User:Asitgoes/Normdis

In statistics and data analysis the application software NormDis is a free and user-friendly calculator for the determination of the cumulative probability Pc(Xr) for any random variable  (X) following the normal distribution. Here, the cumulative probability Pc(Xr) stands for the probability P that X is less than a reference value Xr of X. Biefly : Pc(Xr) = P(X<Xr).

Reversely, the calculator can give the value of Xr given Pc. Hence, it is a two-way calculator. The data required are the mean and the standard deviation of the distribution of X.

Intervals


The probability (Pi) that X occurs in an interval between an upper limit (U) and a lower limit (L) can be found from:


 * Pi = P(L<X<U) = Pc(U)  - Pc(L).

Thus, using the calculator twice, namely for Xr=U and Xr=L, and subtracting the results, one finds the value of Pi that L<X<U.

Numerical method
The cumulative distribution function of the normal distribution cannot be calculated analytically and a numerical approximation has to be used. NormDis uses the Hastings method, as follows :
 * $$ Pc(x) = 1 - \phi(x)\left(b_1t + b_2t^2 + b_3t^3 + b_4t^4 + b_5t^5\right) $$

where
 * $$ t = \frac{1}{1+b_0x} $$

and
 * b0 = 0.2316419, b1 = 0.319381530, b2 = −0.356563782, b3 = 1.781477937, b4 = −1.821255978, b5 = 1.330274429.

Here, $$\phi(x)$$ is the standard normal probability density function (PDF):
 * $$ \phi(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-x^2/2} $$

When the distribution is standard normal, one can use $$x$$ = Xr, otherwise $$x$$ = (Xr - M) / S, where M is the mean and S the standard deviation.



Graphics
The NormDis program provides graphics for the various values computed with the calculator. See the examples to left and right.