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THEORETICAL, & HISTORICAL BASIS of Reliability & Distribution Plotting

following quotations and annotations are useful. Each quote is followed by its author's name; the full

references are provide in the "References" table below. Note that "Reliability Plotting" is referred to by

various names in reliability literature, such as "Probability Plotting", "Rectification", "Linear

Regression Analysis", as well as "Reliability Plotting" and others.

"

during World War II and the Korean War. The basic underlying premise of the reliability theory is that

products of the same design and manufacture are essentially homogeneous and that their failures

follow a definite pattern. Once this premise is accepted, the next task is to define that pattern. This

pattern is known as the distribution and lends itself to statistical resolution. Much of the early efforts in

the development of the reliability discipline were along statistical lines. Attempts were made at curve

fitting of empirical data. There is no such thing as a perfect fit so the search is for a representative

distribution. The statistical techniques used in reliability discipline are varied and depend on the

specific application. ....There are normal (Gaussian) distributions, binomial distributions, Wiebull

distributions, and other distributions ad infinitum. Many of these distributions are subsets of each

other and, in most cases are used to fit a unique situation. ....These statistical relationships are

approximate models that attempt to match empirical data. The word approximate must be

emphasized; it must be remembered that none of these models precisely fit any specific case."

Pyzdek, pp. 389-390.

Such curves represent a model of the reliability "life" of the product, a "life distribution" so to speak.

"How do we pick a

recommend is as follows: Use a life distribution model primarily because it works; that is, fits the

data well and leads to reasonable projection when extrapolating beyond the range of data. Look for a

new model when the one previously used no longer 'works'. " Tobias, p. 91.

The purpose in identifying a life distribution model for a component is to predict the component's

reliability under conditions that are impractical to test. "If we have 100 units on test, the smallest

percentile we can actually observe is the 1 percent point. We need a parametric model to project to

the earlier time when 0.001 percent of the population might fail, or

failures for a time much smaller than the time of the first observed test failure" or beyond the last

observed test failure.

"In many cases...

as a linear relation

grouped about a straight line in such a manner that [they are] normally distributed...then the theory of

linear regression may be applied to the transformed observations. The transformation functions are

chosen on the basis of a graphical analysis of the observations..." Hald, pp. 558-559.

"Many physical situations produce data that are normally distributed; others, data that follow some

other known distribution; and still others, data that can be transformed to normal data." Crow, p. 88.

"

can be used to determine reliability-confidence relationships.

In contrast to our understanding of curved lines, our understanding of confidence limits for

extrapolated straight lines is very well developed. Almost every advanced statistical textbook, and

some quality control textbooks (e.g., Juran), provide the basic equations for establishing confidence

limits on the mean of the data at a point on the X-axis --- that is, at the stress level for which a

reliability statement is sought. Such an equation provides the equivalent of the standard error of the

mean at the extrapolated point.

"Probability paper was developed initially to allow data analysts to plot data, obtain estimates, and

assess fit of a particular model by comparing with a straight line....However, because there are many

different combinations of possible probability and data axes that might be needed, it is useful to have

a computer implementation of probability plotting methods...." Meeker, p.123.

"From a purely statistical point of view, the regression curve provides a description of the interrelation

between the two variables within the limited range of the observations....It is therefore absolutely

necessary that extrapolations [to values outside this range] be based upon professional knowledge

concerning the data...." Hald, p. 559. "These convenient methods do require judgment (e.g., how

straight must the line be?) because the sample is never a perfect fit...." Juran, p. 23.44.

equation mentioned above), the %Failure transformation is reversed, the result of which, when

subtracted from 1.000, yields the Reliability at the extrapolated stress level, with a confidence equal

to whatever was used to determine the "t-table" value in the calculation of the one-sided confidence

limit.

statistical technique that can save time and money by providing reliability data

that is adequate to the task but is based on smaller sample sizes or is

collected in a shorter time-frame than would otherwise be possible.

Amstadter, B. L.,

Crow, E. L.,

Hald, A.,

Juran, J. M., and Gryna, F. M., editors,

Meeker, W. Q., and Escobar, L. A.,

Pyzdek, T., and Berger, R. W., editors,

Tobias, P. A., and Trinidade, D. C.,

Reliability & Distribution Plotting(and how it can save a company a lot of time and money)2010 by John Zorich ( johnzorich@yahoo.com ) |

Stress that a sample of product was subjected to, vs. % of sample that failed. |

Same plot after transformation and extrapolation to area of interest. |