scipy.special.ndtri#

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

Inverse of ndtr.

Returns the quantile x such that the cumulative distribution function of the standard normal distribution evaluated at x equals p, that is, ndtr(x) == p.

Parameters:

parray_like: Probability values.
outndarray, optional: Optional output array for the function results.

Returns:

xscalar or ndarray: Quantile(s) corresponding to the probabilitie(s) in p.

See also

ndtr: Standard normal cumulative distribution function
ndtri_exp: Inverse of log_ndtr

Notes

Array API Standard Support

ndtri has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	✅	✅
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable.

See Support for the array API standard for more information.

Examples

ndtri is the percentile (quantile) function of the standard normal distribution, i.e., the inverse of the cumulative distribution function ndtr.

First, compute a cumulative distribution value:

>>> import numpy as np
>>> from scipy.special import ndtri, ndtr
>>> cdf_val = ndtr(2)
>>> cdf_val
0.9772498680518208

Verify that ndtri yields the original value for x up to floating point errors.

>>> ndtri(cdf_val)
2.0000000000000004

Plot the percentile function over a range of probabilities.

>>> import matplotlib.pyplot as plt
>>> p = np.linspace(1e-3, 1 - 1e-3, 201)
>>> fig, ax = plt.subplots()
>>> ax.plot(p, ndtri(p))
>>> ax.set_title("Standard normal percentile function")
>>> plt.show()