# `stats.stats`¶

A collection of basic statistical functions for Python. The function names appear below.

Some scalar functions defined here are also available in the scipy.special package where they work on arbitrary sized arrays.

Disclaimers: The function list is obviously incomplete and, worse, the functions are not optimized. All functions have been tested (some more so than others), but they are far from bulletproof. Thus, as with any free software, no warranty or guarantee is expressed or implied. :-) A few extra functions that don’t appear in the list below can be found by interested treasure-hunters. These functions don’t necessarily have both list and array versions but were deemed useful.

## Correlation Functions¶

 `pearsonr` `fisher_exact` `spearmanr` `pointbiserialr` `kendalltau` `weightedtau` `linregress` `theilslopes`

## References¶

 [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000.

## Module Contents¶

### Functions¶

 `_chk_asarray`(a,axis) `_chk2_asarray`(a,b,axis) `_contains_nan`(a,nan_policy=”propagate”) `gmean`(a,axis=0,dtype=None) Compute the geometric mean along the specified axis. `hmean`(a,axis=0,dtype=None) Calculate the harmonic mean along the specified axis. `mode`(a,axis=0,nan_policy=”propagate”) Return an array of the modal (most common) value in the passed array. `_mask_to_limits`(a,limits,inclusive) Mask an array for values outside of given limits. `tmean`(a,limits=None,inclusive=tuple,axis=None) Compute the trimmed mean. `tvar`(a,limits=None,inclusive=tuple,axis=0,ddof=1) Compute the trimmed variance. `tmin`(a,lowerlimit=None,axis=0,inclusive=True,nan_policy=”propagate”) Compute the trimmed minimum. `tmax`(a,upperlimit=None,axis=0,inclusive=True,nan_policy=”propagate”) Compute the trimmed maximum. `tstd`(a,limits=None,inclusive=tuple,axis=0,ddof=1) Compute the trimmed sample standard deviation. `tsem`(a,limits=None,inclusive=tuple,axis=0,ddof=1) Compute the trimmed standard error of the mean. `moment`(a,moment=1,axis=0,nan_policy=”propagate”) r `_moment`(a,moment,axis) `variation`(a,axis=0,nan_policy=”propagate”) Compute the coefficient of variation, the ratio of the biased standard `skew`(a,axis=0,bias=True,nan_policy=”propagate”) Compute the skewness of a data set. `kurtosis`(a,axis=0,fisher=True,bias=True,nan_policy=”propagate”) Compute the kurtosis (Fisher or Pearson) of a dataset. `describe`(a,axis=0,ddof=1,bias=True,nan_policy=”propagate”) Compute several descriptive statistics of the passed array. `skewtest`(a,axis=0,nan_policy=”propagate”) Test whether the skew is different from the normal distribution. `kurtosistest`(a,axis=0,nan_policy=”propagate”) Test whether a dataset has normal kurtosis. `normaltest`(a,axis=0,nan_policy=”propagate”) Test whether a sample differs from a normal distribution. `jarque_bera`(x) Perform the Jarque-Bera goodness of fit test on sample data. `itemfreq`(a) Return a 2-D array of item frequencies. `scoreatpercentile`(a,per,limit=tuple,interpolation_method=”fraction”,axis=None) Calculate the score at a given percentile of the input sequence. `_compute_qth_percentile`(sorted,per,interpolation_method,axis) `percentileofscore`(a,score,kind=”rank”) The percentile rank of a score relative to a list of scores. `_histogram`(a,numbins=10,defaultlimits=None,weights=None,printextras=False) Separate the range into several bins and return the number of instances `cumfreq`(a,numbins=10,defaultreallimits=None,weights=None) Return a cumulative frequency histogram, using the histogram function. `relfreq`(a,numbins=10,defaultreallimits=None,weights=None) Return a relative frequency histogram, using the histogram function. `obrientransform`(*args) Compute the O’Brien transform on input data (any number of arrays). `sem`(a,axis=0,ddof=1,nan_policy=”propagate”) Calculate the standard error of the mean (or standard error of `zscore`(a,axis=0,ddof=0) Calculate the z score of each value in the sample, relative to the `zmap`(scores,compare,axis=0,ddof=0) Calculate the relative z-scores. `iqr`(x,axis=None,rng=tuple,scale=”raw”,nan_policy=”propagate”,interpolation=”linear”,keepdims=False) Compute the interquartile range of the data along the specified axis. `_iqr_percentile`(x,q,axis=None,interpolation=”linear”,keepdims=False,contains_nan=False) Private wrapper that works around older versions of numpy. `_iqr_nanpercentile`(x,q,axis=None,interpolation=”linear”,keepdims=False,contains_nan=False) Private wrapper that works around the following: `sigmaclip`(a,low=4.0,high=4.0) Iterative sigma-clipping of array elements. `trimboth`(a,proportiontocut,axis=0) Slices off a proportion of items from both ends of an array. `trim1`(a,proportiontocut,tail=”right”,axis=0) Slices off a proportion from ONE end of the passed array distribution. `trim_mean`(a,proportiontocut,axis=0) Return mean of array after trimming distribution from both tails. `f_oneway`(*args) Performs a 1-way ANOVA. `pearsonr`(x,y) r `fisher_exact`(table,alternative=”two-sided”) Performs a Fisher exact test on a 2x2 contingency table. `spearmanr`(a,b=None,axis=0,nan_policy=”propagate”) Calculate a Spearman rank-order correlation coefficient and the p-value `pointbiserialr`(x,y) r `kendalltau`(x,y,initial_lexsort=None,nan_policy=”propagate”) Calculate Kendall’s tau, a correlation measure for ordinal data. `weightedtau`(x,y,rank=True,weigher=None,additive=True) r `ttest_1samp`(a,popmean,axis=0,nan_policy=”propagate”) Calculate the T-test for the mean of ONE group of scores. `_ttest_finish`(df,t) Common code between all 3 t-test functions. `_ttest_ind_from_stats`(mean1,mean2,denom,df) `_unequal_var_ttest_denom`(v1,n1,v2,n2) `_equal_var_ttest_denom`(v1,n1,v2,n2) `ttest_ind_from_stats`(mean1,std1,nobs1,mean2,std2,nobs2,equal_var=True) T-test for means of two independent samples from descriptive statistics. `ttest_ind`(a,b,axis=0,equal_var=True,nan_policy=”propagate”) Calculate the T-test for the means of two independent samples of scores. `ttest_rel`(a,b,axis=0,nan_policy=”propagate”) Calculate the T-test on TWO RELATED samples of scores, a and b. `kstest`(rvs,cdf,args=tuple,N=20,alternative=”two-sided”,mode=”approx”) Perform the Kolmogorov-Smirnov test for goodness of fit. `_count`(a,axis=None) Count the number of non-masked elements of an array. `power_divergence`(f_obs,f_exp=None,ddof=0,axis=0,lambda_=None) Cressie-Read power divergence statistic and goodness of fit test. `chisquare`(f_obs,f_exp=None,ddof=0,axis=0) Calculate a one-way chi square test. `ks_2samp`(data1,data2) Compute the Kolmogorov-Smirnov statistic on 2 samples. `tiecorrect`(rankvals) Tie correction factor for ties in the Mann-Whitney U and `mannwhitneyu`(x,y,use_continuity=True,alternative=None) Compute the Mann-Whitney rank test on samples x and y. `ranksums`(x,y) Compute the Wilcoxon rank-sum statistic for two samples. `kruskal`(*args,**kwargs) Compute the Kruskal-Wallis H-test for independent samples `friedmanchisquare`(*args) Compute the Friedman test for repeated measurements `combine_pvalues`(pvalues,method=”fisher”,weights=None) Methods for combining the p-values of independent tests bearing upon the `_betai`(a,b,x) `wasserstein_distance`(u_values,v_values,u_weights=None,v_weights=None) r `energy_distance`(u_values,v_values,u_weights=None,v_weights=None) r `_cdf_distance`(p,u_values,v_values,u_weights=None,v_weights=None) r `_validate_distribution`(values,weights) Validate the values and weights from a distribution input of cdf_distance `find_repeats`(arr) Find repeats and repeat counts. `_sum_of_squares`(a,axis=0) Square each element of the input array, and return the sum(s) of that. `_square_of_sums`(a,axis=0) Sum elements of the input array, and return the square(s) of that sum. `rankdata`(a,method=”average”) Assign ranks to data, dealing with ties appropriately.
`_chk_asarray`(a, axis)
`_chk2_asarray`(a, b, axis)
`_contains_nan`(a, nan_policy="propagate")
`gmean`(a, axis=0, dtype=None)

Compute the geometric mean along the specified axis.

Return the geometric average of the array elements. That is: n-th root of (x1 * x2 * … * xn)

a : array_like
Input array or object that can be converted to an array.
axis : int or None, optional
Axis along which the geometric mean is computed. Default is 0. If None, compute over the whole array a.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the elements are summed. If dtype is not specified, it defaults to the dtype of a, unless a has an integer dtype with a precision less than that of the default platform integer. In that case, the default platform integer is used.
gmean : ndarray
see dtype parameter above

numpy.mean : Arithmetic average numpy.average : Weighted average hmean : Harmonic mean

The geometric average is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs.

Use masked arrays to ignore any non-finite values in the input or that arise in the calculations such as Not a Number and infinity because masked arrays automatically mask any non-finite values.

```>>> from scipy.stats import gmean
>>> gmean([1, 4])
2.0
>>> gmean([1, 2, 3, 4, 5, 6, 7])
3.3800151591412964
```
`hmean`(a, axis=0, dtype=None)

Calculate the harmonic mean along the specified axis.

That is: n / (1/x1 + 1/x2 + … + 1/xn)

a : array_like
Input array, masked array or object that can be converted to an array.
axis : int or None, optional
Axis along which the harmonic mean is computed. Default is 0. If None, compute over the whole array a.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the elements are summed. If dtype is not specified, it defaults to the dtype of a, unless a has an integer dtype with a precision less than that of the default platform integer. In that case, the default platform integer is used.
hmean : ndarray
see dtype parameter above

numpy.mean : Arithmetic average numpy.average : Weighted average gmean : Geometric mean

The harmonic mean is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs.

Use masked arrays to ignore any non-finite values in the input or that arise in the calculations such as Not a Number and infinity.

```>>> from scipy.stats import hmean
>>> hmean([1, 4])
1.6000000000000001
>>> hmean([1, 2, 3, 4, 5, 6, 7])
2.6997245179063363
```
`mode`(a, axis=0, nan_policy="propagate")

Return an array of the modal (most common) value in the passed array.

If there is more than one such value, only the smallest is returned. The bin-count for the modal bins is also returned.

a : array_like
n-dimensional array of which to find mode(s).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
mode : ndarray
Array of modal values.
count : ndarray
Array of counts for each mode.
```>>> a = np.array([[6, 8, 3, 0],
...               [3, 2, 1, 7],
...               [8, 1, 8, 4],
...               [5, 3, 0, 5],
...               [4, 7, 5, 9]])
>>> from scipy import stats
>>> stats.mode(a)
(array([[3, 1, 0, 0]]), array([[1, 1, 1, 1]]))
```

To get mode of whole array, specify `axis=None`:

```>>> stats.mode(a, axis=None)
(array([3]), array([3]))
```
`_mask_to_limits`(a, limits, inclusive)

Mask an array for values outside of given limits.

This is primarily a utility function.

a : array limits : (float or None, float or None)

A tuple consisting of the (lower limit, upper limit). Values in the input array less than the lower limit or greater than the upper limit will be masked out. None implies no limit.
inclusive : (bool, bool)
A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to lower or upper are allowed.

A ValueError if there are no values within the given limits.

`tmean`(a, limits=None, inclusive=tuple, axis=None)

Compute the trimmed mean.

This function finds the arithmetic mean of given values, ignoring values outside the given limits.

a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None (default), then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True).
axis : int or None, optional
Axis along which to compute test. Default is None.

tmean : float

trim_mean : returns mean after trimming a proportion from both tails.

