Skip to main content
`Ctrl`+`K`

# scipy.special.ndtri

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

  Inverse of `ndtr` vs x

  Returns the argument x for which the area under the standard normal probability density function (integrated from minus infinity to x) is equal to y.

  Parameters:
    parray_like

      Probability

    outndarray, optional

      Optional output array for the function results

  Returns:
    xscalar or ndarray

      Value of x such that `ndtr(x) == p`.

  See also

  `ndtr`

    Standard normal cumulative probability distribution

  `ndtri_exp`

    Inverse of log_ndtr

  Notes

  `ndtri` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_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

  `ndtri` is the percentile function of the standard normal distribution. This means it returns the inverse of the cumulative density `ndtr`. First, let us compute a cumulative density 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 function. For that purpose, we provide a NumPy array as argument.

      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(0.01, 1, 200)
      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, ndtri(x))
      >>> ax.set_title("Standard normal percentile function")
      >>> plt.show()