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# Statistical functions (`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):

`rv_continuous`([momtype, a, b, xtol, ...]) | A generic continuous random variable class meant for subclassing.
`rv_discrete`([a, b, name, badvalue, ...]) | A generic discrete random variable class meant for subclassing.
`rv_histogram`(histogram, *args[, density]) | Generates a distribution given by a histogram.

### Continuous distributions

`alpha` | An alpha continuous random variable.
`anglit` | An anglit continuous random variable.
`arcsine` | An arcsine continuous random variable.
`argus` | Argus distribution
`beta` | A beta continuous random variable.
`betaprime` | A beta prime continuous random variable.
`bradford` | A Bradford continuous random variable.
`burr` | A Burr (Type III) continuous random variable.
`burr12` | A Burr (Type XII) continuous random variable.
`cauchy` | A Cauchy continuous random variable.
`chi` | A chi continuous random variable.
`chi2` | A chi-squared continuous random variable.
`cosine` | A cosine continuous random variable.
`crystalball` | Crystalball distribution
`dgamma` | A double gamma continuous random variable.
`dpareto_lognorm` | A double Pareto lognormal continuous random variable.
`dweibull` | A double Weibull continuous random variable.
`erlang` | An Erlang continuous random variable.
`expon` | An exponential continuous random variable.
`exponnorm` | An exponentially modified Normal continuous random variable.
`exponweib` | An exponentiated Weibull continuous random variable.
`exponpow` | An exponential power continuous random variable.
`f` | An F continuous random variable.
`fatiguelife` | A fatigue-life (Birnbaum-Saunders) continuous random variable.
`fisk` | A Fisk continuous random variable.
`foldcauchy` | A folded Cauchy continuous random variable.
`foldnorm` | A folded normal continuous random variable.
`genlogistic` | A generalized logistic continuous random variable.
`gennorm` | A generalized normal continuous random variable.
`genpareto` | A generalized Pareto continuous random variable.
`genexpon` | A generalized exponential continuous random variable.
`genextreme` | A generalized extreme value continuous random variable.
`gausshyper` | A Gauss hypergeometric continuous random variable.
`gamma` | A gamma continuous random variable.
`gengamma` | A generalized gamma continuous random variable.
`genhalflogistic` | A generalized half-logistic continuous random variable.
`genhyperbolic` | A generalized hyperbolic continuous random variable.
`geninvgauss` | A Generalized Inverse Gaussian continuous random variable.
`gibrat` | A Gibrat continuous random variable.
`gompertz` | A Gompertz (or truncated Gumbel) continuous random variable.
`gumbel_r` | A right-skewed Gumbel continuous random variable.
`gumbel_l` | A left-skewed Gumbel continuous random variable.
`halfcauchy` | A Half-Cauchy continuous random variable.
`halflogistic` | A half-logistic continuous random variable.
`halfnorm` | A half-normal continuous random variable.
`halfgennorm` | The upper half of a generalized normal continuous random variable.
`hypsecant` | A hyperbolic secant continuous random variable.
`invgamma` | An inverted gamma continuous random variable.
`invgauss` | An inverse Gaussian continuous random variable.
`invweibull` | An inverted Weibull continuous random variable.
`irwinhall` | An Irwin-Hall (Uniform Sum) continuous random variable.