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmean(x)
9.5
>>> stats.tmean(x, (3,17))
10.0
```
`tvar`(a, limits=None, inclusive=tuple, axis=0, ddof=1)

Compute the trimmed variance.

This function computes the sample variance of an array of values, while ignoring values which are outside of given limits.

a : array_like
Array of values.
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
ddof : int, optional
Delta degrees of freedom. Default is 1.
tvar : float
Trimmed variance.

tvar computes the unbiased sample variance, i.e. it uses a correction factor `n / (n - 1)`.

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tvar(x)
35.0
>>> stats.tvar(x, (3,17))
20.0
```
`tmin`(a, lowerlimit=None, axis=0, inclusive=True, nan_policy="propagate")

Compute the trimmed minimum.

This function finds the miminum value of an array a along the specified axis, but only considering values greater than a specified lower limit.

a : array_like
array of values
lowerlimit : None or float, optional
Values in the input array less than the given limit will be ignored. When lowerlimit is None, then all values are used. The default value is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the lower limit are included. The default value is True.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.

tmin : float, int or ndarray

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmin(x)
0
```
```>>> stats.tmin(x, 13)
13
```
```>>> stats.tmin(x, 13, inclusive=False)
14
```
`tmax`(a, upperlimit=None, axis=0, inclusive=True, nan_policy="propagate")

Compute the trimmed maximum.

This function computes the maximum value of an array along a given axis, while ignoring values larger than a specified upper limit.

a : array_like
array of values
upperlimit : None or float, optional
Values in the input array greater than the given limit will be ignored. When upperlimit is None, then all values are used. The default value is None.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
inclusive : {True, False}, optional
This flag determines whether values exactly equal to the upper limit are included. The default value is True.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.

tmax : float, int or ndarray

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tmax(x)
19
```
```>>> stats.tmax(x, 13)
13
```
```>>> stats.tmax(x, 13, inclusive=False)
12
```
`tstd`(a, limits=None, inclusive=tuple, axis=0, ddof=1)

Compute the trimmed sample standard deviation.

This function finds the sample standard deviation of given values, ignoring values outside the given limits.

a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
ddof : int, optional
Delta degrees of freedom. Default is 1.

tstd : float

tstd computes the unbiased sample standard deviation, i.e. it uses a correction factor `n / (n - 1)`.

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tstd(x)
5.9160797830996161
>>> stats.tstd(x, (3,17))
4.4721359549995796
```
`tsem`(a, limits=None, inclusive=tuple, axis=0, ddof=1)

Compute the trimmed standard error of the mean.

This function finds the standard error of the mean for given values, ignoring values outside the given limits.

a : array_like
array of values
limits : None or (lower limit, upper limit), optional
Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None.
inclusive : (bool, bool), optional
A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True).
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
ddof : int, optional
Delta degrees of freedom. Default is 1.

tsem : float

tsem uses unbiased sample standard deviation, i.e. it uses a correction factor `n / (n - 1)`.

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.tsem(x)
1.3228756555322954
>>> stats.tsem(x, (3,17))
1.1547005383792515
```
`moment`(a, moment=1, axis=0, nan_policy="propagate")

r Calculate the nth moment about the mean for a sample.

A moment is a specific quantitative measure of the shape of a set of points. It is often used to calculate coefficients of skewness and kurtosis due to its close relationship with them.

a : array_like
data
moment : int or array_like of ints, optional
order of central moment that is returned. Default is 1.
axis : int or None, optional
Axis along which the central moment is computed. Default is 0. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
n-th central moment : ndarray or float
The appropriate moment along the given axis or over all values if axis is None. The denominator for the moment calculation is the number of observations, no degrees of freedom correction is done.

kurtosis, skew, describe

The k-th central moment of a data sample is:

Where n is the number of samples and x-bar is the mean. This function uses exponentiation by squares [1]_ for efficiency.

```>>> from scipy.stats import moment
>>> moment([1, 2, 3, 4, 5], moment=1)
0.0
>>> moment([1, 2, 3, 4, 5], moment=2)
2.0
```
`_moment`(a, moment, axis)
`variation`(a, axis=0, nan_policy="propagate")

Compute the coefficient of variation, the ratio of the biased standard deviation to the mean.

a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate the coefficient of variation. Default is 0. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
variation : ndarray
The calculated variation along the requested axis.
 [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000.
```>>> from scipy.stats import variation
>>> variation([1, 2, 3, 4, 5])
0.47140452079103173
```
`skew`(a, axis=0, bias=True, nan_policy="propagate")

Compute the skewness of a data set.

For normally distributed data, the skewness should be about 0. For unimodal continuous distributions, a skewness value > 0 means that there is more weight in the right tail of the distribution. The function skewtest can be used to determine if the skewness value is close enough to 0, statistically speaking.

a : ndarray
data
axis : int or None, optional
Axis along which skewness is calculated. Default is 0. If None, compute over the whole array a.
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
skewness : ndarray
The skewness of values along an axis, returning 0 where all values are equal.
 [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 2.2.24.1
```>>> from scipy.stats import skew
>>> skew([1, 2, 3, 4, 5])
0.0
>>> skew([2, 8, 0, 4, 1, 9, 9, 0])
0.2650554122698573
```
`kurtosis`(a, axis=0, fisher=True, bias=True, nan_policy="propagate")

Compute the kurtosis (Fisher or Pearson) of a dataset.

Kurtosis is the fourth central moment divided by the square of the variance. If Fisher’s definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution.

If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators

Use kurtosistest to see if result is close enough to normal.

a : array
data for which the kurtosis is calculated
axis : int or None, optional
Axis along which the kurtosis is calculated. Default is 0. If None, compute over the whole array a.
fisher : bool, optional
If True, Fisher’s definition is used (normal ==> 0.0). If False, Pearson’s definition is used (normal ==> 3.0).
bias : bool, optional
If False, then the calculations are corrected for statistical bias.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
kurtosis : array
The kurtosis of values along an axis. If all values are equal, return -3 for Fisher’s definition and 0 for Pearson’s definition.
 [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000.
```>>> from scipy.stats import kurtosis
>>> kurtosis([1, 2, 3, 4, 5])
-1.3
```
`describe`(a, axis=0, ddof=1, bias=True, nan_policy="propagate")

Compute several descriptive statistics of the passed array.

a : array_like
Input data.
axis : int or None, optional
Axis along which statistics are calculated. Default is 0. If None, compute over the whole array a.
ddof : int, optional
Delta degrees of freedom (only for variance). Default is 1.
bias : bool, optional
If False, then the skewness and kurtosis calculations are corrected for statistical bias.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
nobs : int or ndarray of ints
Number of observations (length of data along axis). When ‘omit’ is chosen as nan_policy, each column is counted separately.
minmax: tuple of ndarrays or floats
Minimum and maximum value of data array.
mean : ndarray or float
Arithmetic mean of data along axis.
variance : ndarray or float
Unbiased variance of the data along axis, denominator is number of observations minus one.
skewness : ndarray or float
Skewness, based on moment calculations with denominator equal to the number of observations, i.e. no degrees of freedom correction.
kurtosis : ndarray or float
Kurtosis (Fisher). The kurtosis is normalized so that it is zero for the normal distribution. No degrees of freedom are used.

skew, kurtosis

```>>> from scipy import stats
>>> a = np.arange(10)
>>> stats.describe(a)
DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.1666666666666661,
skewness=0.0, kurtosis=-1.2242424242424244)
>>> b = [[1, 2], [3, 4]]
>>> stats.describe(b)
DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])),
mean=array([ 2., 3.]), variance=array([ 2., 2.]),
skewness=array([ 0., 0.]), kurtosis=array([-2., -2.]))
```
`skewtest`(a, axis=0, nan_policy="propagate")

Test whether the skew is different from the normal distribution.

This function tests the null hypothesis that the skewness of the population that the sample was drawn from is the same as that of a corresponding normal distribution.

a : array
The data to be tested
axis : int or None, optional
Axis along which statistics are calculated. Default is 0. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float
The computed z-score for this test.
pvalue : float
a 2-sided p-value for the hypothesis test

The sample size must be at least 8.

 [1] R. B. D’Agostino, A. J. Belanger and R. B. D’Agostino Jr., “A suggestion for using powerful and informative tests of normality”, American Statistician 44, pp. 316-321, 1990.
```>>> from scipy.stats import skewtest
>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8])
SkewtestResult(statistic=1.0108048609177787, pvalue=0.31210983614218968)
>>> skewtest([2, 8, 0, 4, 1, 9, 9, 0])
SkewtestResult(statistic=0.44626385374196975, pvalue=0.65540666312754592)
>>> skewtest([1, 2, 3, 4, 5, 6, 7, 8000])
SkewtestResult(statistic=3.5717735103604071, pvalue=0.00035457199058231331)
>>> skewtest([100, 100, 100, 100, 100, 100, 100, 101])
SkewtestResult(statistic=3.5717766638478072, pvalue=0.000354567720281634)
```
`kurtosistest`(a, axis=0, nan_policy="propagate")

Test whether a dataset has normal kurtosis.

This function tests the null hypothesis that the kurtosis of the population from which the sample was drawn is that of the normal distribution: `kurtosis = 3(n-1)/(n+1)`.

a : array
array of the sample data
axis : int or None, optional
Axis along which to compute test. Default is 0. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float
The computed z-score for this test.
pvalue : float
The 2-sided p-value for the hypothesis test

Valid only for n>20. The Z-score is set to 0 for bad entries. This function uses the method described in [1]_.

 [1] see e.g. F. J. Anscombe, W. J. Glynn, “Distribution of the kurtosis statistic b2 for normal samples”, Biometrika, vol. 70, pp. 227-234, 1983.
```>>> from scipy.stats import kurtosistest
>>> kurtosistest(list(range(20)))
KurtosistestResult(statistic=-1.7058104152122062, pvalue=0.088043383325283484)
```
```>>> np.random.seed(28041990)
>>> s = np.random.normal(0, 1, 1000)
>>> kurtosistest(s)
KurtosistestResult(statistic=1.2317590987707365, pvalue=0.21803908613450895)
```
`normaltest`(a, axis=0, nan_policy="propagate")

Test whether a sample differs from a normal distribution.

This function tests the null hypothesis that a sample comes from a normal distribution. It is based on D’Agostino and Pearson’s [1]_, [2]_ test that combines skew and kurtosis to produce an omnibus test of normality.

a : array_like
The array containing the sample to be tested.
axis : int or None, optional
Axis along which to compute test. Default is 0. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float or array
`s^2 + k^2`, where `s` is the z-score returned by skewtest and `k` is the z-score returned by kurtosistest.
pvalue : float or array
A 2-sided chi squared probability for the hypothesis test.
 [1] D’Agostino, R. B. (1971), “An omnibus test of normality for moderate and large sample size”, Biometrika, 58, 341-348
 [2] D’Agostino, R. and Pearson, E. S. (1973), “Tests for departure from normality”, Biometrika, 60, 613-622
```>>> from scipy import stats
>>> pts = 1000
>>> np.random.seed(28041990)
>>> a = np.random.normal(0, 1, size=pts)
>>> b = np.random.normal(2, 1, size=pts)
>>> x = np.concatenate((a, b))
>>> k2, p = stats.normaltest(x)
>>> alpha = 1e-3
>>> print("p = {:g}".format(p))
p = 3.27207e-11
>>> if p < alpha:  # null hypothesis: x comes from a normal distribution
...     print("The null hypothesis can be rejected")
... else:
...     print("The null hypothesis cannot be rejected")
The null hypothesis can be rejected
```
`jarque_bera`(x)

Perform the Jarque-Bera goodness of fit test on sample data.

The Jarque-Bera test tests whether the sample data has the skewness and kurtosis matching a normal distribution.

