User:Asitgoes/Normdis

Developer(s)	Institute for Land Reclamation and Improvement (ILRI)
Written in	Delphi
Operating system	Microsoft Windows
Available in	English
Type	Statistical software
License	Proprietary Freeware
Website	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, as follows :

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

where

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

and

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

Here, $\phi (x)$ 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.

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

Category:Statistical software Category:Data analysis software Category:Freeware