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

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("Standard normal cumulative distribution function $\Phi$")
>>> plt.show()