Standard C or Python libraries to compute standard deviation of normal distribution

Question

Say we have normal distribution n(x): mean=0 and \int_{-a}^{a} n(x) = P.

What is the easiest way to compute standard deviation of such distribution? May be there are standard libraries for python or C, that are suitable for that task?

Answer 1

If X is normal with mean 0 and standard deviation sigma, it must hold

P = Prob[ -a <= X <= a ] = Prob[ -a/sigma <= N <= a/sigma ]
  = 2 Prob[ 0 <= N <= a/sigma ]
  = 2 ( Prob[ N <= a/sigma ] - 1/2 )

where N is normal with mean 0 and standard deviation 1. Hence

P/2 + 1/2 = Prob[ N <= a/sigma ] = Phi(a/sigma)

Where Phi is the cumulative distribution function (cdf) of a normal variable with mean 0 and stddev 1. Now we need the inverse normal cdf (or the "percent point function"), which in Python is scipy.stats.norm.ppf(). Sample code:

from scipy.stats import norm
P = 0.3456
a = 3.0

a_sigma = float(norm.ppf(P/2 + 0.5))   # a/sigma
sigma = a/a_sigma   # Here is the standard deviation

For example, we know that the probability of a N(0,1) variable falling int the interval [-1.1] is ~ 0.682 (the dark blue area in this figure). If you set P = 0.682 and a = 1.0 you obtain sigma ~ 1.0, which is indeed the standard deviation.

Answer 2

The standard deviation of a mean-zero gaussian distribution with Pr(-a < X < a) = P is

a/(sqrt(2)*inverseErf(P))

which is the expression you're looking for, where inverseErf is the inverse of the error function (commonly known as erf).

For C, the Gnu Scientific Library (GSL) is a good resource. However it only has erf, not inverseErf, so you'd have to invert it yourself (a simple binary search would do the trick). Alternatively, here's a nice way to approximate erf and inverseErf:

http://homepages.physik.uni-muenchen.de/~Winitzki/erf-approx.pdf

For Python, inverseErf is available as erfinv in the SciPy library, so the following gives the standard deviation:

a/(math.sqrt(2)*erfinv(P))

PS: There's some kind of bug in Stackoverflow's URL rendering and it wouldn't let me link to GSL above: http://www.gnu.org/software/gsl. It also renders wrong when I make the URL above with a pdf a proper link.