Convert Z-score (Z-value, standard score) to p-value for normal distribution in Python

Question

How does one convert a Z-score from the Z-distribution (standard normal distribution, Gaussian distribution) to a p-value? I have yet to find the magical function in Scipy\'

Answer 1

For Scipy lovers, Tough this is old question but relevant, and we can have not only normal but other distributions as well so here is solution for few more distributions:

def get_p_value_normal(z_score: float) -> float:
    """get p value for normal(Gaussian) distribution 

    Args:
        z_score (float): z score

    Returns:
        float: p value
    """
    return round(norm.sf(z_score), decimal_limit)


def get_p_value_t(z_score: float) -> float:
    """get p value for t distribution 

    Args:
        z_score (float): z score

    Returns:
        float: p value
    """
    return round(t.sf(z_score), decimal_limit)


def get_p_value_chi2(z_score: float) -> float:
    """get p value for chi2 distribution 

    Args:
        z_score (float): z score

    Returns:
        float: p value
    """
    return round(chi2.ppf(z_score, df), decimal_limit)

Answer 2

From formula:

import numpy as np
import scipy.special as scsp
def z2p(z):
    """From z-score return p-value."""
    return 0.5 * (1 + scsp.erf(z / np.sqrt(2)))

Answer 3

I think the cumulative distribution function (cdf) is preferred to the survivor function. The survivor function is defined as 1-cdf, and may communicate improperly the assumptions the language model uses for directional percentiles. Also, the percentage point function (ppf) is the inverse of the cdf, which is very convenient.

>>> import scipy.stats as st
>>> st.norm.ppf(.95)
1.6448536269514722
>>> st.norm.cdf(1.64)
0.94949741652589625

Answer 4

Starting Python 3.8, the standard library provides the NormalDist object as part of the statistics module.

It can be used to apply the inverse cumulative distribution function (inv_cdf, also known as the quantile function or the percent-point function) and the cumulative distribution function (cdf):

NormalDist().inv_cdf(0.95)
# 1.6448536269514715
NormalDist().cdf(1.64)
# 0.9494974165258963

Answer 5

I like the survival function (upper tail probability) of the normal distribution a bit better, because the function name is more informative:

p_values = scipy.stats.norm.sf(abs(z_scores)) #one-sided

p_values = scipy.stats.norm.sf(abs(z_scores))*2 #twosided

normal distribution "norm" is one of around 90 distributions in scipy.stats

norm.sf also calls the corresponding function in scipy.special as in gotgenes example

small advantage of survival function, sf: numerical precision should better for quantiles close to 1 than using the cdf

Answer 6

Aha! I found it: scipy.special.ndtr! This also appears to be under scipy.stats.stats.zprob as well (which is just a pointer to ndtr).

Specifically, given a one-dimensional numpy.array instance z_scores, one can obtain the p-values as

p_values = 1 - scipy.special.ndtr(z_scores)

or alternatively

p_values = scipy.special.ndtr(-z_scores)