large number of probability distributions, summary and frequency statistics, correlation functions and statistical tests, masked statistics, kernel density estimation, quasi-Monte Carlo functionality, and more. Statistics is a very large area, and there are topics that are out of scope for SciPy and are covered by other packages. Some of the most important ones are: - `statsmodels `__: regression, linear models, time series analysis, extensions to topics also covered by ``scipy.stats``. - `Pandas `__: tabular data, time series functionality, interfaces to other statistical languages. - `PyMC `__: Bayesian statistical modeling, probabilistic machine learning. - `scikit-learn `__: classification, regression, model selection. - `Seaborn `__: statistical data visualization. - `rpy2 `__: Python to R bridge. Probability distributions ========================= Each univariate distribution is an instance of a subclass of `rv_continuous` (`rv_discrete` for discrete distributions): .. autosummary:: :toctree: generated/ rv_continuous rv_discrete rv_histogram Continuous distributions ------------------------ .. autosummary:: :toctree: generated/ alpha -- Alpha anglit -- Anglit arcsine -- Arcsine argus -- Argus beta -- Beta betaprime -- Beta Prime bradford -- Bradford burr -- Burr (Type III) burr12 -- Burr (Type XII) cauchy -- Cauchy chi -- Chi chi2 -- Chi-squared cosine -- Cosine crystalball -- Crystalball dgamma -- Double Gamma dweibull -- Double Weibull erlang -- Erlang expon -- Exponential exponnorm -- Exponentially Modified Normal exponweib -- Exponentiated Weibull exponpow -- Exponential Power f -- F (Snecdor F) fatiguelife -- Fatigue Life (Birnbaum-Saunders) fisk -- Fisk foldcauchy -- Folded Cauchy foldnorm -- Folded Normal genlogistic -- Generalized Logistic gennorm -- Generalized normal genpareto -- Generalized Pareto genexpon -- Generalized Exponential genextreme -- Generalized Extreme Value gausshyper -- Gauss Hypergeometric gamma -- Gamma gengamma -- Generalized gamma genhalflogistic -- Generalized Half Logistic genhyperbolic -- Generalized Hyperbolic geninvgauss -- Generalized Inverse Gaussian gibrat -- Gibrat gompertz -- Gompertz (Truncated Gumbel) gumbel_r -- Right Sided Gumbel, Log-Weibull, Fisher-Tippett, Extreme Value Type I gumbel_l -- Left Sided Gumbel, etc. halfcauchy -- Half Cauchy halflogistic -- Half Logistic halfnorm -- Half Normal halfgennorm -- Generalized Half Normal hypsecant -- Hyperbolic Secant invgamma -- Inverse Gamma invgauss -- Inverse Gaussian invweibull -- Inverse Weibull johnsonsb -- Johnson SB johnsonsu -- Johnson SU kappa4 -- Kappa 4 parameter kappa3 -- Kappa 3 parameter ksone -- Distribution of Kolmogorov-Smirnov one-sided test statistic kstwo -- Distribution of Kolmogorov-Smirnov two-sided test statistic kstwobign -- Limiting Distribution of scaled Kolmogorov-Smirnov two-sided test statistic. laplace -- Laplace laplace_asymmetric -- Asymmetric Laplace levy -- Levy levy_l levy_stable logistic -- Logistic loggamma -- Log-Gamma loglaplace -- Log-Laplace (Log Double Exponential) lognorm -- Log-Normal loguniform -- Log-Uniform lomax -- Lomax (Pareto of the second kind) maxwell -- Maxwell mielke -- Mielke's Beta-Kappa moyal -- Moyal nakagami -- Nakagami ncx2 -- Non-central chi-squared ncf -- Non-central F nct -- Non-central Student's T norm -- Normal (Gaussian) norminvgauss -- Normal Inverse Gaussian pareto -- Pareto pearson3 -- Pearson type III powerlaw -- Power-function powerlognorm -- Power log normal powernorm -- Power normal rdist -- R-distribution rayleigh -- Rayleigh rel_breitwigner -- Relativistic Breit-Wigner rice -- Rice recipinvgauss -- Reciprocal Inverse Gaussian semicircular -- Semicircular skewcauchy -- Skew Cauchy skewnorm -- Skew normal studentized_range -- Studentized Range t -- Student's T trapezoid -- Trapezoidal triang -- Triangular truncexpon -- Truncated Exponential truncnorm -- Truncated Normal truncpareto -- Truncated Pareto truncweibull_min -- Truncated minimum Weibull distribution tukeylambda -- Tukey-Lambda uniform -- Uniform vonmises -- Von-Mises (Circular) vonmises_line -- Von-Mises (Line) wald -- Wald weibull_min -- Minimum Weibull (see Frechet) weibull_max -- Maximum Weibull (see Frechet) wrapcauchy -- Wrapped Cauchy The ``fit`` method of the univariate continuous distributions uses maximum likelihood estimation to fit the distribution to a data set. The ``fit`` method can accept regular data or *censored data*. Censored data is represented with instances of the `CensoredData` class. .. autosummary:: :toctree: generated/ CensoredData Multivariate distributions -------------------------- .. autosummary:: :toctree: generated/ multivariate_normal -- Multivariate normal distribution matrix_normal -- Matrix normal distribution dirichlet -- Dirichlet dirichlet_multinomial -- Dirichlet multinomial distribution wishart -- Wishart invwishart -- Inverse Wishart multinomial -- Multinomial distribution special_ortho_group -- SO(N) group ortho_group -- O(N) group