Note that this test only works for a large enough number of data samples (>2000) as the test statistic asymptotically has a Chi-squared distribution with 2 degrees of freedom.

x : array_like
Observations of a random variable.
jb_value : float
The test statistic.
p : float
The p-value for the hypothesis test.
 [1] Jarque, C. and Bera, A. (1980) “Efficient tests for normality, homoscedasticity and serial independence of regression residuals”, 6 Econometric Letters 255-259.
```>>> from scipy import stats
>>> np.random.seed(987654321)
>>> x = np.random.normal(0, 1, 100000)
>>> y = np.random.rayleigh(1, 100000)
>>> stats.jarque_bera(x)
(4.7165707989581342, 0.09458225503041906)
>>> stats.jarque_bera(y)
(6713.7098548143422, 0.0)
```
`itemfreq`(a)

Return a 2-D array of item frequencies.

a : (N,) array_like
Input array.
itemfreq : (K, 2) ndarray
A 2-D frequency table. Column 1 contains sorted, unique values from a, column 2 contains their respective counts.
```>>> from scipy import stats
>>> a = np.array([1, 1, 5, 0, 1, 2, 2, 0, 1, 4])
>>> stats.itemfreq(a)
array([[ 0.,  2.],
[ 1.,  4.],
[ 2.,  2.],
[ 4.,  1.],
[ 5.,  1.]])
>>> np.bincount(a)
array([2, 4, 2, 0, 1, 1])
```
```>>> stats.itemfreq(a/10.)
array([[ 0. ,  2. ],
[ 0.1,  4. ],
[ 0.2,  2. ],
[ 0.4,  1. ],
[ 0.5,  1. ]])
```
`scoreatpercentile`(a, per, limit=tuple, interpolation_method="fraction", axis=None)

Calculate the score at a given percentile of the input sequence.

For example, the score at per=50 is the median. If the desired quantile lies between two data points, we interpolate between them, according to the value of interpolation. If the parameter limit is provided, it should be a tuple (lower, upper) of two values.

a : array_like
A 1-D array of values from which to extract score.
per : array_like
Percentile(s) at which to extract score. Values should be in range [0,100].
limit : tuple, optional
Tuple of two scalars, the lower and upper limits within which to compute the percentile. Values of a outside this (closed) interval will be ignored.
interpolation_method : {‘fraction’, ‘lower’, ‘higher’}, optional

This optional parameter specifies the interpolation method to use, when the desired quantile lies between two data points i and j

• fraction: `i + (j - i) * fraction` where `fraction` is the fractional part of the index surrounded by `i` and `j`.
• lower: `i`.
• higher: `j`.
axis : int, optional
Axis along which the percentiles are computed. Default is None. If None, compute over the whole array a.
score : float or ndarray
Score at percentile(s).

percentileofscore, numpy.percentile

This function will become obsolete in the future. For Numpy 1.9 and higher, numpy.percentile provides all the functionality that scoreatpercentile provides. And it’s significantly faster. Therefore it’s recommended to use numpy.percentile for users that have numpy >= 1.9.

```>>> from scipy import stats
>>> a = np.arange(100)
>>> stats.scoreatpercentile(a, 50)
49.5
```
`_compute_qth_percentile`(sorted, per, interpolation_method, axis)
`percentileofscore`(a, score, kind="rank")

The percentile rank of a score relative to a list of scores.

A percentileofscore of, for example, 80% means that 80% of the scores in a are below the given score. In the case of gaps or ties, the exact definition depends on the optional keyword, kind.

a : array_like
Array of scores to which score is compared.
score : int or float
Score that is compared to the elements in a.
kind : {‘rank’, ‘weak’, ‘strict’, ‘mean’}, optional

This optional parameter specifies the interpretation of the resulting score:

• “rank”: Average percentage ranking of score. In case of

multiple matches, average the percentage rankings of all matching scores.

• “weak”: This kind corresponds to the definition of a cumulative

distribution function. A percentileofscore of 80% means that 80% of values are less than or equal to the provided score.

• “strict”: Similar to “weak”, except that only values that are

strictly less than the given score are counted.

• “mean”: The average of the “weak” and “strict” scores, often used in

testing. See

http://en.wikipedia.org/wiki/Percentile_rank

pcos : float
Percentile-position of score (0-100) relative to a.

numpy.percentile

Three-quarters of the given values lie below a given score:

```>>> from scipy import stats
>>> stats.percentileofscore([1, 2, 3, 4], 3)
75.0
```

With multiple matches, note how the scores of the two matches, 0.6 and 0.8 respectively, are averaged:

```>>> stats.percentileofscore([1, 2, 3, 3, 4], 3)
70.0
```

Only 2/5 values are strictly less than 3:

```>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='strict')
40.0
```

But 4/5 values are less than or equal to 3:

```>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='weak')
80.0
```

The average between the weak and the strict scores is

```>>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='mean')
60.0
```
`_histogram`(a, numbins=10, defaultlimits=None, weights=None, printextras=False)

Separate the range into several bins and return the number of instances in each bin.

a : array_like
Array of scores which will be put into bins.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultlimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically `(a.min() - s, a.max() + s)`, where `s = (1/2)(a.max() - a.min()) / (numbins - 1)`.
weights : array_like, optional
The weights for each value in a. Default is None, which gives each value a weight of 1.0
printextras : bool, optional
If True, if there are extra points (i.e. the points that fall outside the bin limits) a warning is raised saying how many of those points there are. Default is False.
count : ndarray
Number of points (or sum of weights) in each bin.
lowerlimit : float
Lowest value of histogram, the lower limit of the first bin.
binsize : float
The size of the bins (all bins have the same size).
extrapoints : int
The number of points outside the range of the histogram.

numpy.histogram

This histogram is based on numpy’s histogram but has a larger range by default if default limits is not set.

`cumfreq`(a, numbins=10, defaultreallimits=None, weights=None)

Return a cumulative frequency histogram, using the histogram function.

A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin.

a : array_like
Input array.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultreallimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically `(a.min() - s, a.max() + s)`, where `s = (1/2)(a.max() - a.min()) / (numbins - 1)`.
weights : array_like, optional
The weights for each value in a. Default is None, which gives each value a weight of 1.0
cumcount : ndarray
Binned values of cumulative frequency.
lowerlimit : float
Lower real limit
binsize : float
Width of each bin.
extrapoints : int
Extra points.
```>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> x = [1, 4, 2, 1, 3, 1]
>>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5))
>>> res.cumcount
array([ 1.,  2.,  3.,  3.])
>>> res.extrapoints
3
```

Create a normal distribution with 1000 random values

```>>> rng = np.random.RandomState(seed=12345)
>>> samples = stats.norm.rvs(size=1000, random_state=rng)
```

Calculate cumulative frequencies

```>>> res = stats.cumfreq(samples, numbins=25)
```

Calculate space of values for x

```>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size,
...                                  res.cumcount.size)
```

Plot histogram and cumulative histogram

```>>> fig = plt.figure(figsize=(10, 4))
>>> ax1 = fig.add_subplot(1, 2, 1)
>>> ax2 = fig.add_subplot(1, 2, 2)
>>> ax1.hist(samples, bins=25)
>>> ax1.set_title('Histogram')
>>> ax2.bar(x, res.cumcount, width=res.binsize)
>>> ax2.set_title('Cumulative histogram')
>>> ax2.set_xlim([x.min(), x.max()])
```
```>>> plt.show()
```
`relfreq`(a, numbins=10, defaultreallimits=None, weights=None)

Return a relative frequency histogram, using the histogram function.

A relative frequency histogram is a mapping of the number of observations in each of the bins relative to the total of observations.

a : array_like
Input array.
numbins : int, optional
The number of bins to use for the histogram. Default is 10.
defaultreallimits : tuple (lower, upper), optional
The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically `(a.min() - s, a.max() + s)`, where `s = (1/2)(a.max() - a.min()) / (numbins - 1)`.
weights : array_like, optional
The weights for each value in a. Default is None, which gives each value a weight of 1.0
frequency : ndarray
Binned values of relative frequency.
lowerlimit : float
Lower real limit
binsize : float
Width of each bin.
extrapoints : int
Extra points.
```>>> import matplotlib.pyplot as plt
>>> from scipy import stats
>>> a = np.array([2, 4, 1, 2, 3, 2])
>>> res = stats.relfreq(a, numbins=4)
>>> res.frequency
array([ 0.16666667, 0.5       , 0.16666667,  0.16666667])
>>> np.sum(res.frequency)  # relative frequencies should add up to 1
1.0
```

Create a normal distribution with 1000 random values

```>>> rng = np.random.RandomState(seed=12345)
>>> samples = stats.norm.rvs(size=1000, random_state=rng)
```

Calculate relative frequencies

```>>> res = stats.relfreq(samples, numbins=25)
```

Calculate space of values for x

```>>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size,
...                                  res.frequency.size)
```

Plot relative frequency histogram

```>>> fig = plt.figure(figsize=(5, 4))
>>> ax = fig.add_subplot(1, 1, 1)
>>> ax.bar(x, res.frequency, width=res.binsize)
>>> ax.set_title('Relative frequency histogram')
>>> ax.set_xlim([x.min(), x.max()])
```
```>>> plt.show()
```
`obrientransform`(*args)

Compute the O’Brien transform on input data (any number of arrays).

Used to test for homogeneity of variance prior to running one-way stats. Each array in `*args` is one level of a factor. If f_oneway is run on the transformed data and found significant, the variances are unequal. From Maxwell and Delaney [1]_, p.112.

args : tuple of array_like
Any number of arrays.
obrientransform : ndarray
Transformed data for use in an ANOVA. The first dimension of the result corresponds to the sequence of transformed arrays. If the arrays given are all 1-D of the same length, the return value is a 2-D array; otherwise it is a 1-D array of type object, with each element being an ndarray.
 [1] S. E. Maxwell and H. D. Delaney, “Designing Experiments and Analyzing Data: A Model Comparison Perspective”, Wadsworth, 1990.

We’ll test the following data sets for differences in their variance.

```>>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10]
>>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15]
```

Apply the O’Brien transform to the data.

```>>> from scipy.stats import obrientransform
>>> tx, ty = obrientransform(x, y)
```

Use scipy.stats.f_oneway to apply a one-way ANOVA test to the transformed data.

```>>> from scipy.stats import f_oneway
>>> F, p = f_oneway(tx, ty)
>>> p
0.1314139477040335
```

If we require that `p < 0.05` for significance, we cannot conclude that the variances are different.

`sem`(a, axis=0, ddof=1, nan_policy="propagate")

Calculate the standard error of the mean (or standard error of measurement) of the values in the input array.

a : array_like
An array containing the values for which the standard error is returned.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
ddof : int, optional
Delta degrees-of-freedom. How many degrees of freedom to adjust for bias in limited samples relative to the population estimate of variance. Defaults to 1.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
s : ndarray or float
The standard error of the mean in the sample(s), along the input axis.

The default value for ddof is different to the default (0) used by other ddof containing routines, such as np.std and np.nanstd.

Find standard error along the first axis:

```>>> from scipy import stats
>>> a = np.arange(20).reshape(5,4)
>>> stats.sem(a)
array([ 2.8284,  2.8284,  2.8284,  2.8284])
```

Find standard error across the whole array, using n degrees of freedom:

```>>> stats.sem(a, axis=None, ddof=0)
1.2893796958227628
```
`zscore`(a, axis=0, ddof=0)

Calculate the z score of each value in the sample, relative to the sample mean and standard deviation.

a : array_like
An array like object containing the sample data.
axis : int or None, optional
Axis along which to operate. Default is 0. If None, compute over the whole array a.
ddof : int, optional
Degrees of freedom correction in the calculation of the standard deviation. Default is 0.
zscore : array_like
The z-scores, standardized by mean and standard deviation of input array a.