`jf_skew_t` | Jones and Faddy skew-t distribution.
`johnsonsb` | A Johnson SB continuous random variable.
`johnsonsu` | A Johnson SU continuous random variable.
`kappa4` | Kappa 4 parameter distribution.
`kappa3` | Kappa 3 parameter distribution.
`ksone` | Kolmogorov-Smirnov one-sided test statistic distribution.
`kstwo` | Kolmogorov-Smirnov two-sided test statistic distribution.
`kstwobign` | Limiting distribution of scaled Kolmogorov-Smirnov two-sided test statistic.
`landau` | A Landau continuous random variable.
`laplace` | A Laplace continuous random variable.
`laplace_asymmetric` | An asymmetric Laplace continuous random variable.
`levy` | A Levy continuous random variable.
`levy_l` | A left-skewed Levy continuous random variable.
`levy_stable` | A Levy-stable continuous random variable.
`logistic` | A logistic (or Sech-squared) continuous random variable.
`loggamma` | A log gamma continuous random variable.
`loglaplace` | A log-Laplace continuous random variable.
`lognorm` | A lognormal continuous random variable.
`loguniform` | A loguniform or reciprocal continuous random variable.
`lomax` | A Lomax (Pareto of the second kind) continuous random variable.
`maxwell` | A Maxwell continuous random variable.
`mielke` | A Mielke Beta-Kappa / Dagum continuous random variable.
`moyal` | A Moyal continuous random variable.
`nakagami` | A Nakagami continuous random variable.
`ncx2` | A non-central chi-squared continuous random variable.
`ncf` | A non-central F distribution continuous random variable.
`nct` | A non-central Student's t continuous random variable.
`norm` | A normal continuous random variable.
`norminvgauss` | A Normal Inverse Gaussian continuous random variable.
`pareto` | A Pareto continuous random variable.
`pearson3` | A pearson type III continuous random variable.
`powerlaw` | A power-function continuous random variable.
`powerlognorm` | A power log-normal continuous random variable.
`powernorm` | A power normal continuous random variable.
`rdist` | An R-distributed (symmetric beta) continuous random variable.
`rayleigh` | A Rayleigh continuous random variable.
`rel_breitwigner` | A relativistic Breit-Wigner random variable.
`rice` | A Rice continuous random variable.
`recipinvgauss` | A reciprocal inverse Gaussian continuous random variable.
`semicircular` | A semicircular continuous random variable.
`skewcauchy` | A skewed Cauchy random variable.
`skewnorm` | A skew-normal random variable.
`studentized_range` | A studentized range continuous random variable.
`t` | A Student's t continuous random variable.
`trapezoid` | A trapezoidal continuous random variable.
`triang` | A triangular continuous random variable.
`truncexpon` | A truncated exponential continuous random variable.
`truncnorm` | A truncated normal continuous random variable.
`truncpareto` | An upper truncated Pareto continuous random variable.
`truncweibull_min` | A doubly truncated Weibull minimum continuous random variable.
`tukeylambda` | A Tukey-Lamdba continuous random variable.
`uniform` | A uniform continuous random variable.
`vonmises` | A Von Mises continuous random variable.
`vonmises_line` | A Von Mises continuous random variable.
`wald` | A Wald continuous random variable.
`weibull_min` | Weibull minimum continuous random variable.
`weibull_max` | Weibull maximum continuous random variable.
`wrapcauchy` | A wrapped Cauchy continuous random variable.

