Additional Resources > Windchill Risk and Reliability Practitioner’s Guide > Weibull Analysis > Select the Distribution Type > Statistical Concerns
Statistical Concerns
Although statisticians oppose the use of extremely small samples, cases of safety and extraordinary financial loss prevent the collection of additional data. When only a few failures exist, Weibull analysis can provide usable results because:
• Wear-out failures tend to occur in the oldest units. This results in most failures being plotted in the B-0.1 to B-1 lives, which is in the lower left corner of the Weibull probability plot, the area in which engineering is most interested.
• Both failures and suspensions are included. Although suspensions are not weighted as heavily as failures, thousands of suspensions may exist, contributing to more accurate engineering predictions in the B-0.1 to B-1 lives.
The Weibull distribution applies to situations where there are multiple opportunities to fail and the first failure is of extreme interest. The Weibull distribution also applies to system deterioration that is linear rather than accelerating. When deterioration is non-linear but rather a function of the current deterioration, the lognormal distribution applies. Table 7-5 describes the normal and lognormal distributions because they are occasionally used for the parametric analysis of life data even though they are not members of the Weibull family. Most Weibull software provides for quickly generating all distributions and automatically picking the best fit for a data set.
Table 7-5. Non-Weibull Distributions for Failure Data Analysis
Normal (or Gaussian)
The two parameters required for the normal distribution are the mean and standard deviation. Normal distributions, which are always symmetric and commonly called bell curves, are important and widely used in the field of statistics and probability. Normal distributions are frequently used to describe equipment that has increasing failure rates with time. The normal distribution is recommended only if failure times can be expressed as a summation of some other random variables. Although the normal distribution is a handy tool for describing all sorts of different data, it allows observations to be negative. Because parts cannot fail before time t = 0, life data is always positive. As a result, the normal distribution does not usually describe life data very well. Most analysts do not even bother to check for a normal fit because life data that follow the normal distribution also generate good Weibull probability plots.
Lognormal
The two parameters required for the lognormal distribution are the mean and standard deviation. Although the lognormal distribution is similar to the normal distribution, it assumes that the logarithm of the values of random variables is normally distributed rather than the values themselves. Thus, all values are positive, and the distribution is skewed to the left. The lognormal distribution is probably the most significant competitor to the Weibull distribution. It is frequently used in engineering for metal-fatigue testing, maintainability data (time to repair), chemical-process equipment failures and repairs, some material characteristics and non-linear, accelerating deterioration. When the time to failure results from the multiplication of effects, the lognormal distribution is recommended. For example, in the case of progressive deterioration, a crack forms due to stress, and the stress increases as the crack grows. Non-engineering applications of the lognormal distribution include the analysis of personal incomes, inheritances and bank deposits.