The function NORMAL CDF (x) is the cumulative distribution function (CDF) of a standard normal (Gaussian) random variable.
NORMAL CDF(x) expresses the probability that a random variable Y distributed N(0,1) is less than x.
Pr[ Y~N(0,1) <= x] = NORMAL CDF(x)
Therefore the probability that Y lies between a and b is given by NORMAL CDF(b)-NORMAL CDF(a), assuming that a<=b.
If Y is distributed N(mean,sd), x can be transformed to a z-score as (x-mean)/sd. Then:
Pr[ Y~N(mean,sd) <= x ] = NORMAL CDF( (x-mean)/sd )
The Normal CDF is closely related to the Error Function, Erf(x), which is scaled differently, implying standard deviation of ½ and return a value from -1 to 1.
Erf(x) = 2*NORMAL CDF(2*x)-1
The approximation is calculated using the algorithm 26.2.17 in Abromowitz and Stegun, Handbook of Mathematical Functions.