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.

`CensoredData`([uncensored, left, right, interval]) | Instances of this class represent censored data.

### Multivariate distributions

`multivariate_normal` | A multivariate normal random variable.
`matrix_normal` | A matrix normal random variable.
`dirichlet` | A Dirichlet random variable.
`dirichlet_multinomial` | A Dirichlet multinomial random variable.
`wishart` | A Wishart random variable.
`invwishart` | An inverse Wishart random variable.
`multinomial` | A multinomial random variable.
`special_ortho_group` | A Special Orthogonal matrix (SO(N)) random variable.
`ortho_group` | An Orthogonal matrix (O(N)) random variable.
`unitary_group` | A matrix-valued U(N) random variable.
`random_correlation` | A random correlation matrix.
`multivariate_t` | A multivariate t-distributed random variable.
`multivariate_hypergeom` | A multivariate hypergeometric random variable.
`normal_inverse_gamma` | Normal-inverse-gamma distribution.
`random_table` | Contingency tables from independent samples with fixed marginal sums.
`uniform_direction` | A vector-valued uniform direction.
`vonmises_fisher` | A von Mises-Fisher variable.

`scipy.stats.multivariate_normal` methods accept instances of the following class to represent the covariance.

`Covariance`() | Representation of a covariance matrix

### Discrete distributions

`bernoulli` | A Bernoulli discrete random variable.
`betabinom` | A beta-binomial discrete random variable.
`betanbinom` | A beta-negative-binomial discrete random variable.
`binom` | A binomial discrete random variable.
`boltzmann` | A Boltzmann (Truncated Discrete Exponential) random variable.
`dlaplace` | A Laplacian discrete random variable.
`geom` | A geometric discrete random variable.
`hypergeom` | A hypergeometric discrete random variable.
`logser` | A Logarithmic (Log-Series, Series) discrete random variable.
`nbinom` | A negative binomial discrete random variable.
`nchypergeom_fisher` | A Fisher's noncentral hypergeometric discrete random variable.
`nchypergeom_wallenius` | A Wallenius' noncentral hypergeometric discrete random variable.
`nhypergeom` | A negative hypergeometric discrete random variable.
`planck` | A Planck discrete exponential random variable.
`poisson` | A Poisson discrete random variable.
`poisson_binom` | A Poisson Binomial discrete random variable.
`randint` | A uniform discrete random variable.
`skellam` | A Skellam discrete random variable.
`yulesimon` | A Yule-Simon discrete random variable.
`zipf` | A Zipf (Zeta) discrete random variable.
`zipfian` | A Zipfian discrete random variable.

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

`describe`(a[, axis, ddof, bias, nan_policy]) | Compute several descriptive statistics of the passed array.
`gmean`(a[, axis, dtype, weights, nan_policy, ...]) | Compute the weighted geometric mean along the specified axis.
`hmean`(a[, axis, dtype, weights, nan_policy, ...]) | Calculate the weighted harmonic mean along the specified axis.
`pmean`(a, p, *[, axis, dtype, weights, ...]) | Calculate the weighted power mean along the specified axis.
`kurtosis`(a[, axis, fisher, bias, ...]) | Compute the kurtosis (Fisher or Pearson) of a dataset.
`mode`(a[, axis, nan_policy, keepdims]) | Return an array of the modal (most common) value in the passed array.
`moment`(a[, order, axis, nan_policy, center, ...]) | Calculate the nth moment about the mean for a sample.
`lmoment`(sample[, order, axis, sorted, ...]) | Compute L-moments of a sample from a continuous distribution
`expectile`(a[, alpha, weights]) | Compute the expectile at the specified level.
`skew`(a[, axis, bias, nan_policy, keepdims]) | Compute the sample skewness of a data set.
`kstat`(data[, n, axis, nan_policy, keepdims]) | Return the n th k-statistic ( `1<=n<=4` so far).
`kstatvar`(data[, n, axis, nan_policy, keepdims]) | Return an unbiased estimator of the variance of the k-statistic.
`tmean`(a[, limits, inclusive, axis, ...]) | Compute the trimmed mean.
`tvar`(a[, limits, inclusive, axis, ddof, ...]) | Compute the trimmed variance.
`tmin`(a[, lowerlimit, axis, inclusive, ...]) | Compute the trimmed minimum.
`tmax`(a[, upperlimit, axis, inclusive, ...]) | Compute the trimmed maximum.
`tstd`(a[, limits, inclusive, axis, ddof, ...]) | Compute the trimmed sample standard deviation.
`tsem`(a[, limits, inclusive, axis, ddof, ...]) | Compute the trimmed standard error of the mean.
`variation`(a[, axis, nan_policy, ddof, keepdims]) | Compute the coefficient of variation.
`find_repeats`(arr) | Find repeats and repeat counts.
`rankdata`(a[, method, axis, nan_policy]) | Assign ranks to data, dealing with ties appropriately.
`tiecorrect`(rankvals) | Tie correction factor for Mann-Whitney U and Kruskal-Wallis H tests.
`trim_mean`(a, proportiontocut[, axis]) | Return mean of array after trimming a specified fraction of extreme values
`gstd`(a[, axis, ddof, keepdims, nan_policy]) | Calculate the geometric standard deviation of an array.
`iqr`(x[, axis, rng, scale, nan_policy, ...]) | Compute the interquartile range of the data along the specified axis.
`sem`(a[, axis, ddof, nan_policy, keepdims]) | Compute standard error of the mean.
`bayes_mvs`(data[, alpha]) | Bayesian confidence intervals for the mean, var, and std.
`mvsdist`(data) | 'Frozen' distributions for mean, variance, and standard deviation of data.
`entropy`(pk[, qk, base, axis, nan_policy, ...]) | Calculate the Shannon entropy/relative entropy of given distribution(s).
`differential_entropy`(values, *[, ...]) | Given a sample of a distribution, estimate the differential entropy.
`median_abs_deviation`(x[, axis, center, ...]) | Compute the median absolute deviation of the data along the given axis.

