8.3.1 __Graphical Methods__

The basic idea of graphical methods is the use of special
probability plotting papers in which the cumulative distribution function
(cdf) or the cumulative hazard function can be plotted as a straight line for
the particular distribution being studied. Since a straight line has two
parameters (slope and intercept), two parameters of the distribution can be
determined. Thus, reliability data can be evaluated quickly, without a
detailed knowledge of the statistical mathematics being necessary. This
facilitates analysis and presentation of data.

Graphical curve-fitting techniques and special
probability-plotting papers have been developed for all of the distributions
commonly associated with reliability analysis (Refs. [4], [5]).

__Ranking of Data__

Probability graph papers are based upon plots of the variable of
interest against the cumulative percentage probability. The data, therefore,
need to be ordered, and the cumulative probability calculated. For reliability
work, the data are ordered from the smallest to largest; this is referred to
as order statistics. For example, consider the data on times-to-failure of 20
items (Table 8.3-1). For the first failure, the cumulative percent failed is
1/20 or 5%. For the second, the cumulative percent failed is 2/20 or 10%, and
so on to 20/20 or 100% for the 20th failure. However, for probability
plotting, it is better to make an adjustment to allow for the fact that each
failure represents a point on a distribution. Thus, considering that the whole
population of 20 items represents a sample, the times by which 5, 10, ...,
100% will have failed in several samples of 20 will be randomly distributed.
However, the data in Table 8.3.1-1 show a bias, in that the first failure is
shown much further from the zero cumulative percentage point than is the last
from 100% (in fact, it coincides). To overcome this, and thus to improve the
accuracy of the estimation, mean or median ranking of cumulative percentages
is used for probability plotting. Mean ranking is used for symmetrical
distributions, e.g., normal; median ranking is used for skewed distributions,
e.g., Weibull.

The usual method for mean ranking is to use (n + 1) in the
denominator, instead of n, when calculating the cumulative percentage
position. Thus in Table 8.3-1 the cumulative percentages (mean ranks) would
be: