NORMAL PLOT:


     This is a fairly standard graph in statistics which is organized so that data which is
normally distributed will fall on a straight line, while non-normal data will deviate from the
straight line.  The basic idea is organized around the equation

(1)  Z = { X - mean(X) } / ( Standard Deviation X )

Clearly, if mean(X) and (Standard Deviation X) are constants, then a graph of Z versus X will
give a straight line.

     Probability is introduced by ranking the X-data in order of increasing size.  Let X[j]
denote the value of X with rank j.  That is, there are j-values of the X-data-set that are less than or
equal to X[j].  Suppose that there are N values in the total data set.  A good estimate of the
cumulative distribution function, F(x), for the population from which the X-data was sampled
is given by the Gumbel formula

(2)  P[j] =  F(X[j]) = j/[N+1]

     If the distribution function for X is actually the normal probability law, then the
standardized variable Z in (1)  will behave acording to the standard normal distribution, 

(3)  P = Phi(Z)

Phi is a strictly increasing function of  Z, so the function can be inverted to give

(4)  Z = InversePhi(P)

or with application to  P[j].

(5)  Z[j] = InversePhi ( j / [N+1] )

This associates a Z-value with every X[j] through the rank of that X

Hence, from (1)

(5)   Z[j] =  { X[j] - mean(X) } / ( Standard Deviation X )

A graph of Z[j] versus X[j] will give a straight line if the original data is normally distributed.
The Normal plot allow the user to examine the water surface elevations visually to see if it is
reasonable to judge that the seas may be Gaussian.