This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses asanyarray instead of asarray for parameters).

```>>> a = np.array([ 0.7972,  0.0767,  0.4383,  0.7866,  0.8091,
...                0.1954,  0.6307,  0.6599,  0.1065,  0.0508])
>>> from scipy import stats
>>> stats.zscore(a)
array([ 1.1273, -1.247 , -0.0552,  1.0923,  1.1664, -0.8559,  0.5786,
0.6748, -1.1488, -1.3324])
```

Computing along a specified axis, using n-1 degrees of freedom (`ddof=1`) to calculate the standard deviation:

```>>> b = np.array([[ 0.3148,  0.0478,  0.6243,  0.4608],
...               [ 0.7149,  0.0775,  0.6072,  0.9656],
...               [ 0.6341,  0.1403,  0.9759,  0.4064],
...               [ 0.5918,  0.6948,  0.904 ,  0.3721],
...               [ 0.0921,  0.2481,  0.1188,  0.1366]])
>>> stats.zscore(b, axis=1, ddof=1)
array([[-0.19264823, -1.28415119,  1.07259584,  0.40420358],
[ 0.33048416, -1.37380874,  0.04251374,  1.00081084],
[ 0.26796377, -1.12598418,  1.23283094, -0.37481053],
[-0.22095197,  0.24468594,  1.19042819, -1.21416216],
[-0.82780366,  1.4457416 , -0.43867764, -0.1792603 ]])
```
`zmap`(scores, compare, axis=0, ddof=0)

Calculate the relative z-scores.

Return an array of z-scores, i.e., scores that are standardized to zero mean and unit variance, where mean and variance are calculated from the comparison array.

scores : array_like
The input for which z-scores are calculated.
compare : array_like
The input from which the mean and standard deviation of the normalization are taken; assumed to have the same dimension as scores.
axis : int or None, optional
Axis over which mean and variance of compare are calculated. Default is 0. If None, compute over the whole array scores.
ddof : int, optional
Degrees of freedom correction in the calculation of the standard deviation. Default is 0.
zscore : array_like
Z-scores, in the same shape as scores.

This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses asanyarray instead of asarray for parameters).

```>>> from scipy.stats import zmap
>>> a = [0.5, 2.0, 2.5, 3]
>>> b = [0, 1, 2, 3, 4]
>>> zmap(a, b)
array([-1.06066017,  0.        ,  0.35355339,  0.70710678])
```
`iqr`(x, axis=None, rng=tuple, scale="raw", nan_policy="propagate", interpolation="linear", keepdims=False)

Compute the interquartile range of the data along the specified axis.

The interquartile range (IQR) is the difference between the 75th and 25th percentile of the data. It is a measure of the dispersion similar to standard deviation or variance, but is much more robust against outliers [2]_.

The `rng` parameter allows this function to compute other percentile ranges than the actual IQR. For example, setting `rng=(0, 100)` is equivalent to numpy.ptp.

The IQR of an empty array is np.nan.

New in version 0.18.0.

x : array_like
Input array or object that can be converted to an array.
axis : int or sequence of int, optional
Axis along which the range is computed. The default is to compute the IQR for the entire array.
rng : Two-element sequence containing floats in range of [0,100] optional
Percentiles over which to compute the range. Each must be between 0 and 100, inclusive. The default is the true IQR: (25, 75). The order of the elements is not important.
scale : scalar or str, optional

The numerical value of scale will be divided out of the final result. The following string values are recognized:

‘raw’ : No scaling, just return the raw IQR. ‘normal’ : Scale by .

The default is ‘raw’. Array-like scale is also allowed, as long as it broadcasts correctly to the output such that `out / scale` is a valid operation. The output dimensions depend on the input array, x, the axis argument, and the keepdims flag.

nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
interpolation : {‘linear’, ‘lower’, ‘higher’, ‘midpoint’, ‘nearest’}, optional

Specifies the interpolation method to use when the percentile boundaries lie between two data points i and j:

• ‘linear’ : i + (j - i) * fraction, where fraction is the

fractional part of the index surrounded by i and j.

• ‘lower’ : i.

• ‘higher’ : j.

• ‘nearest’ : i or j whichever is nearest.

• ‘midpoint’ : (i + j) / 2.

Default is ‘linear’.

keepdims : bool, optional
If this is set to True, the reduced axes are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array x.
iqr : scalar or ndarray
If `axis=None`, a scalar is returned. If the input contains integers or floats of smaller precision than `np.float64`, then the output data-type is `np.float64`. Otherwise, the output data-type is the same as that of the input.

numpy.std, numpy.var

```>>> from scipy.stats import iqr
>>> x = np.array([[10, 7, 4], [3, 2, 1]])
>>> x
array([[10,  7,  4],
[ 3,  2,  1]])
>>> iqr(x)
4.0
>>> iqr(x, axis=0)
array([ 3.5,  2.5,  1.5])
>>> iqr(x, axis=1)
array([ 3.,  1.])
>>> iqr(x, axis=1, keepdims=True)
array([[ 3.],
[ 1.]])
```

This function is heavily dependent on the version of numpy that is installed. Versions greater than 1.11.0b3 are highly recommended, as they include a number of enhancements and fixes to numpy.percentile and numpy.nanpercentile that affect the operation of this function. The following modifications apply:

Below 1.10.0 : nan_policy is poorly defined.
The default behavior of numpy.percentile is used for ‘propagate’. This is a hybrid of ‘omit’ and ‘propagate’ that mostly yields a skewed version of ‘omit’ since NaNs are sorted to the end of the data. A warning is raised if there are NaNs in the data.
Below 1.9.0: numpy.nanpercentile does not exist.
This means that numpy.percentile is used regardless of nan_policy and a warning is issued. See previous item for a description of the behavior.
Below 1.9.0: keepdims and interpolation are not supported.
The keywords get ignored with a warning if supplied with non-default values. However, multiple axes are still supported.
 [1] “Interquartile range” https://en.wikipedia.org/wiki/Interquartile_range
 [2] “Robust measures of scale” https://en.wikipedia.org/wiki/Robust_measures_of_scale
 [3] “Quantile” https://en.wikipedia.org/wiki/Quantile
`_iqr_percentile`(x, q, axis=None, interpolation="linear", keepdims=False, contains_nan=False)

Private wrapper that works around older versions of numpy.

While this function is pretty much necessary for the moment, it should be removed as soon as the minimum supported numpy version allows.

`_iqr_nanpercentile`(x, q, axis=None, interpolation="linear", keepdims=False, contains_nan=False)

Private wrapper that works around the following:

1. A bug in np.nanpercentile that was around until numpy version 1.11.0.
2. A bug in np.percentile NaN handling that was fixed in numpy version 1.10.0.
3. The non-existence of np.nanpercentile before numpy version 1.9.0.

While this function is pretty much necessary for the moment, it should be removed as soon as the minimum supported numpy version allows.

`sigmaclip`(a, low=4.0, high=4.0)

Iterative sigma-clipping of array elements.

The output array contains only those elements of the input array c that satisfy the conditions

```mean(c) - std(c)*low < c < mean(c) + std(c)*high
```

Starting from the full sample, all elements outside the critical range are removed. The iteration continues with a new critical range until no elements are outside the range.

a : array_like
Data array, will be raveled if not 1-D.
low : float, optional
Lower bound factor of sigma clipping. Default is 4.
high : float, optional
Upper bound factor of sigma clipping. Default is 4.
clipped : ndarray
Input array with clipped elements removed.
lower : float
Lower threshold value use for clipping.
upper : float
Upper threshold value use for clipping.
```>>> from scipy.stats import sigmaclip
>>> a = np.concatenate((np.linspace(9.5, 10.5, 31),
...                     np.linspace(0, 20, 5)))
>>> fact = 1.5
>>> c, low, upp = sigmaclip(a, fact, fact)
>>> c
array([  9.96666667,  10.        ,  10.03333333,  10.        ])
>>> c.var(), c.std()
(0.00055555555555555165, 0.023570226039551501)
>>> low, c.mean() - fact*c.std(), c.min()
(9.9646446609406727, 9.9646446609406727, 9.9666666666666668)
>>> upp, c.mean() + fact*c.std(), c.max()
(10.035355339059327, 10.035355339059327, 10.033333333333333)
```
```>>> a = np.concatenate((np.linspace(9.5, 10.5, 11),
...                     np.linspace(-100, -50, 3)))
>>> c, low, upp = sigmaclip(a, 1.8, 1.8)
>>> (c == np.linspace(9.5, 10.5, 11)).all()
True
```
`trimboth`(a, proportiontocut, axis=0)

Slices off a proportion of items from both ends of an array.

Slices off the passed proportion of items from both ends of the passed array (i.e., with proportiontocut = 0.1, slices leftmost 10% and rightmost 10% of scores). The trimmed values are the lowest and highest ones. Slices off less if proportion results in a non-integer slice index (i.e., conservatively slices off`proportiontocut`).

a : array_like
Data to trim.
proportiontocut : float
Proportion (in range 0-1) of total data set to trim of each end.
axis : int or None, optional
Axis along which to trim data. Default is 0. If None, compute over the whole array a.
out : ndarray
Trimmed version of array a. The order of the trimmed content is undefined.

trim_mean

```>>> from scipy import stats
>>> a = np.arange(20)
>>> b = stats.trimboth(a, 0.1)
>>> b.shape
(16,)
```
`trim1`(a, proportiontocut, tail="right", axis=0)

Slices off a proportion from ONE end of the passed array distribution.

If proportiontocut = 0.1, slices off ‘leftmost’ or ‘rightmost’ 10% of scores. The lowest or highest values are trimmed (depending on the tail). Slices off less if proportion results in a non-integer slice index (i.e., conservatively slices off proportiontocut ).

a : array_like
Input array
proportiontocut : float
Fraction to cut off of ‘left’ or ‘right’ of distribution
tail : {‘left’, ‘right’}, optional
Defaults to ‘right’.
axis : int or None, optional
Axis along which to trim data. Default is 0. If None, compute over the whole array a.
trim1 : ndarray
Trimmed version of array a. The order of the trimmed content is undefined.
`trim_mean`(a, proportiontocut, axis=0)

Return mean of array after trimming distribution from both tails.

If proportiontocut = 0.1, slices off ‘leftmost’ and ‘rightmost’ 10% of scores. The input is sorted before slicing. Slices off less if proportion results in a non-integer slice index (i.e., conservatively slices off proportiontocut ).

a : array_like
Input array
proportiontocut : float
Fraction to cut off of both tails of the distribution
axis : int or None, optional
Axis along which the trimmed means are computed. Default is 0. If None, compute over the whole array a.
trim_mean : ndarray
Mean of trimmed array.

trimboth tmean : compute the trimmed mean ignoring values outside given limits.

```>>> from scipy import stats
>>> x = np.arange(20)
>>> stats.trim_mean(x, 0.1)
9.5
>>> x2 = x.reshape(5, 4)
>>> x2
array([[ 0,  1,  2,  3],
[ 4,  5,  6,  7],
[ 8,  9, 10, 11],
[12, 13, 14, 15],
[16, 17, 18, 19]])
>>> stats.trim_mean(x2, 0.25)
array([  8.,   9.,  10.,  11.])
>>> stats.trim_mean(x2, 0.25, axis=1)
array([  1.5,   5.5,   9.5,  13.5,  17.5])
```
`f_oneway`(*args)

Performs a 1-way ANOVA.

The one-way ANOVA tests the null hypothesis that two or more groups have the same population mean. The test is applied to samples from two or more groups, possibly with differing sizes.

sample1, sample2, … : array_like
The sample measurements for each group.
statistic : float
The computed F-value of the test.
pvalue : float
The associated p-value from the F-distribution.

The ANOVA test has important assumptions that must be satisfied in order for the associated p-value to be valid.

1. The samples are independent.
2. Each sample is from a normally distributed population.
3. The population standard deviations of the groups are all equal. This property is known as homoscedasticity.

If these assumptions are not true for a given set of data, it may still be possible to use the Kruskal-Wallis H-test (scipy.stats.kruskal) although with some loss of power.

The algorithm is from Heiman[2], pp.394-7.

 [1] Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 14. http://faculty.vassar.edu/lowry/ch14pt1.html
 [2] Heiman, G.W. Research Methods in Statistics. 2002.
 [3] McDonald, G. H. “Handbook of Biological Statistics”, One-way ANOVA. http://www.biostathandbook.com/onewayanova.html
```>>> import scipy.stats as stats
```

[3]_ Here are some data on a shell measurement (the length of the anterior adductor muscle scar, standardized by dividing by length) in the mussel Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon; Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a much larger data set used in McDonald et al. (1991).

```>>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735,
...              0.0659, 0.0923, 0.0836]
>>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835,
...            0.0725]
>>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105]
>>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764,
...            0.0689]
>>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045]
>>> stats.f_oneway(tillamook, newport, petersburg, magadan, tvarminne)
(7.1210194716424473, 0.00028122423145345439)
```
`pearsonr`(x, y)

r Calculate a Pearson correlation coefficient and the p-value for testing non-correlation.

