scipy.stats.chi2 = <scipy.stats._continuous_distns.chi2_gen object>[source]#

A chi-squared continuous random variable.

For the noncentral chi-square distribution, see ncx2.

As an instance of the rv_continuous class, chi2 object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

See also



The probability density function for chi2 is:

\[f(x, k) = \frac{1}{2^{k/2} \Gamma \left( k/2 \right)} x^{k/2-1} \exp \left( -x/2 \right)\]

for \(x > 0\) and \(k > 0\) (degrees of freedom, denoted df in the implementation).

chi2 takes df as a shape parameter.

The chi-squared distribution is a special case of the gamma distribution, with gamma parameters a = df/2, loc = 0 and scale = 2.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, chi2.pdf(x, df, loc, scale) is identically equivalent to chi2.pdf(y, df) / scale with y = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.


>>> import numpy as np
>>> from scipy.stats import chi2
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> df = 55
>>> mean, var, skew, kurt = chi2.stats(df, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(chi2.ppf(0.01, df),
...                 chi2.ppf(0.99, df), 100)
>>> ax.plot(x, chi2.pdf(x, df),
...        'r-', lw=5, alpha=0.6, label='chi2 pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

>>> rv = chi2(df)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = chi2.ppf([0.001, 0.5, 0.999], df)
>>> np.allclose([0.001, 0.5, 0.999], chi2.cdf(vals, df))

Generate random numbers:

>>> r = chi2.rvs(df, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)


rvs(df, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, df, loc=0, scale=1)

Probability density function.

logpdf(x, df, loc=0, scale=1)

Log of the probability density function.

cdf(x, df, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, df, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, df, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, df, loc=0, scale=1)

Log of the survival function.

ppf(q, df, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, df, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, df, loc=0, scale=1)

Non-central moment of the specified order.

stats(df, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(df, loc=0, scale=1)

(Differential) entropy of the RV.


Parameter estimates for generic data. See for detailed documentation of the keyword arguments.

expect(func, args=(df,), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)

Expected value of a function (of one argument) with respect to the distribution.

median(df, loc=0, scale=1)

Median of the distribution.

mean(df, loc=0, scale=1)

Mean of the distribution.

var(df, loc=0, scale=1)

Variance of the distribution.

std(df, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, df, loc=0, scale=1)

Confidence interval with equal areas around the median.