## Frequency statistics

`cumfreq`(a[, numbins, defaultreallimits, weights]) | Return a cumulative frequency histogram, using the histogram function.
`quantile`(x, p, *[, method, axis, ...]) | Compute the p-th quantile of the data along the specified axis.
`percentileofscore`(a, score[, kind, nan_policy]) | Compute the percentile rank of a score relative to a list of scores.
`scoreatpercentile`(a, per[, limit, ...]) | Calculate the score at a given percentile of the input sequence.
`relfreq`(a[, numbins, defaultreallimits, weights]) | Return a relative frequency histogram, using the histogram function.

`binned_statistic`(x, values[, statistic, ...]) | Compute a binned statistic for one or more sets of data.
`binned_statistic_2d`(x, y, values[, ...]) | Compute a bidimensional binned statistic for one or more sets of data.
`binned_statistic_dd`(sample, values[, ...]) | Compute a multidimensional 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).

`ttest_1samp`(a, popmean[, axis, nan_policy, ...]) | Calculate the T-test for the mean of ONE group of scores.
`binomtest`(k, n[, p, alternative]) | Perform a test that the probability of success is p.
`quantile_test`(x, *[, q, p, alternative]) | Perform a quantile test and compute a confidence interval of the quantile.
`skewtest`(a[, axis, nan_policy, alternative, ...]) | Test whether the skew is different from the normal distribution.
`kurtosistest`(a[, axis, nan_policy, ...]) | Test whether a dataset has normal kurtosis.
`normaltest`(a[, axis, nan_policy, keepdims]) | Test whether a sample differs from a normal distribution.
`jarque_bera`(x, *[, axis, nan_policy, keepdims]) | Perform the Jarque-Bera goodness of fit test on sample data.
`shapiro`(x, *[, axis, nan_policy, keepdims]) | Perform the Shapiro-Wilk test for normality.
`anderson`(x[, dist]) | Anderson-Darling test for data coming from a particular distribution.
`cramervonmises`(rvs, cdf[, args, axis, ...]) | Perform the one-sample Cramér-von Mises test for goodness of fit.
`ks_1samp`(x, cdf[, args, alternative, ...]) | Performs the one-sample Kolmogorov-Smirnov test for goodness of fit.
`goodness_of_fit`(dist, data, *[, ...]) | Perform a goodness of fit test comparing data to a distribution family.
`chisquare`(f_obs[, f_exp, ddof, axis, ...]) | Perform Pearson's chi-squared test.
`power_divergence`(f_obs[, f_exp, ddof, axis, ...]) | Cressie-Read power divergence statistic and goodness of fit test.

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).

`ttest_rel`(a, b[, axis, nan_policy, ...]) | Calculate the t-test on TWO RELATED samples of scores, a and b.
`wilcoxon`(x[, y, zero_method, correction, ...]) | Calculate the Wilcoxon signed-rank test.

