Statistical functions (scipy.stats)

This module contains a large number of probability distributions as well as a growing library of statistical functions.

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, **kwargs)	Generates a distribution given by a histogram.

Continuous distributions

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

Multivariate distributions

multivariate_normal([mean, cov, …])	A multivariate normal random variable.
matrix_normal([mean, rowcov, colcov, seed])	A matrix normal random variable.
dirichlet(alpha[, seed])	A Dirichlet random variable.
wishart([df, scale, seed])	A Wishart random variable.
invwishart([df, scale, seed])	An inverse Wishart random variable.
multinomial(n, p[, seed])	A multinomial random variable.
special_ortho_group([dim, seed])	A matrix-valued SO(N) random variable.
ortho_group	A matrix-valued O(N) random variable.
unitary_group	A matrix-valued U(N) random variable.
random_correlation	A random correlation matrix.
multivariate_t([loc, shape, df, …])	A multivariate t-distributed random variable.
multivariate_hypergeom(m, n[, seed])	A multivariate hypergeometric random variable.

Discrete distributions

bernoulli(*args, **kwds)	A Bernoulli discrete random variable.
betabinom(*args, **kwds)	A beta-binomial discrete random variable.
binom(*args, **kwds)	A binomial discrete random variable.
boltzmann(*args, **kwds)	A Boltzmann (Truncated Discrete Exponential) random variable.
dlaplace(*args, **kwds)	A Laplacian discrete random variable.
geom(*args, **kwds)	A geometric discrete random variable.
hypergeom(*args, **kwds)	A hypergeometric discrete random variable.
logser(*args, **kwds)	A Logarithmic (Log-Series, Series) discrete random variable.
nbinom(*args, **kwds)	A negative binomial discrete random variable.
nhypergeom(*args, **kwds)	A negative hypergeometric discrete random variable.
planck(*args, **kwds)	A Planck discrete exponential random variable.
poisson(*args, **kwds)	A Poisson discrete random variable.
randint(*args, **kwds)	A uniform discrete random variable.
skellam(*args, **kwds)	A Skellam discrete random variable.
zipf(*args, **kwds)	A Zipf discrete random variable.
yulesimon(*args, **kwds)	A Yule-Simon discrete random variable.

An overview of statistical functions is given below. Several 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])	Compute the geometric mean along the specified axis.
hmean(a[, axis, dtype])	Calculate the harmonic mean along the specified axis.
kurtosis(a[, axis, fisher, bias, nan_policy])	Compute the kurtosis (Fisher or Pearson) of a dataset.
mode(a[, axis, nan_policy])	Return an array of the modal (most common) value in the passed array.
moment(a[, moment, axis, nan_policy])	Calculate the nth moment about the mean for a sample.
skew(a[, axis, bias, nan_policy])	Compute the sample skewness of a data set.
kstat(data[, n])	Return the nth k-statistic (1<=n<=4 so far).
kstatvar(data[, n])	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])	Compute the coefficient of variation.
find_repeats(arr)	Find repeats and repeat counts.
trim_mean(a, proportiontocut[, axis])	Return mean of array after trimming distribution from both tails.
gstd(a[, axis, ddof])	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])	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])	Calculate the entropy of a distribution for given probability values.
median_absolute_deviation(*args, **kwds)	median_absolute_deviation is deprecated, use median_abs_deviation instead!
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.
itemfreq(*args, **kwds)	itemfreq is deprecated! itemfreq is deprecated and will be removed in a future version.
percentileofscore(a, score[, kind])	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.

Correlation functions

f_oneway(*args[, axis])	Perform one-way ANOVA.
pearsonr(x, y)	Pearson correlation coefficient and p-value for testing non-correlation.
spearmanr(a[, b, axis, nan_policy])	Calculate a Spearman correlation coefficient with associated p-value.
pointbiserialr(x, y)	Calculate a point biserial correlation coefficient and its p-value.
kendalltau(x, y[, initial_lexsort, …])	Calculate Kendall’s tau, a correlation measure for ordinal data.
weightedtau(x, y[, rank, weigher, additive])	Compute a weighted version of Kendall’s \(\tau\).
linregress(x[, y])	Calculate a linear least-squares regression for two sets of measurements.
siegelslopes(y[, x, method])	Computes the Siegel estimator for a set of points (x, y).
theilslopes(y[, x, alpha])	Computes the Theil-Sen estimator for a set of points (x, y).
multiscale_graphcorr(x, y[, …])	Computes the Multiscale Graph Correlation (MGC) test statistic.

