# User:Asitgoes/Normdis

Developer(s) Institute for Land Reclamation and Improvement (ILRI) Delphi Microsoft Windows English Statistical software Proprietary Freeware 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

Values of Pi in % for different intervals based on a unit length equal to the value of the standard deviation σ.

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,[1] as follows :

${\displaystyle Pc(x)=1-\phi (x)\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\right)}$

where

${\displaystyle t={\frac {1}{1+b_{0}x}}}$

and

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

Here, ${\displaystyle \phi (x)}$ is the standard normal probability density function (PDF):

${\displaystyle \phi (x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-x^{2}/2}}$

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

Cumulative probability given the value of a normally distributed variable
Total probability as a surface area under the normal probability density function given lower and upper limit of an interval of a normally distributed variable

## Graphics

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

## References

1. ^ Zelen, Marvin; Severo, Norman C. (1964). Probability Functions (chapter 26). Handbook of mathematical functions with formulas, graphs, and mathematical tables, by Abramowitz, M.; and Stegun, I. A.: National Bureau of Standards. New York, NY: Dover. ISBN 0-486-61272-4.