# Statistical functions (scipy.stats)¶

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

Each included distribution is an instance of the class rv_continous: For each given name the following methods are available:

 rv_continuous([momtype, a, b, xtol, ...]) A generic continuous random variable class meant for subclassing. rv_continuous.pdf(x, *args, **kwds) Probability density function at x of the given RV. rv_continuous.logpdf(x, *args, **kwds) Log of the probability density function at x of the given RV. rv_continuous.cdf(x, *args, **kwds) Cumulative distribution function of the given RV. rv_continuous.logcdf(x, *args, **kwds) Log of the cumulative distribution function at x of the given RV. rv_continuous.sf(x, *args, **kwds) Survival function (1-cdf) at x of the given RV. rv_continuous.logsf(x, *args, **kwds) Log of the survival function of the given RV. rv_continuous.ppf(q, *args, **kwds) Percent point function (inverse of cdf) at q of the given RV. rv_continuous.isf(q, *args, **kwds) Inverse survival function at q of the given RV. rv_continuous.moment(n, *args, **kwds) n’th order non-central moment of distribution. rv_continuous.stats(*args, **kwds) Some statistics of the given RV rv_continuous.entropy(*args, **kwds) Differential entropy of the RV. rv_continuous.fit(data, *args, **kwds) Return MLEs for shape, location, and scale parameters from data. rv_continuous.expect([func, args, loc, ...]) Calculate expected value of a function with respect to the distribution.

Calling the instance as a function returns a frozen pdf whose shape, location, and scale parameters are fixed.

Similarly, each discrete distribution is an instance of the class rv_discrete:

 rv_discrete([a, b, name, badvalue, ...]) A generic discrete random variable class meant for subclassing. rv_discrete.rvs(*args, **kwargs) Random variates of given type. rv_discrete.pmf(k, *args, **kwds) Probability mass function at k of the given RV. rv_discrete.logpmf(k, *args, **kwds) Log of the probability mass function at k of the given RV. rv_discrete.cdf(k, *args, **kwds) Cumulative distribution function of the given RV. rv_discrete.logcdf(k, *args, **kwds) Log of the cumulative distribution function at k of the given RV rv_discrete.sf(k, *args, **kwds) Survival function (1-cdf) at k of the given RV. rv_discrete.logsf(k, *args, **kwds) Log of the survival function of the given RV. rv_discrete.ppf(q, *args, **kwds) Percent point function (inverse of cdf) at q of the given RV rv_discrete.isf(q, *args, **kwds) Inverse survival function (1-sf) at q of the given RV. rv_discrete.stats(*args, **kwds) Some statistics of the given RV rv_discrete.moment(n, *args, **kwds) n’th order non-central moment of distribution. rv_discrete.entropy(*args, **kwds) Differential entropy of the RV. rv_discrete.expect([func, args, loc, lb, ...]) Calculate expected value of a function with respect to the distribution

## Continuous distributions¶

 alpha An alpha continuous random variable. anglit An anglit continuous random variable. arcsine An arcsine continuous random variable. beta A beta continuous random variable. betaprime A beta prime continuous random variable. bradford A Bradford continuous random variable. burr A Burr 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. dgamma A double gamma continuous random variable. dweibull A double Weibull continuous random variable. erlang An Erlang continuous random variable. expon An exponential 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-Sanders) continuous random variable. fisk A Fisk continuous random variable. foldcauchy A folded Cauchy continuous random variable. foldnorm A folded normal continuous random variable. frechet_r A Frechet right (or Weibull minimum) continuous random variable. frechet_l A Frechet left (or Weibull maximum) continuous random variable. genlogistic A generalized logistic 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. gilbrat A Gilbrat 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. 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. johnsonsb A Johnson SB continuous random variable. johnsonsu A Johnson SU continuous random variable. ksone General Kolmogorov-Smirnov one-sided test. kstwobign Kolmogorov-Smirnov two-sided test for large N. laplace A Laplace 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. lomax A Lomax (Pareto of the second kind) continuous random variable. maxwell A Maxwell continuous random variable. mielke A Mielke’s Beta-Kappa 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. 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 continuous random variable. reciprocal A reciprocal continuous random variable. rayleigh A Rayleigh continuous random variable. rice A Rice continuous random variable. recipinvgauss A reciprocal inverse Gaussian continuous random variable. semicircular A semicircular continuous random variable. t A Student’s T continuous random variable. triang A triangular continuous random variable. truncexpon A truncated exponential continuous random variable. truncnorm A truncated normal continuous random variable. tukeylambda A Tukey-Lamdba continuous random variable. uniform A uniform continuous random variable. vonmises A Von Mises continuous random variable. wald A Wald continuous random variable. weibull_min A Frechet right (or Weibull minimum) continuous random variable. weibull_max A Frechet left (or Weibull maximum) continuous random variable. wrapcauchy A wrapped Cauchy continuous random variable.

## Multivariate distributions¶

 multivariate_normal A multivariate normal random variable.

## Discrete distributions¶

 bernoulli A Bernoulli 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. planck A Planck discrete exponential random variable. poisson A Poisson discrete random variable. randint A uniform discrete random variable. skellam A Skellam discrete random variable. zipf A Zipf discrete random variable.