The Pearson correlation coefficient measures the linear relationship between two datasets. Strictly speaking, Pearson’s correlation requires that each dataset be normally distributed, and not necessarily zero-mean. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases.

The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Pearson correlation at least as extreme as the one computed from these datasets. The p-values are not entirely reliable but are probably reasonable for datasets larger than 500 or so.

x : (N,) array_like
Input
y : (N,) array_like
Input
r : float
Pearson’s correlation coefficient
p-value : float
2-tailed p-value

The correlation coefficient is calculated as follows:

where is the mean of the vector and is the mean of the vector .

http://www.statsoft.com/textbook/glosp.html#Pearson%20Correlation

```>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pearsonr(a, b)
(0.8660254037844386, 0.011724811003954654)
```
```>>> stats.pearsonr([1,2,3,4,5], [5,6,7,8,7])
(0.83205029433784372, 0.080509573298498519)
```
`fisher_exact`(table, alternative="two-sided")

Performs a Fisher exact test on a 2x2 contingency table.

table : array_like of ints
A 2x2 contingency table. Elements should be non-negative integers.
alternative : {‘two-sided’, ‘less’, ‘greater’}, optional
Which alternative hypothesis to the null hypothesis the test uses. Default is ‘two-sided’.
oddsratio : float
This is prior odds ratio and not a posterior estimate.
p_value : float
P-value, the probability of obtaining a distribution at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.
chi2_contingency : Chi-square test of independence of variables in a
contingency table.

The calculated odds ratio is different from the one R uses. This scipy implementation returns the (more common) “unconditional Maximum Likelihood Estimate”, while R uses the “conditional Maximum Likelihood Estimate”.

For tables with large numbers, the (inexact) chi-square test implemented in the function chi2_contingency can also be used.

Say we spend a few days counting whales and sharks in the Atlantic and Indian oceans. In the Atlantic ocean we find 8 whales and 1 shark, in the Indian ocean 2 whales and 5 sharks. Then our contingency table is:

```        Atlantic  Indian
whales     8        2
sharks     1        5
```

We use this table to find the p-value:

```>>> import scipy.stats as stats
>>> oddsratio, pvalue = stats.fisher_exact([[8, 2], [1, 5]])
>>> pvalue
0.0349...
```

The probability that we would observe this or an even more imbalanced ratio by chance is about 3.5%. A commonly used significance level is 5%–if we adopt that, we can therefore conclude that our observed imbalance is statistically significant; whales prefer the Atlantic while sharks prefer the Indian ocean.

`spearmanr`(a, b=None, axis=0, nan_policy="propagate")

Calculate a Spearman rank-order correlation coefficient and the p-value to test for non-correlation.

The Spearman correlation is a nonparametric measure of the monotonicity of the relationship between two datasets. Unlike the Pearson correlation, the Spearman correlation does not assume that both datasets are normally distributed. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact monotonic relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases.

The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Spearman correlation at least as extreme as the one computed from these datasets. The p-values are not entirely reliable but are probably reasonable for datasets larger than 500 or so.

a, b : 1D or 2D array_like, b is optional
One or two 1-D or 2-D arrays containing multiple variables and observations. When these are 1-D, each represents a vector of observations of a single variable. For the behavior in the 2-D case, see under `axis`, below. Both arrays need to have the same length in the `axis` dimension.
axis : int or None, optional
If axis=0 (default), then each column represents a variable, with observations in the rows. If axis=1, the relationship is transposed: each row represents a variable, while the columns contain observations. If axis=None, then both arrays will be raveled.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
correlation : float or ndarray (2-D square)
Spearman correlation matrix or correlation coefficient (if only 2 variables are given as parameters. Correlation matrix is square with length equal to total number of variables (columns or rows) in a and b combined.
pvalue : float
The two-sided p-value for a hypothesis test whose null hypothesis is that two sets of data are uncorrelated, has same dimension as rho.

Changes in scipy 0.8.0: rewrite to add tie-handling, and axis.

 [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 14.7
```>>> from scipy import stats
>>> stats.spearmanr([1,2,3,4,5], [5,6,7,8,7])
(0.82078268166812329, 0.088587005313543798)
>>> np.random.seed(1234321)
>>> x2n = np.random.randn(100, 2)
>>> y2n = np.random.randn(100, 2)
>>> stats.spearmanr(x2n)
(0.059969996999699973, 0.55338590803773591)
>>> stats.spearmanr(x2n[:,0], x2n[:,1])
(0.059969996999699973, 0.55338590803773591)
>>> rho, pval = stats.spearmanr(x2n, y2n)
>>> rho
array([[ 1.        ,  0.05997   ,  0.18569457,  0.06258626],
[ 0.05997   ,  1.        ,  0.110003  ,  0.02534653],
[ 0.18569457,  0.110003  ,  1.        ,  0.03488749],
[ 0.06258626,  0.02534653,  0.03488749,  1.        ]])
>>> pval
array([[ 0.        ,  0.55338591,  0.06435364,  0.53617935],
[ 0.55338591,  0.        ,  0.27592895,  0.80234077],
[ 0.06435364,  0.27592895,  0.        ,  0.73039992],
[ 0.53617935,  0.80234077,  0.73039992,  0.        ]])
>>> rho, pval = stats.spearmanr(x2n.T, y2n.T, axis=1)
>>> rho
array([[ 1.        ,  0.05997   ,  0.18569457,  0.06258626],
[ 0.05997   ,  1.        ,  0.110003  ,  0.02534653],
[ 0.18569457,  0.110003  ,  1.        ,  0.03488749],
[ 0.06258626,  0.02534653,  0.03488749,  1.        ]])
>>> stats.spearmanr(x2n, y2n, axis=None)
(0.10816770419260482, 0.1273562188027364)
>>> stats.spearmanr(x2n.ravel(), y2n.ravel())
(0.10816770419260482, 0.1273562188027364)
```
```>>> xint = np.random.randint(10, size=(100, 2))
>>> stats.spearmanr(xint)
(0.052760927029710199, 0.60213045837062351)
```
`pointbiserialr`(x, y)

r Calculate a point biserial correlation coefficient and its p-value.

The point biserial correlation is used to measure the relationship between a binary variable, x, and a continuous variable, y. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply a determinative relationship.

This function uses a shortcut formula but produces the same result as pearsonr.

x : array_like of bools
Input array.
y : array_like
Input array.
correlation : float
R value
pvalue : float
2-tailed p-value

pointbiserialr uses a t-test with `n-1` degrees of freedom. It is equivalent to pearsonr.

The value of the point-biserial correlation can be calculated from:

Where and are means of the metric observations coded 0 and 1 respectively; and are number of observations coded 0 and 1 respectively; is the total number of observations and is the standard deviation of all the metric observations.

A value of that is significantly different from zero is completely equivalent to a significant difference in means between the two groups. Thus, an independent groups t Test with degrees of freedom may be used to test whether is nonzero. The relation between the t-statistic for comparing two independent groups and is given by:

 [1] J. Lev, “The Point Biserial Coefficient of Correlation”, Ann. Math. Statist., Vol. 20, no.1, pp. 125-126, 1949.
 [2] R.F. Tate, “Correlation Between a Discrete and a Continuous Variable. Point-Biserial Correlation.”, Ann. Math. Statist., Vol. 25, np. 3, pp. 603-607, 1954.
```>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pointbiserialr(a, b)
(0.8660254037844386, 0.011724811003954652)
>>> stats.pearsonr(a, b)
(0.86602540378443871, 0.011724811003954626)
>>> np.corrcoef(a, b)
array([[ 1.       ,  0.8660254],
[ 0.8660254,  1.       ]])
```
`kendalltau`(x, y, initial_lexsort=None, nan_policy="propagate")

Calculate Kendall’s tau, a correlation measure for ordinal data.

Kendall’s tau is a measure of the correspondence between two rankings. Values close to 1 indicate strong agreement, values close to -1 indicate strong disagreement. This is the 1945 “tau-b” version of Kendall’s tau [2]_, which can account for ties and which reduces to the 1938 “tau-a” version [1]_ in absence of ties.

x, y : array_like
Arrays of rankings, of the same shape. If arrays are not 1-D, they will be flattened to 1-D.
initial_lexsort : bool, optional
Unused (deprecated).
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’. Note that if the input contains nan ‘omit’ delegates to mstats_basic.kendalltau(), which has a different implementation.
correlation : float
The tau statistic.
pvalue : float
The two-sided p-value for a hypothesis test whose null hypothesis is an absence of association, tau = 0.

spearmanr : Calculates a Spearman rank-order correlation coefficient. theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). weightedtau : Computes a weighted version of Kendall’s tau.

The definition of Kendall’s tau that is used is [2]_:

```tau = (P - Q) / sqrt((P + Q + T) * (P + Q + U))
```

where P is the number of concordant pairs, Q the number of discordant pairs, T the number of ties only in x, and U the number of ties only in y. If a tie occurs for the same pair in both x and y, it is not added to either T or U.

 [1] Maurice G. Kendall, “A New Measure of Rank Correlation”, Biometrika Vol. 30, No. 1/2, pp. 81-93, 1938.
 [2] Maurice G. Kendall, “The treatment of ties in ranking problems”, Biometrika Vol. 33, No. 3, pp. 239-251. 1945.
 [3] Gottfried E. Noether, “Elements of Nonparametric Statistics”, John Wiley & Sons, 1967.
 [4] Peter M. Fenwick, “A new data structure for cumulative frequency tables”, Software: Practice and Experience, Vol. 24, No. 3, pp. 327-336, 1994.
```>>> from scipy import stats
>>> x1 = [12, 2, 1, 12, 2]
>>> x2 = [1, 4, 7, 1, 0]
>>> tau, p_value = stats.kendalltau(x1, x2)
>>> tau
-0.47140452079103173
>>> p_value
0.2827454599327748
```
`weightedtau`(x, y, rank=True, weigher=None, additive=True)

r Compute a weighted version of Kendall’s .

The weighted is a weighted version of Kendall’s in which exchanges of high weight are more influential than exchanges of low weight. The default parameters compute the additive hyperbolic version of the index, , which has been shown to provide the best balance between important and unimportant elements [1]_.

The weighting is defined by means of a rank array, which assigns a nonnegative rank to each element, and a weigher function, which assigns a weight based from the rank to each element. The weight of an exchange is then the sum or the product of the weights of the ranks of the exchanged elements. The default parameters compute : an exchange between elements with rank and (starting from zero) has weight .

Specifying a rank array is meaningful only if you have in mind an external criterion of importance. If, as it usually happens, you do not have in mind a specific rank, the weighted is defined by averaging the values obtained using the decreasing lexicographical rank by (x, y) and by (y, x). This is the behavior with default parameters.

Note that if you are computing the weighted on arrays of ranks, rather than of scores (i.e., a larger value implies a lower rank) you must negate the ranks, so that elements of higher rank are associated with a larger value.

x, y : array_like
Arrays of scores, of the same shape. If arrays are not 1-D, they will be flattened to 1-D.
rank: array_like of ints or bool, optional
A nonnegative rank assigned to each element. If it is None, the decreasing lexicographical rank by (x, y) will be used: elements of higher rank will be those with larger x-values, using y-values to break ties (in particular, swapping x and y will give a different result). If it is False, the element indices will be used directly as ranks. The default is True, in which case this function returns the average of the values obtained using the decreasing lexicographical rank by (x, y) and by (y, x).
weigher : callable, optional
The weigher function. Must map nonnegative integers (zero representing the most important element) to a nonnegative weight. The default, None, provides hyperbolic weighing, that is, rank is mapped to weight .
If True, the weight of an exchange is computed by adding the weights of the ranks of the exchanged elements; otherwise, the weights are multiplied. The default is True.
correlation : float
The weighted correlation index.
pvalue : float
Presently `np.nan`, as the null statistics is unknown (even in the additive hyperbolic case).

kendalltau : Calculates Kendall’s tau. spearmanr : Calculates a Spearman rank-order correlation coefficient. theilslopes : Computes the Theil-Sen estimator for a set of points (x, y).

