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:


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