## Statistical functions¶

Several of these functions have a similar version in scipy.stats.mstats which work for masked arrays.

 describe(a[, axis]) Computes several descriptive statistics of the passed array. gmean(a[, axis, dtype]) Compute the geometric mean along the specified axis. hmean(a[, axis, dtype]) Calculates the harmonic mean along the specified axis. kurtosis(a[, axis, fisher, bias]) Computes the kurtosis (Fisher or Pearson) of a dataset. kurtosistest(a[, axis]) Tests whether a dataset has normal kurtosis mode(a[, axis]) Returns an array of the modal (most common) value in the passed array. moment(a[, moment, axis]) Calculates the nth moment about the mean for a sample. normaltest(a[, axis]) Tests whether a sample differs from a normal distribution. skew(a[, axis, bias]) Computes the skewness of a data set. skewtest(a[, axis]) Tests whether the skew is different from the normal distribution. tmean(a[, limits, inclusive]) Compute the trimmed mean. tvar(a[, limits, inclusive]) 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]) Compute the trimmed sample standard deviation tsem(a[, limits, inclusive]) Compute the trimmed standard error of the mean. nanmean(x[, axis]) Compute the mean over the given axis ignoring nans. nanstd(x[, axis, bias]) Compute the standard deviation over the given axis, ignoring nans. nanmedian(x[, axis]) Compute the median along the given axis ignoring nan values. variation(a[, axis]) Computes the coefficient of variation, the ratio of the biased standard deviation to the mean.
 cumfreq(a[, numbins, defaultreallimits, weights]) Returns a cumulative frequency histogram, using the histogram function. histogram2(a, bins) Compute histogram using divisions in bins. histogram(a[, numbins, defaultlimits, ...]) Separates the range into several bins and returns the number of instances in each bin. itemfreq(a) Returns a 2-D array of item frequencies. percentileofscore(a, score[, kind]) 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]) Returns a relative frequency histogram, using the histogram function.
 binned_statistic(x, values[, statistic, ...]) Compute a binned statistic for a set of data. binned_statistic_2d(x, y, values[, ...]) Compute a bidimensional binned statistic for a set of data. binned_statistic_dd(sample, values[, ...]) Compute a multidimensional binned statistic for a set of data.
 obrientransform(*args) Computes the O’Brien transform on input data (any number of arrays). signaltonoise(a[, axis, ddof]) The signal-to-noise ratio of the input data. bayes_mvs(data[, alpha]) Bayesian confidence intervals for the mean, var, and std. sem(a[, axis, ddof]) Calculates the standard error of the mean (or standard error of measurement) of the values in the input array. zmap(scores, compare[, axis, ddof]) Calculates the relative z-scores. zscore(a[, axis, ddof]) Calculates the z score of each value in the sample, relative to the sample mean and standard deviation.
 sigmaclip(a[, low, high]) Iterative sigma-clipping of array elements. threshold(a[, threshmin, threshmax, newval]) Clip array to a given value. trimboth(a, proportiontocut[, axis]) Slices off a proportion of items from both ends of an array. trim1(a, proportiontocut[, tail]) Slices off a proportion of items from ONE end of the passed array distribution.
 f_oneway(*args) Performs a 1-way ANOVA. pearsonr(x, y) Calculates a Pearson correlation coefficient and the p-value for testing non-correlation. spearmanr(a[, b, axis]) Calculates a Spearman rank-order correlation coefficient and the p-value to test for non-correlation. pointbiserialr(x, y) Calculates a point biserial correlation coefficient and the associated p-value. kendalltau(x, y[, initial_lexsort]) Calculates Kendall’s tau, a correlation measure for ordinal data. linregress(x[, y]) Calculate a regression line
 ttest_1samp(a, popmean[, axis]) Calculates the T-test for the mean of ONE group of scores. ttest_ind(a, b[, axis, equal_var]) Calculates the T-test for the means of TWO INDEPENDENT samples of scores. ttest_rel(a, b[, axis]) Calculates the T-test on TWO RELATED samples of scores, a and b. kstest(rvs, cdf[, args, N, alternative, mode]) Perform the Kolmogorov-Smirnov test for goodness of fit. chisquare(f_obs[, f_exp, ddof, axis]) Calculates a one-way chi square test. power_divergence(f_obs[, f_exp, ddof, axis, ...]) Cressie-Read power divergence statistic and goodness of fit test. ks_2samp(data1, data2) Computes the Kolmogorov-Smirnov statistic on 2 samples. mannwhitneyu(x, y[, use_continuity]) Computes the Mann-Whitney rank test on samples x and y. tiecorrect(rankvals) Tie correction factor for ties in the Mann-Whitney U and Kruskal-Wallis H tests. rankdata(a[, method]) 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) Compute the Kruskal-Wallis H-test for independent samples friedmanchisquare(*args) Computes the Friedman test for repeated measurements
 ansari(x, y) Perform the Ansari-Bradley test for equal scale parameters bartlett(*args) Perform Bartlett’s test for equal variances levene(*args, **kwds) Perform Levene test for equal variances. shapiro(x[, a, reta]) 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]) Perform a test that the probability of success is p. fligner(*args, **kwds) Perform Fligner’s test for equal variances. mood(x, y[, axis]) Perform Mood’s test for equal scale parameters.
 boxcox(x[, lmbda, alpha]) Return a positive 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. entropy(pk[, qk, base]) Calculate the entropy of a distribution for given probability values.

## 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]) Performs a Fisher exact test on a 2x2 contingency table.

## Plot-tests¶

 ppcc_max(x[, brack, dist]) Returns the shape parameter that maximizes the probability plot correlation coefficient for the given data to a one-parameter family of distributions. ppcc_plot(x, a, b[, dist, plot, N]) Returns (shape, ppcc), and optionally plots shape vs. probplot(x[, sparams, dist, fit, plot]) 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.

## Univariate and multivariate kernel density estimation (scipy.stats.kde)¶

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

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