the Normal Distribution   ... an approximation
motivated by e-mail from Andre P.

I got this e-mail asking if I knew of a formula for generating the cumulative Normal Distribution function (which looks like Figure 1).
 >And you knew, eh? Uh ... no. As far as I knew, there was no such formula. However I googled and found a neat approximation due to Bagby. Bagby, R. J. "Calculating Normal Probabilities." Amer. Math. Monthly 102, 46-49, 1995 It looks like this:   >And is it any good ... Bagby's approximation? It's excellent ... as indicated in Figure 2. In Excel, you can use: =0.5+IF(x>Mean,1,-1)*0.5*SQRT(1-(7*EXP(-0.5*z*z)+16*EXP(-(2-SQRT(2))*z*z)+EXP(-z*z)*(7+0.25*PI()*z^2))/30) where z = (x - Mean) / StandardDeviation >How about a lognormal distribution? For a Lognormal distribution, the logarithm of the variable has a Normal distribution. So we need only change z, above, to: where z = (LN(x) - Mean) / StandardDeviation where Mean and StandardDeviation are the Mean and SD of the logarithm of x ... and x is a variable greater than 1 (example: x = 1 + StockReturn). >The natural logarithm, eh? Yes ... that's LN(x). Figure 1 Figure 2

In Excel, you can use:

= 0.5+IF(LN(x)>Mean,1,-1)*0.5*SQRT(1-(7*EXP(-0.5*z*z)+16*EXP(-(2-SQRT(2))*z*z)+EXP(-z*z)*(7+0.25*PI()*z^2))/30)
where z = (LN(x) - Mean) / StandardDeviation