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 experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variableSCIPY_ARRAY_API=1
and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. 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
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()