### 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.

`linregress`(x, y[, alternative, axis, ...]) | Calculate a linear least-squares regression for two sets of measurements.
`pearsonr`(x, y, *[, alternative, method, axis]) | Pearson correlation coefficient and p-value for testing non-correlation.
`spearmanr`(a[, b, axis, nan_policy, alternative]) | Calculate a Spearman correlation coefficient with associated p-value.
`pointbiserialr`(x, y, *[, axis, nan_policy, ...]) | Calculate a point biserial correlation coefficient and its p-value.
`kendalltau`(x, y, *[, nan_policy, method, ...]) | Calculate Kendall's tau, a correlation measure for ordinal data.
`chatterjeexi`(x, y, *[, axis, y_continuous, ...]) | Compute the xi correlation and perform a test of independence
`weightedtau`(x, y[, rank, weigher, additive, ...]) | Compute a weighted version of Kendall's \(\tau\).
`somersd`(x[, y, alternative]) | Calculates Somers' D, an asymmetric measure of ordinal association.
`siegelslopes`(y[, x, method, axis, ...]) | Computes the Siegel estimator for a set of points (x, y).
`theilslopes`(y[, x, alpha, method, axis, ...]) | Computes the Theil-Sen estimator for a set of points (x, y).
`page_trend_test`(data[, ranked, ...]) | Perform Page's Test, a measure of trend in observations between treatments.
`multiscale_graphcorr`(x, y[, ...]) | Computes the Multiscale Graph Correlation (MGC) test statistic.

These association tests and are to work with samples in the form of contingency tables. Supporting functions are available in `scipy.stats.contingency`.

`chi2_contingency`(observed[, correction, ...]) | Chi-square test of independence of variables in a contingency table.
`fisher_exact`(table[, alternative, method]) | Perform a Fisher exact test on a contingency table.
`barnard_exact`(table[, alternative, pooled, n]) | Perform a Barnard exact test on a 2x2 contingency table.
`boschloo_exact`(table[, alternative, n]) | Perform Boschloo's exact test on a 2x2 contingency table.

### 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.

`ttest_ind_from_stats`(mean1, std1, nobs1, ...) | T-test for means of two independent samples from descriptive statistics.
`poisson_means_test`(k1, n1, k2, n2, *[, ...]) | Performs the Poisson means test, AKA the "E-test".
`ttest_ind`(a, b, *[, axis, equal_var, ...]) | Calculate the T-test for the means of two independent samples of scores.
`mannwhitneyu`(x, y[, use_continuity, ...]) | Perform the Mann-Whitney U rank test on two independent samples.
`bws_test`(x, y, *[, alternative, method]) | Perform the Baumgartner-Weiss-Schindler test on two independent samples.
`ranksums`(x, y[, alternative, axis, ...]) | Compute the Wilcoxon rank-sum statistic for two samples.
`brunnermunzel`(x, y[, alternative, ...]) | Compute the Brunner-Munzel test on samples x and y.
`mood`(x, y[, axis, alternative, nan_policy, ...]) | Perform Mood's test for equal scale parameters.
`ansari`(x, y[, alternative, axis, ...]) | Perform the Ansari-Bradley test for equal scale parameters.
`cramervonmises_2samp`(x, y[, method, axis, ...]) | Perform the two-sample Cramér-von Mises test for goodness of fit.
`epps_singleton_2samp`(x, y[, t, axis, ...]) | Compute the Epps-Singleton (ES) test statistic.
`ks_2samp`(data1, data2[, alternative, ...]) | Performs the two-sample Kolmogorov-Smirnov test for goodness of fit.
`kstest`(rvs, cdf[, args, N, alternative, ...]) | Performs the (one-sample or two-sample) Kolmogorov-Smirnov test for goodness of fit.

Others are generalized to multiple samples.