This function uses an , mergesort-based algorithm [1]_ that is a weighted extension of Knight’s algorithm for Kendall’s [2]_. It can compute Shieh’s weighted [3]_ between rankings without ties (i.e., permutations) by setting additive and rank to False, as the definition given in [1]_ is a generalization of Shieh’s.

NaNs are considered the smallest possible score.

New in version 0.19.0.

 [1] Sebastiano Vigna, “A weighted correlation index for rankings with ties”, Proceedings of the 24th international conference on World Wide Web, pp. 1166-1176, ACM, 2015.
 [2] W.R. Knight, “A Computer Method for Calculating Kendall’s Tau with Ungrouped Data”, Journal of the American Statistical Association, Vol. 61, No. 314, Part 1, pp. 436-439, 1966.
 [3] Grace S. Shieh. “A weighted Kendall’s tau statistic”, Statistics & Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998.
```>>> from scipy import stats
>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> tau, p_value = stats.weightedtau(x, y)
>>> tau
-0.56694968153682723
>>> p_value
nan
>>> tau, p_value = stats.weightedtau(x, y, additive=False)
>>> tau
-0.62205716951801038
```

NaNs are considered the smallest possible score:

```>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, np.nan]
>>> tau, _ = stats.weightedtau(x, y)
>>> tau
-0.56694968153682723
```

This is exactly Kendall’s tau:

```>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> tau, _ = stats.weightedtau(x, y, weigher=lambda x: 1)
>>> tau
-0.47140452079103173
```
```>>> x = [12, 2, 1, 12, 2]
>>> y = [1, 4, 7, 1, 0]
>>> stats.weightedtau(x, y, rank=None)
WeightedTauResult(correlation=-0.4157652301037516, pvalue=nan)
>>> stats.weightedtau(y, x, rank=None)
WeightedTauResult(correlation=-0.71813413296990281, pvalue=nan)
```
`ttest_1samp`(a, popmean, axis=0, nan_policy="propagate")

Calculate the T-test for the mean of ONE group of scores.

This is a two-sided test for the null hypothesis that the expected value (mean) of a sample of independent observations a is equal to the given population mean, popmean.

a : array_like
sample observation
popmean : float or array_like
expected value in null hypothesis, if array_like than it must have the same shape as a excluding the axis dimension
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole array a.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value
```>>> from scipy import stats
```
```>>> np.random.seed(7654567)  # fix seed to get the same result
>>> rvs = stats.norm.rvs(loc=5, scale=10, size=(50,2))
```

Test if mean of random sample is equal to true mean, and different mean. We reject the null hypothesis in the second case and don’t reject it in the first case.

```>>> stats.ttest_1samp(rvs,5.0)
(array([-0.68014479, -0.04323899]), array([ 0.49961383,  0.96568674]))
>>> stats.ttest_1samp(rvs,0.0)
(array([ 2.77025808,  4.11038784]), array([ 0.00789095,  0.00014999]))
```

Examples using axis and non-scalar dimension for population mean.

```>>> stats.ttest_1samp(rvs,[5.0,0.0])
(array([-0.68014479,  4.11038784]), array([  4.99613833e-01,   1.49986458e-04]))
>>> stats.ttest_1samp(rvs.T,[5.0,0.0],axis=1)
(array([-0.68014479,  4.11038784]), array([  4.99613833e-01,   1.49986458e-04]))
>>> stats.ttest_1samp(rvs,[[5.0],[0.0]])
(array([[-0.68014479, -0.04323899],
[ 2.77025808,  4.11038784]]), array([[  4.99613833e-01,   9.65686743e-01],
[  7.89094663e-03,   1.49986458e-04]]))
```
`_ttest_finish`(df, t)

Common code between all 3 t-test functions.

`_ttest_ind_from_stats`(mean1, mean2, denom, df)
`_unequal_var_ttest_denom`(v1, n1, v2, n2)
`_equal_var_ttest_denom`(v1, n1, v2, n2)
`ttest_ind_from_stats`(mean1, std1, nobs1, mean2, std2, nobs2, equal_var=True)

T-test for means of two independent samples from descriptive statistics.

This is a two-sided test for the null hypothesis that two independent samples have identical average (expected) values.

mean1 : array_like
The mean(s) of sample 1.
std1 : array_like
The standard deviation(s) of sample 1.
nobs1 : array_like
The number(s) of observations of sample 1.
mean2 : array_like
The mean(s) of sample 2
std2 : array_like
The standard deviations(s) of sample 2.
nobs2 : array_like
The number(s) of observations of sample 2.
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]_. If False, perform Welch’s t-test, which does not assume equal population variance [2]_.
statistic : float or array
The calculated t-statistics
pvalue : float or array
The two-tailed p-value.

scipy.stats.ttest_ind

New in version 0.16.0.

Suppose we have the summary data for two samples, as follows:

```                 Sample   Sample
Size   Mean   Variance
Sample 1    13    15.0     87.5
Sample 2    11    12.0     39.0
```

Apply the t-test to this data (with the assumption that the population variances are equal):

```>>> from scipy.stats import ttest_ind_from_stats
>>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13,
...                      mean2=12.0, std2=np.sqrt(39.0), nobs2=11)
Ttest_indResult(statistic=0.90513580933102689, pvalue=0.37519967975814872)
```

For comparison, here is the data from which those summary statistics were taken. With this data, we can compute the same result using scipy.stats.ttest_ind:

```>>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26])
>>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21])
>>> from scipy.stats import ttest_ind
>>> ttest_ind(a, b)
Ttest_indResult(statistic=0.905135809331027, pvalue=0.37519967975814861)
```
`ttest_ind`(a, b, axis=0, equal_var=True, nan_policy="propagate")

Calculate the T-test for the means of two independent samples of scores.

This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values. This test assumes that the populations have identical variances by default.

a, b : array_like
The arrays must have the same shape, except in the dimension corresponding to axis (the first, by default).
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole arrays, a, and b.
equal_var : bool, optional

If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]_. If False, perform Welch’s t-test, which does not assume equal population variance [2]_.

New in version 0.11.0.

nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float or array
The calculated t-statistic.
pvalue : float or array
The two-tailed p-value.

We can use this test, if we observe two independent samples from the same or different population, e.g. exam scores of boys and girls or of two ethnic groups. The test measures whether the average (expected) value differs significantly across samples. If we observe a large p-value, for example larger than 0.05 or 0.1, then we cannot reject the null hypothesis of identical average scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, then we reject the null hypothesis of equal averages.

```>>> from scipy import stats
>>> np.random.seed(12345678)
```

Test with sample with identical means:

```>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> rvs2 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> stats.ttest_ind(rvs1,rvs2)
(0.26833823296239279, 0.78849443369564776)
>>> stats.ttest_ind(rvs1,rvs2, equal_var = False)
(0.26833823296239279, 0.78849452749500748)
```

ttest_ind underestimates p for unequal variances:

```>>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500)
>>> stats.ttest_ind(rvs1, rvs3)
(-0.46580283298287162, 0.64145827413436174)
>>> stats.ttest_ind(rvs1, rvs3, equal_var = False)
(-0.46580283298287162, 0.64149646246569292)
```

When n1 != n2, the equal variance t-statistic is no longer equal to the unequal variance t-statistic:

```>>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100)
>>> stats.ttest_ind(rvs1, rvs4)
(-0.99882539442782481, 0.3182832709103896)
>>> stats.ttest_ind(rvs1, rvs4, equal_var = False)
(-0.69712570584654099, 0.48716927725402048)
```

T-test with different means, variance, and n:

```>>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100)
>>> stats.ttest_ind(rvs1, rvs5)
(-1.4679669854490653, 0.14263895620529152)
>>> stats.ttest_ind(rvs1, rvs5, equal_var = False)
(-0.94365973617132992, 0.34744170334794122)
```
`ttest_rel`(a, b, axis=0, nan_policy="propagate")

Calculate the T-test on TWO RELATED samples of scores, a and b.

This is a two-sided test for the null hypothesis that 2 related or repeated samples have identical average (expected) values.

a, b : array_like
The arrays must have the same shape.
axis : int or None, optional
Axis along which to compute test. If None, compute over the whole arrays, a, and b.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float or array
t-statistic
pvalue : float or array
two-tailed p-value

Examples for the use are scores of the same set of student in different exams, or repeated sampling from the same units. The test measures whether the average score differs significantly across samples (e.g. exams). If we observe a large p-value, for example greater than 0.05 or 0.1 then we cannot reject the null hypothesis of identical average scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, then we reject the null hypothesis of equal averages. Small p-values are associated with large t-statistics.

https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples

```>>> from scipy import stats
>>> np.random.seed(12345678) # fix random seed to get same numbers
```
```>>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500)
>>> rvs2 = (stats.norm.rvs(loc=5,scale=10,size=500) +
...         stats.norm.rvs(scale=0.2,size=500))
>>> stats.ttest_rel(rvs1,rvs2)
(0.24101764965300962, 0.80964043445811562)
>>> rvs3 = (stats.norm.rvs(loc=8,scale=10,size=500) +
...         stats.norm.rvs(scale=0.2,size=500))
>>> stats.ttest_rel(rvs1,rvs3)
(-3.9995108708727933, 7.3082402191726459e-005)
```
`kstest`(rvs, cdf, args=tuple, N=20, alternative="two-sided", mode="approx")

Perform the Kolmogorov-Smirnov test for goodness of fit.

This performs a test of the distribution G(x) of an observed random variable against a given distribution F(x). Under the null hypothesis the two distributions are identical, G(x)=F(x). The alternative hypothesis can be either ‘two-sided’ (default), ‘less’ or ‘greater’. The KS test is only valid for continuous distributions.

rvs : str, array or callable
If a string, it should be the name of a distribution in scipy.stats. If an array, it should be a 1-D array of observations of random variables. If a callable, it should be a function to generate random variables; it is required to have a keyword argument size.
cdf : str or callable
If a string, it should be the name of a distribution in scipy.stats. If rvs is a string then cdf can be False or the same as rvs. If a callable, that callable is used to calculate the cdf.
args : tuple, sequence, optional
Distribution parameters, used if rvs or cdf are strings.
N : int, optional
Sample size if rvs is string or callable. Default is 20.
alternative : {‘two-sided’, ‘less’,’greater’}, optional
Defines the alternative hypothesis (see explanation above). Default is ‘two-sided’.
mode : ‘approx’ (default) or ‘asymp’, optional

Defines the distribution used for calculating the p-value.

• ‘approx’ : use approximation to exact distribution of test statistic
• ‘asymp’ : use asymptotic distribution of test statistic
statistic : float
KS test statistic, either D, D+ or D-.
pvalue : float
One-tailed or two-tailed p-value.

