scipy.stats.t = <scipy.stats._continuous_distns.t_gen object>[source]#

A Student’s t continuous random variable.

For the noncentral t distribution, see nct.

As an instance of the rv_continuous class, t 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 t is:

\[f(x, \nu) = \frac{\Gamma((\nu+1)/2)} {\sqrt{\pi \nu} \Gamma(\nu/2)} (1+x^2/\nu)^{-(\nu+1)/2}\]

where \(x\) is a real number and the degrees of freedom parameter \(\nu\) (denoted df in the implementation) satisfies \(\nu > 0\). \(\Gamma\) is the gamma function (scipy.special.gamma).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, t.pdf(x, df, loc, scale) is identically equivalent to t.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 t
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> df = 2.74
>>> mean, var, skew, kurt = t.stats(df, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(t.ppf(0.01, df),
...                 t.ppf(0.99, df), 100)
>>> ax.plot(x, t.pdf(x, df),
...        'r-', lw=5, alpha=0.6, label='t 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 = t(df)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = t.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.