# scipy.special.stdtrit#

scipy.special.stdtrit(df, p, out=None) = <ufunc 'stdtrit'>#

The p-th quantile of the student t distribution.

This function is the inverse of the student t distribution cumulative distribution function (CDF), returning t such that stdtr(df, t) = p.

Returns the argument t such that stdtr(df, t) is equal to p.

Parameters:
dfarray_like

Degrees of freedom

parray_like

Probability

outndarray, optional

Optional output array for the function results

Returns:
tscalar or ndarray

Value of t such that `stdtr(df, t) == p`

`stdtr`

Student t CDF

`stdtridf`

inverse of stdtr with respect to df

`scipy.stats.t`

Student t distribution

Notes

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

Examples

`stdtrit` represents the inverse of the student t distribution CDF which is available as `stdtr`. Here, we calculate the CDF for `df` at `x=1`. `stdtrit` then returns `1` up to floating point errors given the same value for df and the computed CDF value.

```>>> import numpy as np
>>> from scipy.special import stdtr, stdtrit
>>> import matplotlib.pyplot as plt
>>> df = 3
>>> x = 1
>>> cdf_value = stdtr(df, x)
>>> stdtrit(df, cdf_value)
0.9999999994418539
```

Plot the function for three different degrees of freedom.

```>>> x = np.linspace(0, 1, 1000)
>>> parameters = [(1, "solid"), (2, "dashed"), (5, "dotted")]
>>> fig, ax = plt.subplots()
>>> for (df, linestyle) in parameters:
...     ax.plot(x, stdtrit(df, x), ls=linestyle, label=f"\$df={df}\$")
>>> ax.legend()
>>> ax.set_ylim(-10, 10)
>>> ax.set_title("Student t distribution quantile function")
>>> plt.show()
```

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

```>>> stdtrit([1, 2, 3], 0.7)
array([0.72654253, 0.6172134 , 0.58438973])
```

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

```>>> dfs = np.array([[1], [2], [3]])
>>> p = np.array([0.2, 0.4, 0.7, 0.8])
>>> dfs.shape, p.shape
((3, 1), (4,))
```
```>>> stdtrit(dfs, p)
array([[-1.37638192, -0.3249197 ,  0.72654253,  1.37638192],
[-1.06066017, -0.28867513,  0.6172134 ,  1.06066017],
[-0.97847231, -0.27667066,  0.58438973,  0.97847231]])
```

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

```>>> from scipy.stats import t
>>> df, x = 3, 0.5
>>> stdtrit_result = stdtrit(df, x)  # this can be faster than below
>>> stats_result = t(df).ppf(x)
>>> stats_result == stdtrit_result  # test that results are equal
True
```