Statistical tests

ttest_1samp(a, popmean[, axis, nan_policy, …])	Calculate the T-test for the mean of ONE group of scores.
ttest_ind(a, b[, axis, equal_var, …])	Calculate the T-test for the means of two independent samples of scores.
ttest_ind_from_stats(mean1, std1, nobs1, …)	T-test for means of two independent samples from descriptive statistics.
ttest_rel(a, b[, axis, nan_policy, alternative])	Calculate the t-test on TWO RELATED samples of scores, a and b.
chisquare(f_obs[, f_exp, ddof, axis])	Calculate a one-way chi-square test.
cramervonmises(rvs, cdf[, args])	Perform the Cramér-von Mises test for goodness of fit.
power_divergence(f_obs[, f_exp, ddof, axis, …])	Cressie-Read power divergence statistic and goodness of fit test.
kstest(rvs, cdf[, args, N, alternative, mode])	Performs the (one sample or two samples) Kolmogorov-Smirnov test for goodness of fit.
ks_1samp(x, cdf[, args, alternative, mode])	Performs the Kolmogorov-Smirnov test for goodness of fit.
ks_2samp(data1, data2[, alternative, mode])	Compute the Kolmogorov-Smirnov statistic on 2 samples.
epps_singleton_2samp(x, y[, t])	Compute the Epps-Singleton (ES) test statistic.
mannwhitneyu(x, y[, use_continuity, alternative])	Compute the Mann-Whitney rank test on samples x and y.
tiecorrect(rankvals)	Tie correction factor for Mann-Whitney U and Kruskal-Wallis H tests.
rankdata(a[, method, axis])	Assign ranks to data, dealing with ties appropriately.
ranksums(x, y)	Compute the Wilcoxon rank-sum statistic for two samples.
wilcoxon(x[, y, zero_method, correction, …])	Calculate the Wilcoxon signed-rank test.
kruskal(*args[, nan_policy])	Compute the Kruskal-Wallis H-test for independent samples.
friedmanchisquare(*args)	Compute the Friedman test for repeated measurements.
brunnermunzel(x, y[, alternative, …])	Compute the Brunner-Munzel test on samples x and y.
combine_pvalues(pvalues[, method, weights])	Combine p-values from independent tests bearing upon the same hypothesis.
jarque_bera(x)	Perform the Jarque-Bera goodness of fit test on sample data.

ansari(x, y)	Perform the Ansari-Bradley test for equal scale parameters.
bartlett(*args)	Perform Bartlett’s test for equal variances.
levene(*args[, center, proportiontocut])	Perform Levene test for equal variances.
shapiro(x)	Perform the Shapiro-Wilk test for normality.
anderson(x[, dist])	Anderson-Darling test for data coming from a particular distribution.
anderson_ksamp(samples[, midrank])	The Anderson-Darling test for k-samples.
binom_test(x[, n, p, alternative])	Perform a test that the probability of success is p.
fligner(*args[, center, proportiontocut])	Perform Fligner-Killeen test for equality of variance.
median_test(*args[, ties, correction, …])	Perform a Mood’s median test.
mood(x, y[, axis])	Perform Mood’s test for equal scale parameters.
skewtest(a[, axis, nan_policy])	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])	Test whether a sample differs from a normal distribution.

Transformations

boxcox(x[, lmbda, alpha])	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)	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(*args)	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])	Calculate the relative z-scores.
zscore(a[, axis, ddof, nan_policy])	Compute the z score.

Statistical distances

wasserstein_distance(u_values, v_values[, …])	Compute the first Wasserstein distance between two 1D distributions.
energy_distance(u_values, v_values[, …])	Compute the energy distance between two 1D distributions.

Random variate generation

rvs_ratio_uniforms(pdf, umax, vmin, vmax[, …])

Generate random samples from a probability density function using the ratio-of-uniforms method.

Circular statistical functions

circmean(samples[, high, low, axis, nan_policy])	Compute the circular mean for samples in a range.
circvar(samples[, high, low, axis, nan_policy])	Compute the circular variance for samples assumed to be in a range.
circstd(samples[, high, low, axis, nan_policy])	Compute the circular standard deviation for samples assumed to be in the range [low to high].

Contingency table functions

chi2_contingency(observed[, correction, lambda_])	Chi-square test of independence of variables in a contingency table.
contingency.expected_freq(observed)	Compute the expected frequencies from a contingency table.
contingency.margins(a)	Return a list of the marginal sums of the array a.
fisher_exact(table[, alternative])	Perform a Fisher exact test on a 2x2 contingency table.

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.

Masked statistics functions

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

Univariate and multivariate kernel density estimation

gaussian_kde(dataset[, bw_method, weights])

Representation of a kernel-density estimate using Gaussian kernels.

Warnings used in scipy.stats

F_onewayConstantInputWarning([msg])	Warning generated by f_oneway when an input is constant, e.g.
F_onewayBadInputSizesWarning	Warning generated by f_oneway when an input has length 0, or if all the inputs have length 1.
PearsonRConstantInputWarning([msg])	Warning generated by pearsonr when an input is constant.
PearsonRNearConstantInputWarning([msg])	Warning generated by pearsonr when an input is nearly constant.
SpearmanRConstantInputWarning([msg])	Warning generated by spearmanr when an input is constant.

For many more stat related functions install the software R and the interface package rpy.