In the one-sided test, the alternative is that the empirical cumulative distribution function of the random variable is “less” or “greater” than the cumulative distribution function F(x) of the hypothesis, `G(x)<=F(x)`, resp. `G(x)>=F(x)`.

```>>> from scipy import stats
```
```>>> x = np.linspace(-15, 15, 9)
>>> stats.kstest(x, 'norm')
(0.44435602715924361, 0.038850142705171065)
```
```>>> np.random.seed(987654321) # set random seed to get the same result
>>> stats.kstest('norm', False, N=100)
(0.058352892479417884, 0.88531190944151261)
```

The above lines are equivalent to:

```>>> np.random.seed(987654321)
>>> stats.kstest(stats.norm.rvs(size=100), 'norm')
(0.058352892479417884, 0.88531190944151261)
```

Test against one-sided alternative hypothesis

Shift distribution to larger values, so that `cdf_dgp(x) < norm.cdf(x)`:

```>>> np.random.seed(987654321)
>>> x = stats.norm.rvs(loc=0.2, size=100)
>>> stats.kstest(x,'norm', alternative = 'less')
(0.12464329735846891, 0.040989164077641749)
```

Reject equal distribution against alternative hypothesis: less

```>>> stats.kstest(x,'norm', alternative = 'greater')
(0.0072115233216311081, 0.98531158590396395)
```

Don’t reject equal distribution against alternative hypothesis: greater

```>>> stats.kstest(x,'norm', mode='asymp')
(0.12464329735846891, 0.08944488871182088)
```

Testing t distributed random variables against normal distribution

With 100 degrees of freedom the t distribution looks close to the normal distribution, and the K-S test does not reject the hypothesis that the sample came from the normal distribution:

```>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(100,size=100),'norm')
(0.072018929165471257, 0.67630062862479168)
```

With 3 degrees of freedom the t distribution looks sufficiently different from the normal distribution, that we can reject the hypothesis that the sample came from the normal distribution at the 10% level:

```>>> np.random.seed(987654321)
>>> stats.kstest(stats.t.rvs(3,size=100),'norm')
(0.131016895759829, 0.058826222555312224)
```
`_count`(a, axis=None)

Count the number of non-masked elements of an array.

This function behaves like np.ma.count(), but is much faster for ndarrays.

`power_divergence`(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None)

Cressie-Read power divergence statistic and goodness of fit test.

This function tests the null hypothesis that the categorical data has the given frequencies, using the Cressie-Read power divergence statistic.

f_obs : array_like
Observed frequencies in each category.
f_exp : array_like, optional
Expected frequencies in each category. By default the categories are assumed to be equally likely.
ddof : int, optional
“Delta degrees of freedom”: adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with `k - 1 - ddof` degrees of freedom, where k is the number of observed frequencies. The default value of ddof is 0.
axis : int or None, optional
The axis of the broadcast result of f_obs and f_exp along which to apply the test. If axis is None, all values in f_obs are treated as a single data set. Default is 0.
lambda_ : float or str, optional

lambda_ gives the power in the Cressie-Read power divergence statistic. The default is 1. For convenience, lambda_ may be assigned one of the following strings, in which case the corresponding numerical value is used:

```String              Value   Description
"pearson"             1     Pearson's chi-squared statistic.
In this case, the function is
equivalent to `stats.chisquare`.
"log-likelihood"      0     Log-likelihood ratio. Also known as
the G-test [3]_.
"freeman-tukey"      -1/2   Freeman-Tukey statistic.
"mod-log-likelihood" -1     Modified log-likelihood ratio.
"neyman"             -2     Neyman's statistic.
"cressie-read"        2/3   The power recommended in [5]_.
```
statistic : float or ndarray
The Cressie-Read power divergence test statistic. The value is a float if axis is None or if` f_obs and f_exp are 1-D.
pvalue : float or ndarray
The p-value of the test. The value is a float if ddof and the return value stat are scalars.

chisquare

This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5.

When lambda_ is less than zero, the formula for the statistic involves dividing by f_obs, so a warning or error may be generated if any value in f_obs is 0.

Similarly, a warning or error may be generated if any value in f_exp is zero when lambda_ >= 0.

The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not a chisquare, in which case this test is not appropriate.

This function handles masked arrays. If an element of f_obs or f_exp is masked, then data at that position is ignored, and does not count towards the size of the data set.

New in version 0.13.0.

 [1] Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html
 [2] “Chi-squared test”, http://en.wikipedia.org/wiki/Chi-squared_test
 [3] “G-test”, http://en.wikipedia.org/wiki/G-test
 [4] Sokal, R. R. and Rohlf, F. J. “Biometry: the principles and practice of statistics in biological research”, New York: Freeman (1981)
 [5] Cressie, N. and Read, T. R. C., “Multinomial Goodness-of-Fit Tests”, J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), pp. 440-464.

(See chisquare for more examples.)

When just f_obs is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies. Here we perform a G-test (i.e. use the log-likelihood ratio statistic):

```>>> from scipy.stats import power_divergence
>>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood')
(2.006573162632538, 0.84823476779463769)
```

The expected frequencies can be given with the f_exp argument:

```>>> power_divergence([16, 18, 16, 14, 12, 12],
...                  f_exp=[16, 16, 16, 16, 16, 8],
...                  lambda_='log-likelihood')
(3.3281031458963746, 0.6495419288047497)
```

When f_obs is 2-D, by default the test is applied to each column.

```>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> power_divergence(obs, lambda_="log-likelihood")
(array([ 2.00657316,  6.77634498]), array([ 0.84823477,  0.23781225]))
```

By setting `axis=None`, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array.

```>>> power_divergence(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> power_divergence(obs.ravel())
(23.31034482758621, 0.015975692534127565)
```

ddof is the change to make to the default degrees of freedom.

```>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)
```

The calculation of the p-values is done by broadcasting the test statistic with ddof.

```>>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504,  0.73575888,  0.5724067 ]))
```

f_obs and f_exp are also broadcast. In the following, f_obs has shape (6,) and f_exp has shape (2, 6), so the result of broadcasting f_obs and f_exp has shape (2, 6). To compute the desired chi-squared statistics, we must use `axis=1`:

```>>> power_divergence([16, 18, 16, 14, 12, 12],
...                  f_exp=[[16, 16, 16, 16, 16, 8],
...                         [8, 20, 20, 16, 12, 12]],
...                  axis=1)
(array([ 3.5 ,  9.25]), array([ 0.62338763,  0.09949846]))
```
`chisquare`(f_obs, f_exp=None, ddof=0, axis=0)

Calculate a one-way chi square test.

The chi square test tests the null hypothesis that the categorical data has the given frequencies.

f_obs : array_like
Observed frequencies in each category.
f_exp : array_like, optional
Expected frequencies in each category. By default the categories are assumed to be equally likely.
ddof : int, optional
“Delta degrees of freedom”: adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with `k - 1 - ddof` degrees of freedom, where k is the number of observed frequencies. The default value of ddof is 0.
axis : int or None, optional
The axis of the broadcast result of f_obs and f_exp along which to apply the test. If axis is None, all values in f_obs are treated as a single data set. Default is 0.
chisq : float or ndarray
The chi-squared test statistic. The value is a float if axis is None or f_obs and f_exp are 1-D.
p : float or ndarray
The p-value of the test. The value is a float if ddof and the return value chisq are scalars.

power_divergence mstats.chisquare

This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5.

The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not a chisquare, in which case this test is not appropriate.

 [1] Lowry, Richard. “Concepts and Applications of Inferential Statistics”. Chapter 8. http://faculty.vassar.edu/lowry/ch8pt1.html
 [2] “Chi-squared test”, http://en.wikipedia.org/wiki/Chi-squared_test

When just f_obs is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies.

```>>> from scipy.stats import chisquare
>>> chisquare([16, 18, 16, 14, 12, 12])
(2.0, 0.84914503608460956)
```

With f_exp the expected frequencies can be given.

```>>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8])
(3.5, 0.62338762774958223)
```

When f_obs is 2-D, by default the test is applied to each column.

```>>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T
>>> obs.shape
(6, 2)
>>> chisquare(obs)
(array([ 2.        ,  6.66666667]), array([ 0.84914504,  0.24663415]))
```

By setting `axis=None`, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array.

```>>> chisquare(obs, axis=None)
(23.31034482758621, 0.015975692534127565)
>>> chisquare(obs.ravel())
(23.31034482758621, 0.015975692534127565)
```

ddof is the change to make to the default degrees of freedom.

```>>> chisquare([16, 18, 16, 14, 12, 12], ddof=1)
(2.0, 0.73575888234288467)
```

The calculation of the p-values is done by broadcasting the chi-squared statistic with ddof.

```>>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2])
(2.0, array([ 0.84914504,  0.73575888,  0.5724067 ]))
```

f_obs and f_exp are also broadcast. In the following, f_obs has shape (6,) and f_exp has shape (2, 6), so the result of broadcasting f_obs and f_exp has shape (2, 6). To compute the desired chi-squared statistics, we use `axis=1`:

```>>> chisquare([16, 18, 16, 14, 12, 12],
...           f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]],
...           axis=1)
(array([ 3.5 ,  9.25]), array([ 0.62338763,  0.09949846]))
```
`ks_2samp`(data1, data2)

Compute the Kolmogorov-Smirnov statistic on 2 samples.

This is a two-sided test for the null hypothesis that 2 independent samples are drawn from the same continuous distribution.

data1, data2 : sequence of 1-D ndarrays
two arrays of sample observations assumed to be drawn from a continuous distribution, sample sizes can be different
statistic : float
KS statistic
pvalue : float
two-tailed p-value

This tests whether 2 samples are drawn from the same distribution. Note that, like in the case of the one-sample K-S test, the distribution is assumed to be continuous.

This is the two-sided test, one-sided tests are not implemented. The test uses the two-sided asymptotic Kolmogorov-Smirnov distribution.

If the K-S statistic is small or the p-value is high, then we cannot reject the hypothesis that the distributions of the two samples are the same.

```>>> from scipy import stats
>>> np.random.seed(12345678)  #fix random seed to get the same result
>>> n1 = 200  # size of first sample
>>> n2 = 300  # size of second sample
```

For a different distribution, we can reject the null hypothesis since the pvalue is below 1%:

```>>> rvs1 = stats.norm.rvs(size=n1, loc=0., scale=1)
>>> rvs2 = stats.norm.rvs(size=n2, loc=0.5, scale=1.5)
>>> stats.ks_2samp(rvs1, rvs2)
(0.20833333333333337, 4.6674975515806989e-005)
```

For a slightly different distribution, we cannot reject the null hypothesis at a 10% or lower alpha since the p-value at 0.144 is higher than 10%

```>>> rvs3 = stats.norm.rvs(size=n2, loc=0.01, scale=1.0)
>>> stats.ks_2samp(rvs1, rvs3)
(0.10333333333333333, 0.14498781825751686)
```

For an identical distribution, we cannot reject the null hypothesis since the p-value is high, 41%:

```>>> rvs4 = stats.norm.rvs(size=n2, loc=0.0, scale=1.0)
>>> stats.ks_2samp(rvs1, rvs4)
(0.07999999999999996, 0.41126949729859719)
```
`tiecorrect`(rankvals)

Tie correction factor for ties in the Mann-Whitney U and Kruskal-Wallis H tests.

rankvals : array_like
A 1-D sequence of ranks. Typically this will be the array returned by stats.rankdata.
factor : float
Correction factor for U or H.

rankdata : Assign ranks to the data mannwhitneyu : Mann-Whitney rank test kruskal : Kruskal-Wallis H test

 [1] Siegel, S. (1956) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill.
```>>> from scipy.stats import tiecorrect, rankdata
>>> tiecorrect([1, 2.5, 2.5, 4])
0.9
>>> ranks = rankdata([1, 3, 2, 4, 5, 7, 2, 8, 4])
>>> ranks
array([ 1. ,  4. ,  2.5,  5.5,  7. ,  8. ,  2.5,  9. ,  5.5])
>>> tiecorrect(ranks)
0.9833333333333333
```
`mannwhitneyu`(x, y, use_continuity=True, alternative=None)

Compute the Mann-Whitney rank test on samples x and y.

x, y : array_like
Array of samples, should be one-dimensional.
use_continuity : bool, optional
Whether a continuity correction (1/2.) should be taken into account. Default is True.
alternative : None (deprecated), ‘less’, ‘two-sided’, or ‘greater’
Whether to get the p-value for the one-sided hypothesis (‘less’ or ‘greater’) or for the two-sided hypothesis (‘two-sided’). Defaults to None, which results in a p-value half the size of the ‘two-sided’ p-value and a different U statistic. The default behavior is not the same as using ‘less’ or ‘greater’: it only exists for backward compatibility and is deprecated.
statistic : float
The Mann-Whitney U statistic, equal to min(U for x, U for y) if alternative is equal to None (deprecated; exists for backward compatibility), and U for y otherwise.
pvalue : float
p-value assuming an asymptotic normal distribution. One-sided or two-sided, depending on the choice of alternative.

