How to calculate the inverse of the normal cumulative distribution function in python?

Question

How do I calculate the inverse of the cumulative distribution function (CDF) of the normal distribution in Python?

Which library should I use? Possibly scipy?

Solution 1

NORMSINV (mentioned in a comment) is the inverse of the CDF of the standard normal distribution. Using scipy, you can compute this with the ppf method of the scipy.stats.norm object. The acronym ppf stands for percent point function, which is another name for the quantile function.

In [20]: from scipy.stats import norm

In [21]: norm.ppf(0.95)
Out[21]: 1.6448536269514722

Check that it is the inverse of the CDF:

In [34]: norm.cdf(norm.ppf(0.95))
Out[34]: 0.94999999999999996

By default, norm.ppf uses mean=0 and stddev=1, which is the "standard" normal distribution. You can use a different mean and standard deviation by specifying the loc and scale arguments, respectively.

In [35]: norm.ppf(0.95, loc=10, scale=2)
Out[35]: 13.289707253902945

If you look at the source code for scipy.stats.norm, you'll find that the ppf method ultimately calls scipy.special.ndtri. So to compute the inverse of the CDF of the standard normal distribution, you could use that function directly:

In [43]: from scipy.special import ndtri

In [44]: ndtri(0.95)
Out[44]: 1.6448536269514722

Solution 2

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

It can be used to get the inverse cumulative distribution function (inv_cdf - inverse of the cdf), also known as the quantile function or the percent-point function for a given mean (mu) and standard deviation (sigma):

from statistics import NormalDist

NormalDist(mu=10, sigma=2).inv_cdf(0.95)
# 13.289707253902943

Which can be simplified for the standard normal distribution (mu = 0 and sigma = 1):

NormalDist().inv_cdf(0.95)
# 1.6448536269514715

Solution 3

# given random variable X (house price) with population muy = 60, sigma = 40
import scipy as sc
import scipy.stats as sct
sc.version.full_version # 0.15.1

#a. Find P(X<50)
sct.norm.cdf(x=50,loc=60,scale=40) # 0.4012936743170763

#b. Find P(X>=50)
sct.norm.sf(x=50,loc=60,scale=40) # 0.5987063256829237

#c. Find P(60<=X<=80)
sct.norm.cdf(x=80,loc=60,scale=40) - sct.norm.cdf(x=60,loc=60,scale=40)

#d. how much top most 5% expensive house cost at least? or find x where P(X>=x) = 0.05
sct.norm.isf(q=0.05,loc=60,scale=40)

#e. how much top most 5% cheapest house cost at least? or find x where P(X<=x) = 0.05
sct.norm.ppf(q=0.05,loc=60,scale=40)

Comments

[Warren Weckesser]

Do you mean the inverse Gaussian distribution (en.wikipedia.org/wiki/Inverse_Gaussian_distribution), or the inverse of the cumulative distribution function of the normal distribution (en.wikipedia.org/wiki/Normal_distribution), or something else?

[Yueyoum]

@WarrenWeckesser the second one: inverse of the cumulative distribution function of the normal distribution

[Yueyoum]

@WarrenWeckesser i mean the python version of "normsinv" function in excel.

[William Zhang]

I always think "percent point function" (ppf) is a terrible name. Most people in statistics just use "quantile function".

[Suresh2692]

PS: You can assume 'loc' as 'mean' and 'scale' as 'standard deviation'

[Jethro Cao]

Great tip! This allows me to drop the dependency on scipy, which I needed just for the single stats.norm.ppf method

[bones.felipe]

Don't you need to specify the mean and the std on both ppf and cdf?

[Warren Weckesser]

@bones.felipe, the "standard" normal distribution has mean 0 and standard deviation 1. These are the default values for the location and scale of the scipy.stats.norm methods.

[bones.felipe]

Right, I thought I saw this norm.cdf(norm.ppf(0.95, loc=10, scale=2)) and I thought it was weird norm.cdf did not have loc=10 and scale=2 too, I guess it should.

[vanetoj]

can you use that to transform data with uniform distribution to normal ?