unitary_group -- U(N) group random_correlation -- random correlation matrices multivariate_t -- Multivariate t-distribution multivariate_hypergeom -- Multivariate hypergeometric distribution random_table -- Distribution of random tables with given marginals uniform_direction -- Uniform distribution on S(N-1) vonmises_fisher -- Von Mises-Fisher distribution `scipy.stats.multivariate_normal` methods accept instances of the following class to represent the covariance. .. autosummary:: :toctree: generated/ Covariance -- Representation of a covariance matrix Discrete distributions ---------------------- .. autosummary:: :toctree: generated/ bernoulli -- Bernoulli betabinom -- Beta-Binomial binom -- Binomial boltzmann -- Boltzmann (Truncated Discrete Exponential) dlaplace -- Discrete Laplacian geom -- Geometric hypergeom -- Hypergeometric logser -- Logarithmic (Log-Series, Series) nbinom -- Negative Binomial nchypergeom_fisher -- Fisher's Noncentral Hypergeometric nchypergeom_wallenius -- Wallenius's Noncentral Hypergeometric nhypergeom -- Negative Hypergeometric planck -- Planck (Discrete Exponential) poisson -- Poisson randint -- Discrete Uniform skellam -- Skellam yulesimon -- Yule-Simon zipf -- Zipf (Zeta) zipfian -- Zipfian An overview of statistical functions is given below. Many of these functions have a similar version in `scipy.stats.mstats` which work for masked arrays. Summary statistics ================== .. autosummary:: :toctree: generated/ describe -- Descriptive statistics gmean -- Geometric mean hmean -- Harmonic mean pmean -- Power mean kurtosis -- Fisher or Pearson kurtosis mode -- Modal value moment -- Central moment expectile -- Expectile skew -- Skewness kstat -- kstatvar -- tmean -- Truncated arithmetic mean tvar -- Truncated variance tmin -- tmax -- tstd -- tsem -- variation -- Coefficient of variation find_repeats rankdata tiecorrect trim_mean gstd -- Geometric Standard Deviation iqr sem bayes_mvs mvsdist entropy differential_entropy median_abs_deviation Frequency statistics ==================== .. autosummary:: :toctree: generated/ cumfreq percentileofscore scoreatpercentile relfreq .. autosummary:: :toctree: generated/ binned_statistic -- Compute a binned statistic for a set of data. binned_statistic_2d -- Compute a 2-D binned statistic for a set of data. binned_statistic_dd -- Compute a d-D binned statistic for a set of data. Hypothesis Tests and related functions ====================================== SciPy has many functions for performing hypothesis tests that return a test statistic and a p-value, and several of them return confidence intervals and/or other related information. The headings below are based on common uses of the functions within, but due to the wide variety of statistical procedures, any attempt at coarse-grained categorization will be imperfect. Also, note that tests within the same heading are not interchangeable in general (e.g. many have different distributional assumptions). One Sample Tests / Paired Sample Tests -------------------------------------- One sample tests are typically used to assess whether a single sample was drawn from a specified distribution or a distribution with specified properties (e.g. zero mean). .. autosummary:: :toctree: generated/ ttest_1samp binomtest skewtest kurtosistest normaltest jarque_bera shapiro anderson cramervonmises ks_1samp goodness_of_fit chisquare power_divergence Paired sample tests are often used to assess whether two samples were drawn from the same distribution; they differ from the independent sample tests below in that each observation in one sample is treated as paired with a closely-related observation in the other sample (e.g. when environmental factors are controlled between observations within a pair but not among pairs). They can also be interpreted or used as one-sample tests (e.g. tests on the mean or median of *differences* between paired observations). .. autosummary:: :toctree: generated/ ttest_rel wilcoxon Association/Correlation Tests ----------------------------- These tests are often used to assess whether there is a relationship (e.g. linear) between paired observations in multiple samples or among the coordinates of multivariate observations. .. autosummary:: :toctree: generated/ linregress pearsonr spearmanr pointbiserialr kendalltau weightedtau somersd siegelslopes theilslopes page_trend_test multiscale_graphcorr These association tests and are to work with samples in the form of contingency tables. Supporting functions are available in `scipy.stats.contingency`. .. autosummary:: :toctree: generated/ chi2_contingency fisher_exact barnard_exact boschloo_exact Independent Sample Tests ------------------------ Independent sample tests are typically used to assess whether multiple samples were independently drawn from the same distribution or different distributions with a shared property (e.g. equal means). Some tests are specifically for comparing two samples. .. autosummary:: :toctree: generated/ ttest_ind_from_stats poisson_means_test ttest_ind mannwhitneyu ranksums brunnermunzel mood ansari cramervonmises_2samp epps_singleton_2samp ks_2samp kstest Others are generalized to multiple samples. .. autosummary:: :toctree: generated/ f_oneway tukey_hsd dunnett kruskal alexandergovern fligner levene bartlett median_test friedmanchisquare anderson_ksamp Resampling and Monte Carlo Methods ---------------------------------- The following functions can reproduce the p-value and confidence interval results of most of the functions above, and often produce accurate results in a wider variety of conditions. They can also be used to perform hypothesis tests and generate confidence intervals for custom statistics. This flexibility comes at the cost of greater computational requirements and stochastic results. .. autosummary:: :toctree: generated/ monte_carlo_test permutation_test bootstrap Instances of the following object can be passed into some hypothesis test functions to perform a resampling or Monte Carlo version of the hypothesis test. .. autosummary:: :toctree: generated/ MonteCarloMethod PermutationMethod BootstrapMethod Multiple Hypothesis Testing and Meta-Analysis --------------------------------------------- These functions are for assessing the results of individual tests as a whole. Functions for performing specific multiple hypothesis tests (e.g. post hoc tests) are listed above. .. autosummary:: :toctree: generated/ combine_pvalues false_discovery_control Deprecated and Legacy Functions ------------------------------- .. autosummary:: :toctree: generated/ binom_test The following functions are related to the tests above but do not belong in the above categories. Quasi-Monte Carlo ================= .. toctree:: :maxdepth: 4 stats.qmc Contingency Tables ================== .. toctree:: :maxdepth: 4 stats.contingency Masked statistics functions =========================== .. toctree:: stats.mstats Other statistical functionality =============================== Transformations --------------- .. autosummary:: :toctree: generated/ boxcox boxcox_normmax boxcox_llf yeojohnson yeojohnson_normmax yeojohnson_llf obrientransform sigmaclip trimboth trim1 zmap zscore gzscore Statistical distances --------------------- .. autosummary:: :toctree: generated/ wasserstein_distance energy_distance Sampling -------- .. toctree:: :maxdepth: 4 stats.sampling Random variate generation / CDF Inversion ----------------------------------------- .. autosummary:: :toctree: generated/ rvs_ratio_uniforms Fitting / Survival Analysis --------------------------- .. autosummary:: :toctree: generated/ fit ecdf logrank Directional statistical functions --------------------------------- .. autosummary:: :toctree: generated/ directional_stats circmean circvar circstd Sensitivity Analysis -------------------- .. autosummary:: :toctree: generated/ sobol_indices Plot-tests ---------- .. autosummary:: :toctree: generated/ ppcc_max ppcc_plot probplot boxcox_normplot yeojohnson_normplot Univariate and multivariate kernel density estimation ----------------------------------------------------- .. autosummary:: :toctree: generated/ gaussian_kde Warnings / Errors used in :mod:`scipy.stats` -------------------------------------------- .. autosummary:: :toctree: generated/ DegenerateDataWarning ConstantInputWarning NearConstantInputWarning FitError Result classes used in :mod:`scipy.stats` ----------------------------------------- .. warning:: These classes are private, but they are included here because instances of them are returned by other statistical functions. User import and instantiation is not supported. .. toctree:: :maxdepth: 2 stats._result_classes """ from ._warnings_errors import (ConstantInputWarning, NearConstantInputWarning, DegenerateDataWarning, FitError) from ._stats_py import * from ._variation import variation from .distributions import * from ._morestats import * from ._multicomp import * from ._binomtest import binomtest from ._binned_statistic import * from ._kde import gaussian_kde from . import mstats from . import qmc from ._multivariate import * from . import contingency from .contingency import chi2_contingency from ._censored_data import CensoredData # noqa from ._resampling import (bootstrap, monte_carlo_test, permutation_test, MonteCarloMethod, PermutationMethod, BootstrapMethod) from ._entropy import * from ._hypotests import * from ._rvs_sampling import rvs_ratio_uniforms from ._page_trend_test import page_trend_test from ._mannwhitneyu import mannwhitneyu from ._fit import fit, goodness_of_fit from ._covariance import Covariance from ._sensitivity_analysis import * from ._survival import * # Deprecated namespaces, to be removed in v2.0.0 from . import ( biasedurn, kde, morestats, mstats_basic, mstats_extras, mvn, statlib, stats ) __all__ = [s for s in dir() if not s.startswith("_")] # Remove dunders. from scipy._lib._testutils import PytestTester test = PytestTester(__name__) del PytestTester