Calculate probability in normal distribution given mean, std in Python

Question

How to calculate probability in normal distribution given mean, std in Python? I can always explicitly code my own function according to the definition like the OP in this question did: Calculating Probability of a Random Variable in a Distribution in Python

Just wondering if there is a library function call will allow you to do this. In my imagine it would like this:

nd = NormalDistribution(mu=100, std=12)
p = nd.prob(98)

There is a similar question in Perl: How can I compute the probability at a point given a normal distribution in Perl?. But I didn't see one in Python.

Numpy has a random.normal function but it's like sampling, not exactly what I want.

Answer 1

There's one in scipy.stats:

>>> import scipy.stats
>>> scipy.stats.norm(0, 1)
<scipy.stats.distributions.rv_frozen object at 0x928352c>
>>> scipy.stats.norm(0, 1).pdf(0)
0.3989422804014327
>>> scipy.stats.norm(0, 1).cdf(0)
0.5
>>> scipy.stats.norm(100, 12)
<scipy.stats.distributions.rv_frozen object at 0x928352c>
>>> scipy.stats.norm(100, 12).pdf(98)
0.032786643008494994
>>> scipy.stats.norm(100, 12).cdf(98)
0.43381616738909634
>>> scipy.stats.norm(100, 12).cdf(100)
0.5

[One thing to beware of -- just a tip -- is that the parameter passing is a little broad. Because of the way the code is set up, if you accidentally write scipy.stats.norm(mean=100, std=12) instead of scipy.stats.norm(100, 12) or scipy.stats.norm(loc=100, scale=12), then it'll accept it, but silently discard those extra keyword arguments and give you the default (0,1).]

Answer 2

Scipy.stats is a great module. Just to offer another approach, you can calculate it directly using

import math
def normpdf(x, mean, sd):
    var = float(sd)**2
    denom = (2*math.pi*var)**.5
    num = math.exp(-(float(x)-float(mean))**2/(2*var))
    return num/denom

This uses the formula found here: http://en.wikipedia.org/wiki/Normal_distribution#Probability_density_function

to test:

>>> normpdf(7,5,5)  
0.07365402806066466
>>> norm(5,5).pdf(7)
0.073654028060664664

Answer 3

Here is more info. First you are dealing with a frozen distribution (frozen in this case means its parameters are set to specific values). To create a frozen distribution:

import scipy.stats
scipy.stats.norm(loc=100, scale=12)
#where loc is the mean and scale is the std dev
#if you wish to pull out a random number from your distribution
scipy.stats.norm.rvs(loc=100, scale=12)

#To find the probability that the variable has a value LESS than or equal
#let's say 113, you'd use CDF cumulative Density Function
scipy.stats.norm.cdf(113,100,12)
Output: 0.86066975255037792
#or 86.07% probability

#To find the probability that the variable has a value GREATER than or
#equal to let's say 125, you'd use SF Survival Function 
scipy.stats.norm.sf(125,100,12)
Output: 0.018610425189886332
#or 1.86%

#To find the variate for which the probability is given, let's say the 
#value which needed to provide a 98% probability, you'd use the 
#PPF Percent Point Function
scipy.stats.norm.ppf(.98,100,12)
Output: 124.64498692758187

Answer 4

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

It can be used to get the probability density function (pdf - likelihood that a random sample X will be near the given value x) for a given mean (mu) and standard deviation (sigma):

from statistics import NormalDist

NormalDist(mu=100, sigma=12).pdf(98)
# 0.032786643008494994

Also note that the NormalDist object also provides the cumulative distribution function (cdf - probability that a random sample X will be less than or equal to x):

NormalDist(mu=100, sigma=12).cdf(98)
# 0.43381616738909634

Answer 5

The formula cited from wikipedia mentioned in the answers cannot be used to calculate normal probabilites. You would have to write a numerical integration approximation function using that formula in order to calculate the probability.

That formula computes the value for the probability density function. Since the normal distribution is continuous, you have to compute an integral to get probabilities. The wikipedia site mentions the CDF, which does not have a closed form for the normal distribution.

Answer 6

I wrote this program to do the math for you. Just enter in the summary statistics. No need to provide an array:

One-Sample Z-Test for a Population Proportion:

To do this for mean rather than proportion, change the formula for z accordingly

Answer 7

You can just use the error function that's built in to the math library, as stated on their website.

Answer 8

In case you would like to find the area between 2 values of x mean = 1; standard deviation = 2; the probability of x between [0.5,2]

import scipy.stats
scipy.stats.norm(1, 2).cdf(2) - scipy.stats.norm(1,2).cdf(0.5)