Use only when the number of observation in each sample is > 20 and you have 2 independent samples of ranks. Mann-Whitney U is significant if the u-obtained is LESS THAN or equal to the critical value of U.

This test corrects for ties and by default uses a continuity correction.

 [2] H.B. Mann and D.R. Whitney, “On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other,” The Annals of Mathematical Statistics, vol. 18, no. 1, pp. 50-60, 1947.
`ranksums`(x, y)

Compute the Wilcoxon rank-sum statistic for two samples.

The Wilcoxon rank-sum test tests the null hypothesis that two sets of measurements are drawn from the same distribution. The alternative hypothesis is that values in one sample are more likely to be larger than the values in the other sample.

This test should be used to compare two samples from continuous distributions. It does not handle ties between measurements in x and y. For tie-handling and an optional continuity correction see scipy.stats.mannwhitneyu.

x,y : array_like
The data from the two samples
statistic : float
The test statistic under the large-sample approximation that the rank sum statistic is normally distributed
pvalue : float
The two-sided p-value of the test
`kruskal`(*args, **kwargs)

Compute the Kruskal-Wallis H-test for independent samples

The Kruskal-Wallis H-test tests the null hypothesis that the population median of all of the groups are equal. It is a non-parametric version of ANOVA. The test works on 2 or more independent samples, which may have different sizes. Note that rejecting the null hypothesis does not indicate which of the groups differs. Post-hoc comparisons between groups are required to determine which groups are different.

sample1, sample2, … : array_like
Two or more arrays with the sample measurements can be given as arguments.
nan_policy : {‘propagate’, ‘raise’, ‘omit’}, optional
Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’.
statistic : float
The Kruskal-Wallis H statistic, corrected for ties
pvalue : float
The p-value for the test using the assumption that H has a chi square distribution

f_oneway : 1-way ANOVA mannwhitneyu : Mann-Whitney rank test on two samples. friedmanchisquare : Friedman test for repeated measurements

Due to the assumption that H has a chi square distribution, the number of samples in each group must not be too small. A typical rule is that each sample must have at least 5 measurements.

 [1] W. H. Kruskal & W. W. Wallis, “Use of Ranks in One-Criterion Variance Analysis”, Journal of the American Statistical Association, Vol. 47, Issue 260, pp. 583-621, 1952.
```>>> from scipy import stats
>>> x = [1, 3, 5, 7, 9]
>>> y = [2, 4, 6, 8, 10]
>>> stats.kruskal(x, y)
KruskalResult(statistic=0.27272727272727337, pvalue=0.60150813444058948)
```
```>>> x = [1, 1, 1]
>>> y = [2, 2, 2]
>>> z = [2, 2]
>>> stats.kruskal(x, y, z)
KruskalResult(statistic=7.0, pvalue=0.030197383422318501)
```
`friedmanchisquare`(*args)

Compute the Friedman test for repeated measurements

The Friedman test tests the null hypothesis that repeated measurements of the same individuals have the same distribution. It is often used to test for consistency among measurements obtained in different ways. For example, if two measurement techniques are used on the same set of individuals, the Friedman test can be used to determine if the two measurement techniques are consistent.

measurements1, measurements2, measurements3… : array_like
Arrays of measurements. All of the arrays must have the same number of elements. At least 3 sets of measurements must be given.
statistic : float
the test statistic, correcting for ties
pvalue : float
the associated p-value assuming that the test statistic has a chi squared distribution

Due to the assumption that the test statistic has a chi squared distribution, the p-value is only reliable for n > 10 and more than 6 repeated measurements.

`combine_pvalues`(pvalues, method="fisher", weights=None)

Methods for combining the p-values of independent tests bearing upon the same hypothesis.

pvalues : array_like, 1-D
Array of p-values assumed to come from independent tests.
method : {‘fisher’, ‘stouffer’}, optional

Name of method to use to combine p-values. The following methods are available:

• “fisher”: Fisher’s method (Fisher’s combined probability test), the default.
• “stouffer”: Stouffer’s Z-score method.
weights : array_like, 1-D, optional
Optional array of weights used only for Stouffer’s Z-score method.
statistic: float
The statistic calculated by the specified method: - “fisher”: The chi-squared statistic - “stouffer”: The Z-score
pval: float
The combined p-value.

Fisher’s method (also known as Fisher’s combined probability test) [1]_ uses a chi-squared statistic to compute a combined p-value. The closely related Stouffer’s Z-score method [2]_ uses Z-scores rather than p-values. The advantage of Stouffer’s method is that it is straightforward to introduce weights, which can make Stouffer’s method more powerful than Fisher’s method when the p-values are from studies of different size [3]_ [4]_.

Fisher’s method may be extended to combine p-values from dependent tests [5]_. Extensions such as Brown’s method and Kost’s method are not currently implemented.

New in version 0.15.0.

 [3] Whitlock, M. C. “Combining probability from independent tests: the weighted Z-method is superior to Fisher’s approach.” Journal of Evolutionary Biology 18, no. 5 (2005): 1368-1373.
 [4] Zaykin, Dmitri V. “Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis.” Journal of Evolutionary Biology 24, no. 8 (2011): 1836-1841.
`_betai`(a, b, x)
`wasserstein_distance`(u_values, v_values, u_weights=None, v_weights=None)

r Compute the first Wasserstein distance between two 1D distributions.

This distance is also known as the earth mover’s distance, since it can be seen as the minimum amount of “work” required to transform into , where “work” is measured as the amount of distribution weight that must be moved, multiplied by the distance it has to be moved.

New in version 1.0.0.

u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same weight. u_weights (resp. v_weights) must have the same length as u_values (resp. v_values). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1.
distance : float
The computed distance between the distributions.

The first Wasserstein distance between the distributions and is:

where is the set of (probability) distributions on whose marginals are and on the first and second factors respectively.

If and are the respective CDFs of and , this distance also equals to:

See [2]_ for a proof of the equivalence of both definitions.

The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values.

 [1] “Wasserstein metric”, http://en.wikipedia.org/wiki/Wasserstein_metric
 [2] Ramdas, Garcia, Cuturi “On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests” (2015). :arXiv:`1509.02237`.
```>>> from scipy.stats import wasserstein_distance
>>> wasserstein_distance([0, 1, 3], [5, 6, 8])
5.0
>>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2])
0.25
>>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4],
...                      [1.4, 0.9, 3.1, 7.2], [3.2, 3.5])
4.0781331438047861
```
`energy_distance`(u_values, v_values, u_weights=None, v_weights=None)

r Compute the energy distance between two 1D distributions.

New in version 1.0.0.

u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same weight. u_weights (resp. v_weights) must have the same length as u_values (resp. v_values). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1.
distance : float
The computed distance between the distributions.

The energy distance between two distributions and , whose respective CDFs are and , equals to:

where and (resp. and ) are independent random variables whose probability distribution is (resp. ).

As shown in [2]_, for one-dimensional real-valued variables, the energy distance is linked to the non-distribution-free version of the Cramer-von Mises distance:

Note that the common Cramer-von Mises criterion uses the distribution-free version of the distance. See [2]_ (section 2), for more details about both versions of the distance.

The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values.

 [1] “Energy distance”, https://en.wikipedia.org/wiki/Energy_distance
 [2] Szekely “E-statistics: The energy of statistical samples.” Bowling Green State University, Department of Mathematics and Statistics, Technical Report 02-16 (2002).
 [3] Rizzo, Szekely “Energy distance.” Wiley Interdisciplinary Reviews: Computational Statistics, 8(1):27-38 (2015).
 [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, Munos “The Cramer Distance as a Solution to Biased Wasserstein Gradients” (2017). :arXiv:`1705.10743`.
```>>> from scipy.stats import energy_distance
>>> energy_distance([0], [2])
2.0000000000000004
>>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2])
1.0000000000000002
>>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ],
...                 [2.1, 4.2, 7.4, 8. ], [7.6, 8.8])
0.88003340976158217
```
`_cdf_distance`(p, u_values, v_values, u_weights=None, v_weights=None)

r Compute, between two one-dimensional distributions and , whose respective CDFs are and , the statistical distance that is defined as:

p is a positive parameter; p = 1 gives the Wasserstein distance, p = 2 gives the energy distance.

u_values, v_values : array_like
Values observed in the (empirical) distribution.
u_weights, v_weights : array_like, optional
Weight for each value. If unspecified, each value is assigned the same weight. u_weights (resp. v_weights) must have the same length as u_values (resp. v_values). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1.
distance : float
The computed distance between the distributions.

The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values.

 [1] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, Munos “The Cramer Distance as a Solution to Biased Wasserstein Gradients” (2017). :arXiv:`1705.10743`.
`_validate_distribution`(values, weights)

Validate the values and weights from a distribution input of cdf_distance and return them as ndarray objects.

values : array_like
Values observed in the (empirical) distribution.
weights : array_like
Weight for each value.
values : ndarray
Values as ndarray.
weights : ndarray
Weights as ndarray.
`find_repeats`(arr)

Find repeats and repeat counts.

arr : array_like
Input array. This is cast to float64.
values : ndarray
The unique values from the (flattened) input that are repeated.
counts : ndarray
Number of times the corresponding ‘value’ is repeated.

In numpy >= 1.9 numpy.unique provides similar functionality. The main difference is that find_repeats only returns repeated values.

```>>> from scipy import stats
>>> stats.find_repeats([2, 1, 2, 3, 2, 2, 5])
RepeatedResults(values=array([ 2.]), counts=array([4]))
```
```>>> stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]])
RepeatedResults(values=array([ 4.,  5.]), counts=array([2, 2]))
```
`_sum_of_squares`(a, axis=0)

Square each element of the input array, and return the sum(s) of that.

a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate. Default is 0. If None, compute over the whole array a.
sum_of_squares : ndarray
The sum along the given axis for (a**2).

_square_of_sums : The square(s) of the sum(s) (the opposite of _sum_of_squares).

`_square_of_sums`(a, axis=0)

Sum elements of the input array, and return the square(s) of that sum.

a : array_like
Input array.
axis : int or None, optional
Axis along which to calculate. Default is 0. If None, compute over the whole array a.
square_of_sums : float or ndarray
The square of the sum over axis.

_sum_of_squares : The sum of squares (the opposite of square_of_sums).

`rankdata`(a, method="average")

Assign ranks to data, dealing with ties appropriately.

Ranks begin at 1. The method argument controls how ranks are assigned to equal values. See [1]_ for further discussion of ranking methods.

a : array_like
The array of values to be ranked. The array is first flattened.
method : str, optional

The method used to assign ranks to tied elements. The options are ‘average’, ‘min’, ‘max’, ‘dense’ and ‘ordinal’.

‘average’:
The average of the ranks that would have been assigned to all the tied values is assigned to each value.
‘min’:
The minimum of the ranks that would have been assigned to all the tied values is assigned to each value. (This is also referred to as “competition” ranking.)
‘max’:
The maximum of the ranks that would have been assigned to all the tied values is assigned to each value.
‘dense’:
Like ‘min’, but the rank of the next highest element is assigned the rank immediately after those assigned to the tied elements.
‘ordinal’:
All values are given a distinct rank, corresponding to the order that the values occur in a.

The default is ‘average’.

ranks : ndarray
An array of length equal to the size of a, containing rank scores.
 [1] “Ranking”, http://en.wikipedia.org/wiki/Ranking
```>>> from scipy.stats import rankdata
>>> rankdata([0, 2, 3, 2])
array([ 1. ,  2.5,  4. ,  2.5])
>>> rankdata([0, 2, 3, 2], method='min')
array([ 1,  2,  4,  2])
>>> rankdata([0, 2, 3, 2], method='max')
array([ 1,  3,  4,  3])
>>> rankdata([0, 2, 3, 2], method='dense')
array([ 1,  2,  3,  2])
>>> rankdata([0, 2, 3, 2], method='ordinal')
array([ 1,  2,  4,  3])
```