Using an appropriate probability grid, plot the cdf (cumulative distribution function)
data. The data will appear as a straight line on the correct grid. Fortunately, there are
two grids that will cover most of the distributions we need, and making additional ones
isn't too complicated.
To make a Normal (or Lognormal) grid notice that the yaxis is in terms of
number of standard deviations, although it's not labeled that way. So the middle of the
graph is at y=0 and that corresponds to cdf, F(x) = 0.5 = 50%. One standard
deviation unit up (or down) is F(x) = 0.8413 (or 0.1587). Two units up (or down) is
0.9772 (or 0.0228). Three units up (down) is 9987 (0.0013). And so on. These values, and
intermediate values chosen for graphing purposes, are tabulated everywhere and can be
found using MS EXCEL also. If the xaxis is to represent a normally distributed x, then
it's Cartesian. If lognormal is what you want, then the xaxis is logarithmic.
To make a grid for the exponential distribution, we can take advantage of
knowing that the exponential distribution is a special case of the Weibull, when the slope
parameter, beta, equals one. The Weibull grid is even easier to make than the Normal grids
because F(x) has a closed form (unlike the Normal), viz.
A little arithmetic shows that
This is a linear equation, Y=M*X+B, where X=ln(x) and Y=ln(ln(1F(x))),
with slope of M = b, and
intercept = b ln(h). (Remember that b and h are constants for a given fit so b ln(h) is
also a constant.)
The grid then is simply Y=ln(ln(1F(x))), and X=ln(x).
(Logarithmic xaxis, and log(logarithmic) yaxis.) Notice that the (0, 0) point occurs at
x=1 (so that ln(x)=0, and y= ~0.632, since ln(ln(10.632))~0. (The actual value is at
y=1exp(1) ~0.6321205588)
While mean rank and median rank plotting
positions are often used, the following rank position has the useful property that the MLE
line goes through the points if there is nothing wrong with the data. The rank for
the ith uncensored observation is
y_{i} = (i0.5)/n
where i = 1, 2, 3, ... n, and n
is the sample size.
That's it. You can program EXCEL to do the plots if you don't have
software to do it already.