`f_oneway`(*samples[, axis, equal_var, ...]) | Perform one-way ANOVA.
`tukey_hsd`(*args[, equal_var]) | Perform Tukey's HSD test for equality of means over multiple treatments.
`dunnett`(*samples, control[, alternative, ...]) | Dunnett's test: multiple comparisons of means against a control group.
`kruskal`(*samples[, nan_policy, axis, keepdims]) | Compute the Kruskal-Wallis H-test for independent samples.
`alexandergovern`(*samples[, nan_policy, ...]) | Performs the Alexander Govern test.
`fligner`(*samples[, center, proportiontocut, ...]) | Perform Fligner-Killeen test for equality of variance.
`levene`(*samples[, center, proportiontocut, ...]) | Perform Levene test for equal variances.
`bartlett`(*samples[, axis, nan_policy, keepdims]) | Perform Bartlett's test for equal variances.
`median_test`(*samples[, ties, correction, ...]) | Perform a Mood's median test.
`friedmanchisquare`(*samples[, axis, ...]) | Compute the Friedman test for repeated samples.
`anderson_ksamp`(samples[, midrank, method]) | The Anderson-Darling test for k-samples.

### 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.

`monte_carlo_test`(data, rvs, statistic, *[, ...]) | Perform a Monte Carlo hypothesis test.
`permutation_test`(data, statistic, *[, ...]) | Performs a permutation test of a given statistic on provided data.
`bootstrap`(data, statistic, *[, n_resamples, ...]) | Compute a two-sided bootstrap confidence interval of a statistic.
`power`(test, rvs, n_observations, *[, ...]) | Simulate the power of a hypothesis test under an alternative hypothesis.

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.

`MonteCarloMethod`([n_resamples, batch, rvs, rng]) | Configuration information for a Monte Carlo hypothesis test.
`PermutationMethod`([n_resamples, batch, ...]) | Configuration information for a permutation hypothesis test.
`BootstrapMethod`([n_resamples, batch, ...]) | Configuration information for a bootstrap confidence interval.

### 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.

`combine_pvalues`(pvalues[, method, weights, ...]) | Combine p-values from independent tests that bear upon the same hypothesis.
`false_discovery_control`(ps, *[, axis, method]) | Adjust p-values to control the false discovery rate.

The following functions are related to the tests above but do not belong in the above categories.

## Random Variables

`make_distribution`(dist) | Generate a UnivariateDistribution class from a compatible object
`Normal`([mu, sigma]) | Normal distribution with prescribed mean and standard deviation.
`Uniform`(*[, a, b]) | Uniform distribution.
`Binomial`(*, n, p, **kwargs) | Binomial distribution with prescribed success probability and number of trials
`Mixture`(components, *[, weights]) | Representation of a mixture distribution.
`order_statistic`(X, /, *, r, n) | Probability distribution of an order statistic
`truncate`(X[, lb, ub]) | Truncate the support of a random variable.
`abs`(X, /) | Absolute value of a random variable
`exp`(X, /) | Natural exponential of a random variable
`log`(X, /) | Natural logarithm of a non-negative random variable

## Quasi-Monte Carlo

- Quasi-Monte Carlo submodule (`scipy.stats.qmc`)
  - Quasi-Monte Carlo
    - Engines
      - QMCEngine
      - Sobol
      - Halton
      - LatinHypercube
      - PoissonDisk
      - MultinomialQMC
      - MultivariateNormalQMC
    - Helpers
      - discrepancy
      - geometric_discrepancy
      - update_discrepancy
      - scale
  - Introduction to Quasi-Monte Carlo
    - References

## Contingency Tables

- Contingency table functions (`scipy.stats.contingency`)
  - chi2_contingency
    - `chi2_contingency`
  - relative_risk
    - `relative_risk`
  - odds_ratio
    - `odds_ratio`
  - crosstab
    - `crosstab`
  - association
    - `association`
  - expected_freq
    - `expected_freq`
  - margins
    - `margins`

