[FIXED] How to get the Upper and Lower Values from a Multivariate Normal Distribution when I only supply a percentage in python?

Issue

In the Image below, for example, if I supply x as a percentage (like 68.2%), how do I get the values at 1Sigma and -1Sigma in Python?

Image

Similarity, x = 100% should give me the extreme endpoints in the data distribution

The upper and lower bounds must be calculated about the mean of the data.

Any help would be appreciated. Thanks!

Solution

In your particular example, the answer is simple for a normal distribution with mean parameter (mu/loc) and standard deviation (std/scale).

upper, lower = loc-scale, loc+scale

so there is no need for elaborate Python code.

However, for a more general case where you wonder what quantiles (centered around the mean) enclose a certain probability p taking values in [0,1], you can use SciPy's stats library for the normal distribution. The code is:

from scipy.stats import norm 

p     = 0.68268949  # prob. in between [mu-std, mu+std]
loc   = 0           # mu
scale = 1           # sigma/std

lower = norm.ppf((1-p)/2., loc, scale)
upper = norm.ppf((1+p)/2., loc, scale)
lower, upper
>(-0.9999999955839989, 0.9999999955839989)

After accounting for floating point arithmetic, this equates to [-1,1] = [loc-scale, loc+scale] as expected.

Since the normal distribution density has support on the entire real line, only that exact interval, namely (-inf, +inf), encloses the probability of 1. The code above produces that result consistently, since

p = 1.0
lower = norm.ppf((1-p)/2., loc, scale)
upper = norm.ppf((1+p)/2., loc, scale)
lower, upper
>(-inf, inf)

Answered By - 7shoe