find sigma for known normcdf parameters

Hi,

Knowing p, x and mu I would like to find the corresponding sigma for p = normcdf(x,mu,sigma). I tried with the symbolic toolbox using the following code but it failed.

syms x mu sigma
p = 1/2 - erf((2^(1/2) * (mu - x)) / (2 * sigma)) / 2
solve(p,sigma)

Thanks in advance for your help,

Philippe

Answer 1

[asm8086]

A friend of mine found the solution :

sigma = 0.5 * (2^(1/2) * (mu - x)) / erfinv(2*(1/2-p))

Answer 2

[Walter Roberson]

That can be rearranged so that the essential part is solving an equation of the form erf(t)=r for t. Unfortunately I have not found a useful analytic method of solving that.

It appears that there is a real solution if and only if p is in the range 0 to 1; however, if mu = x then any non-zero sigma satisfies the equation.