scipy.special.stdtr(df, t, out=None) = <ufunc 'stdtr'>#

Student t distribution cumulative distribution function

Returns the integral:

\[\frac{\Gamma((df+1)/2)}{\sqrt{\pi df} \Gamma(df/2)} \int_{-\infty}^t (1+x^2/df)^{-(df+1)/2}\, dx\]

Degrees of freedom


Upper bound of the integral

outndarray, optional

Optional output array for the function results

scalar or ndarray

Value of the Student t CDF at t

See also


inverse of stdtr with respect to df


inverse of stdtr with respect to t


student t distribution


The student t distribution is also available as scipy.stats.t. Calling stdtr directly can improve performance compared to the cdf method of scipy.stats.t (see last example below).


Calculate the function for df=3 at t=1.

>>> import numpy as np
>>> from scipy.special import stdtr
>>> import matplotlib.pyplot as plt
>>> stdtr(3, 1)

Plot the function for three different degrees of freedom.

>>> x = np.linspace(-10, 10, 1000)
>>> fig, ax = plt.subplots()
>>> parameters = [(1, "solid"), (3, "dashed"), (10, "dotted")]
>>> for (df, linestyle) in parameters:
...     ax.plot(x, stdtr(df, x), ls=linestyle, label=f"$df={df}$")
>>> ax.legend()
>>> ax.set_title("Student t distribution cumulative distribution function")

The function can be computed for several degrees of freedom at the same time by providing a NumPy array or list for df:

>>> stdtr([1, 2, 3], 1)
array([0.75      , 0.78867513, 0.80449889])

It is possible to calculate the function at several points for several different degrees of freedom simultaneously by providing arrays for df and t with shapes compatible for broadcasting. Compute stdtr at 4 points for 3 degrees of freedom resulting in an array of shape 3x4.

>>> dfs = np.array([[1], [2], [3]])
>>> t = np.array([2, 4, 6, 8])
>>> dfs.shape, t.shape
((3, 1), (4,))
>>> stdtr(dfs, t)
array([[0.85241638, 0.92202087, 0.94743154, 0.96041658],
       [0.90824829, 0.97140452, 0.98666426, 0.99236596],
       [0.93033702, 0.98599577, 0.99536364, 0.99796171]])

The t distribution is also available as scipy.stats.t. Calling stdtr directly can be much faster than calling the cdf method of scipy.stats.t. To get the same results, one must use the following parametrization: scipy.stats.t(df).cdf(x) = stdtr(df, x).

>>> from scipy.stats import t
>>> df, x = 3, 1
>>> stdtr_result = stdtr(df, x)  # this can be faster than below
>>> stats_result = t(df).cdf(x)
>>> stats_result == stdtr_result  # test that results are equal