## Masked statistics functions

- Statistical functions for masked arrays (`scipy.stats.mstats`)
  - Summary statistics
    - describe
      - `describe`
    - gmean
      - `gmean`
    - hmean
      - `hmean`
    - kurtosis
      - `kurtosis`
    - mode
      - `mode`
    - mquantiles
      - `mquantiles`
    - hdmedian
      - `hdmedian`
    - hdquantiles
      - `hdquantiles`
    - hdquantiles_sd
      - `hdquantiles_sd`
    - idealfourths
      - `idealfourths`
    - plotting_positions
      - `plotting_positions`
    - meppf
      - `meppf`
    - moment
      - `moment`
    - skew
      - `skew`
    - tmean
      - `tmean`
    - tvar
      - `tvar`
    - tmin
      - `tmin`
    - tmax
      - `tmax`
    - tsem
      - `tsem`
    - variation
      - `variation`
    - find_repeats
      - `find_repeats`
    - sem
      - `sem`
    - trimmed_mean
      - `trimmed_mean`
    - trimmed_mean_ci
      - `trimmed_mean_ci`
    - trimmed_std
      - `trimmed_std`
    - trimmed_var
      - `trimmed_var`
  - Frequency statistics
    - scoreatpercentile
      - `scoreatpercentile`
  - Correlation functions
    - f_oneway
      - `f_oneway`
    - pearsonr
      - `pearsonr`
    - spearmanr
      - `spearmanr`
    - pointbiserialr
      - `pointbiserialr`
    - kendalltau
      - `kendalltau`
    - kendalltau_seasonal
      - `kendalltau_seasonal`
    - linregress
      - `linregress`
    - siegelslopes
      - `siegelslopes`
    - theilslopes
      - `theilslopes`
    - sen_seasonal_slopes
      - `sen_seasonal_slopes`
  - Statistical tests
    - ttest_1samp
      - `ttest_1samp`
    - ttest_onesamp
      - `ttest_onesamp`
    - ttest_ind
      - `ttest_ind`
    - ttest_rel
      - `ttest_rel`
    - chisquare
      - `chisquare`
    - kstest
      - `kstest`
    - ks_2samp
      - `ks_2samp`
    - ks_1samp
      - `ks_1samp`
    - ks_twosamp
      - `ks_twosamp`
    - mannwhitneyu
      - `mannwhitneyu`
    - rankdata
      - `rankdata`
    - kruskal
      - `kruskal`
    - kruskalwallis
      - `kruskalwallis`
    - friedmanchisquare
      - `friedmanchisquare`
    - brunnermunzel
      - `brunnermunzel`
    - skewtest
      - `skewtest`
    - kurtosistest
      - `kurtosistest`
    - normaltest
      - `normaltest`
  - Transformations
    - obrientransform
      - `obrientransform`
    - trim
      - `trim`
    - trima
      - `trima`
    - trimmed_stde
      - `trimmed_stde`
    - trimr
      - `trimr`
    - trimtail
      - `trimtail`
    - trimboth
      - `trimboth`
    - winsorize
      - `winsorize`
    - zmap
      - `zmap`
    - zscore
      - `zscore`
  - Other
    - argstoarray
      - `argstoarray`
    - count_tied_groups
      - `count_tied_groups`
    - msign
      - `msign`
    - compare_medians_ms
      - `compare_medians_ms`
    - median_cihs
      - `median_cihs`
    - mjci
      - `mjci`
    - mquantiles_cimj
      - `mquantiles_cimj`
    - rsh
      - `rsh`

## Other statistical functionality

### Transformations

`boxcox`(x[, lmbda, alpha, optimizer]) | Return a dataset transformed by a Box-Cox power transformation.
`boxcox_normmax`(x[, brack, method, ...]) | Compute optimal Box-Cox transform parameter for input data.
`boxcox_llf`(lmb, data, *[, axis, keepdims, ...]) | The boxcox log-likelihood function.
`yeojohnson`(x[, lmbda]) | Return a dataset transformed by a Yeo-Johnson power transformation.
`yeojohnson_normmax`(x[, brack]) | Compute optimal Yeo-Johnson transform parameter.
`yeojohnson_llf`(lmb, data) | The yeojohnson log-likelihood function.
`obrientransform`(*samples) | Compute the O'Brien transform on input data (any number of arrays).
`sigmaclip`(a[, low, high]) | Perform iterative sigma-clipping of array elements.
`trimboth`(a, proportiontocut[, axis]) | Slice off a proportion of items from both ends of an array.
`trim1`(a, proportiontocut[, tail, axis]) | Slice off a proportion from ONE end of the passed array distribution.
`zmap`(scores, compare[, axis, ddof, nan_policy]) | Calculate the relative z-scores.
`zscore`(a[, axis, ddof, nan_policy]) | Compute the z score.
`gzscore`(a, *[, axis, ddof, nan_policy]) | Compute the geometric standard score.

