scipy.special.ndtr#

scipy.special.ndtr(x, out=None) = <ufunc 'ndtr'>#

Cumulative distribution of the standard normal distribution.

Returns the area under the standard Gaussian probability density function, integrated from minus infinity to x

\[\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp(-t^2/2) dt\]

Parameters:

xarray_like, real or complex: Argument
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: The value of the normal CDF evaluated at x

See also

log_ndtr: Logarithm of ndtr
ndtri: Inverse of ndtr, standard normal percentile function
erf: Error function
erfc: 1 - erf
scipy.stats.norm: Normal distribution

Notes

Array API Standard Support

ndtr 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

Evaluate ndtr at one point.

>>> import numpy as np
>>> from scipy.special import ndtr
>>> ndtr(0.5)
0.6914624612740131

Evaluate the function at several points by providing a NumPy array or list for x.

>>> ndtr([0, 0.5, 2])
array([0.5       , 0.69146246, 0.97724987])

Plot the function.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-5, 5, 100)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, ndtr(x))
>>> ax.set_title(r"Standard normal cumulative distribution function $\Phi$")
>>> plt.show()