If X is a normal variable with parameters mu and sigma, i.e. X is distributed N(mu, sigma^2), then the pdf(x) is given by
pdf(x) = exp{-0.5[(x - mu)/sigma]^2}/[sigma*sqrt(2*pi)]where mu is the mean of the distribution, sigma^2 is the variance and sigma is the standard deviation.
The cumulative distribution function cdf(x) is the integral of pdf(u) from u = minus infinity to u = x.
Upper and lower tail probabilities are defined for an alpha (0 =< alpha =< 1) and some x-value (say x = x-critical) by
lower tail probability = P(X =< x-critical) = 1 - alpha, upper tail probability = P(X >= x-critical) = alpha,where P(E) = probability of event E in the sample space. Sometimes percentage points 100(1 - alpha)%, or 100*alpha% are preferred.
You input parameters mu and sigma for all the subsequent
options and the variance is set equal to sigma squared.
Option 1 is selected to change mu and sigma.
You input x-values and the program calculates probability
density functions, i.e. pdf(x) values.
You input x-values and the program calculates cumulative
distribution functions, i.e. cdf(x) values.
You input significance levels alpha and obtain x-critical
values such that P(X >= x-critical) = alpha, i.e. inverses
of the normal distribution.
You input measured values V1, V2, ..., VM and the program
calculates transforms W1, W2, ... , WM, where Wi = cdf(Vi)
under the null hypothesis that V is N(mu, sigma^2).
Kolmogorov-Smirnov and chi-square tests are performed on
the transforms to test the hypothesis that W is uniformly
distributed on the interval (0,1), which is equivalent to
the null hypothesis.
If you have some numbers V1, V2, ..., VM and you want to see if it is reasonable to regard these as coming from a normal distribution, e.g. before doing a t test, then use option 5 and check the normal scores plot for nonlinearities.
There are two different ways to use option 5.
You input a sigma then explore power and sample size.
Always check you are using the correct mu and sigma values. Use options 1, 2, 3 and 4 just as you would look up tables except that you do not need to transform x-values to unit normal values, i.e.
Z = (X - mu)/sigma.For an upper tail 5% point use option 4 with 100*alpha% = 5, but for two tailed values for 95% confidence limits set 100*alpha% = 2.5. If unsure about option 5 use sample estimates for mu, sigma but consider using tables for more accurate significance levels. You can input x-values for option 5 from a console, but this is not recommended since it is error prone, and all input is lost after the program has been run. If you prepare a data file, this is a permanent store for the data that can be run repeatedly through the same, or different, programs and can be easily edited to add, delete or change values. Files (like the test file, normal.tf1) must have a one line title, a header (n 1 for n by 1 matrix) then a data column. Use program MAKMAT to prepare data files and program EDITMT for editing such files.