### Statistical distances

`wasserstein_distance`(u_values, v_values[, ...]) | Compute the Wasserstein-1 distance between two 1D discrete distributions.
`wasserstein_distance_nd`(u_values, v_values) | Compute the Wasserstein-1 distance between two N-D discrete distributions.
`energy_distance`(u_values, v_values[, ...]) | Compute the energy distance between two 1D distributions.

### Sampling

- Random Number Generators (`scipy.stats.sampling`)
  - Generators Wrapped
    - For continuous distributions
      - NumericalInverseHermite
      - NumericalInversePolynomial
      - TransformedDensityRejection
      - SimpleRatioUniforms
      - RatioUniforms
    - For discrete distributions
      - DiscreteAliasUrn
      - DiscreteGuideTable
    - Warnings / Errors used in `scipy.stats.sampling`
      - scipy.stats.sampling.UNURANError
  - Generators for pre-defined distributions
    - FastGeneratorInversion
      - `FastGeneratorInversion`
      - evaluate_error
      - ppf
      - qrvs
      - rvs
      - support

### Fitting / Survival Analysis

`fit`(dist, data[, bounds, guess, method, ...]) | Fit a discrete or continuous distribution to data
`ecdf`(sample) | Empirical cumulative distribution function of a sample.
`logrank`(x, y[, alternative]) | Compare the survival distributions of two samples via the logrank test.

### Directional statistical functions

`directional_stats`(samples, *[, axis, normalize]) | Computes sample statistics for directional data.
`circmean`(samples[, high, low, axis, ...]) | Compute the circular mean of a sample of angle observations.
`circvar`(samples[, high, low, axis, ...]) | Compute the circular variance of a sample of angle observations.
`circstd`(samples[, high, low, axis, ...]) | Compute the circular standard deviation of a sample of angle observations.

### Sensitivity Analysis

`sobol_indices`(*, func, n[, dists, method, ...]) | Global sensitivity indices of Sobol'.

### Plot-tests

`ppcc_max`(x[, brack, dist]) | Calculate the shape parameter that maximizes the PPCC.
`ppcc_plot`(x, a, b[, dist, plot, N]) | Calculate and optionally plot probability plot correlation coefficient.
`probplot`(x[, sparams, dist, fit, plot, rvalue]) | Calculate quantiles for a probability plot, and optionally show the plot.
`boxcox_normplot`(x, la, lb[, plot, N]) | Compute parameters for a Box-Cox normality plot, optionally show it.
`yeojohnson_normplot`(x, la, lb[, plot, N]) | Compute parameters for a Yeo-Johnson normality plot, optionally show it.

### Univariate and multivariate kernel density estimation

`gaussian_kde`(dataset[, bw_method, weights]) | Representation of a kernel-density estimate using Gaussian kernels.

### Warnings / Errors used in `scipy.stats`

`DegenerateDataWarning`([msg]) | Warns when data is degenerate and results may not be reliable.
`ConstantInputWarning`([msg]) | Warns when all values in data are exactly equal.
`NearConstantInputWarning`([msg]) | Warns when all values in data are nearly equal.
`FitError`([msg]) | Represents an error condition when fitting a distribution to data.

### Result classes used in `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.

- Result classes
  - RelativeRiskResult
  - BinomTestResult
  - TukeyHSDResult
  - DunnettResult
  - PearsonRResult
  - FitResult
  - OddsRatioResult
  - TtestResult
  - ECDFResult
  - EmpiricalDistributionFunction