Distribution Types
About Normal Distribution
A Normal Distribution describes the spread of data values through the calculation of two parameters: mean and standard deviation. When using the Normal Distribution on time to failure data, the mean exactly equals MTBF and is a straight arithmetic average of failure data. Standard deviation (denoted by sigma) gives estimate of data spread or variance.
A Normal Distribution uses the following parameters:
 Mean: The arithmetic average of the datapoints.
 Standard Deviation: A value that represents the scatter (how tightly the datapoints are clustered around the mean).
About Weibull Distribution
A Weibull Distribution describes the type of failure mode experienced by the population (infant mortality, early wear out, random failures, rapid wearout). Estimates are given for Beta (shape factor) and Eta (scale). MTBF (Mean Time Between Failures) is based on characteristic life curve, not straight arithmetic average.
A Weibull Distribution uses the following parameters:
 Beta: Beta, also called the shape factor, controls the type of failure of the element (infant mortality, wearout, or random).
 Eta: Eta is the scale factor, representing the time when 63.2 % of the total population is failed.
 Gamma: Gamma is the location parameter that allows offsetting the Weibull distribution on time. The Gamma parameter should be used if the datapoints on the Weibull plot do not fall on a straight line.
If the value of Beta is greater than one (1), you can perform Preventative Maintenance (PM) Optimizations. A Gamma different from a value zero (0) means that the distribution is shifted to fit the datapoints more closely.
Weibull Analysis Information
You can use the following information to compare the results of individual Weibull analyses. The following results are for good populations of equipment.
Beta Values Weibull Shape Factor  

Components 
Low 
Typical 
High 
Low (days) 
Typical (days) 
High (days) 
Ball bearing 
0.7 
1.3 
3.5 
583 
1667 
10417 
Roller bearings 
0.7 
1.3 
3.5 
375 
2083 
5208 
Sleeve bearing 
0.7 
1 
3 
417 
2083 
5958 
Belts drive 
0.5 
1.2 
2.8 
375 
1250 
3792 
Bellows hydraulic 
0.5 
1.3 
3 
583 
2083 
4167 
Bolts 
0.5 
3 
10 
5208 
12500 
4166667 
Clutches friction 
0.5 
1.4 
3 
2792 
4167 
20833 
Clutches magnetic 
0.8 
1 
1.6 
4167 
6250 
13875 
Couplings 
0.8 
2 
6 
1042 
3125 
13875 
Couplings gear 
0.8 
2.5 
4 
1042 
3125 
52083 
Cylinders hydraulic 
1 
2 
3.8 
375000 
37500 
8333333 
Diaphragm metal 
0.5 
3 
6 
2083 
2708 
20833 
Diaphragm rubber 
0.5 
1.1 
1.4 
2083 
2500 
12500 
Gaskets hydraulics 
0.5 
1.1 
1.4 
29167 
3125 
137500 
Filter oil 
0.5 
1.1 
1.4 
833 
1042 
5208 
Gears 
0.5 
2 
6 
1375 
3125 
20833 
Impellers pumps 
0.5 
2.5 
6 
5208 
6250 
58333 
Joints mechanical 
0.5 
1.2 
6 
58333 
6250 
416667 
Knife edges fulcrum 
0.5 
1 
6 
70833 
83333 
695833 
Liner recip. comp. cyl. 
0.5 
1.8 
3 
833 
2083 
12500 
Nuts 
0.5 
1.1 
1.4 
583 
2083 
20833 
"O"rings elastomeric 
0.5 
1.1 
1.4 
208 
833 
1375 
Packings recip. comp. rod 
0.5 
1.1 
1.4 
208 
833 
1375 
Pins 
0.5 
1.4 
5 
708 
2083 
7083 
Pivots 
0.5 
1.4 
5 
12500 
16667 
58333 
Pistons engines 
0.5 
1.4 
3 
833 
3125 
7083 
Pumps lubricators 
0.5 
1.1 
1.4 
542 
2083 
5208 
Seals mechanical 
0.8 
1.4 
4 
125 
1042 
2083 
Shafts cent. pumps 
0.8 
1.2 
3 
2083 
2083 
12500 
Springs 
0.5 
1.1 
3 
583 
1042 
208333 
Vibration mounts 
0.5 
1.1 
2.2 
708 
2083 
8333 
Wear rings cent. pumps 
0.5 
1.1 
4 
417 
2083 
3750 
Valves recip comp. 
0.5 
1.4 
4 
125 
1667 
3333 
Equipment Assemblies 
Low 
Typical 
High 
Low (days) 
Typical (days) 
High (days) 
Circuit breakers 
0.5 
1.5 
3 
2792 
4167 
58333 
Compressors centrifugal 
0.5 
1.9 
3 
833 
2500 
5000 
Compressor blades 
0.5 
2.5 
3 
16667 
33333 
62500 
Compressor vanes 
0.5 
3 
4 
20833 
41667 
83333 
Diaphgram couplings 
0.5 
2 
4 
5208 
12500 
25000 
Gas turb. comp. blades/vanes 
1.2 
2.5 
6.6 
417 
10417 
12500 
Gas turb. blades/vanes 
0.9 
1.6 
2.7 
417 
5208 
6667 
Motors AC 
0.5 
1.2 
3 
42 
4167 
8333 
Motors DC 
0.5 
1.2 
3 
4 
2083 
4167 
Pumps centrifugal 
0.5 
1.2 
3 
42 
1458 
5208 
Steam turbines 
0.5 
1.7 
3 
458 
2708 
7083 
Steam turbine blades 
0.5 
2.5 
3 
16667 
33333 
62500 
Steam turbine vanes 
0.5 
3 
3 
20833 
37500 
75000 
Transformers 
0.5 
1.1 
3 
583 
8333 
591667 
Instrumentation 
Low 
Typical 
High 
Low (days) 
Typical (days) 
High (days) 
Controllers pneumatic 
0.5 
1.1 
2 
42 
1042 
41667 
Controllers solid state 
0.5 
0.7 
1.1 
833 
4167 
8333 
Control valves 
0.5 
1 
2 
583 
4167 
13875 
Motorized valves 
0.5 
1.1 
3 
708 
1042 
41667 
Solenoid valves 
0.5 
1.1 
3 
2083 
3125 
41667 
Transducers 
0.5 
1 
3 
458 
833 
3750 
Transmitters 
0.5 
1 
2 
4167 
6250 
45833 
Temperature indicators 
0.5 
1 
2 
5833 
6250 
137500 
Pressure indicators 
0.5 
1.2 
3 
4583 
5208 
137500 
Flow instrumentation 
0.5 
1 
3 
4167 
5208 
416667 
Level instrumentation 
0.5 
1 
3 
583 
1042 
20833 
Electromechanical parts 
0.5 
1 
3 
542 
1042 
41667 
Static Equipment 
Low 
Typical 
High 
Low (days) 
Typical (days) 
High (days) 
Boilers condensers 
0.5 
1.2 
3 
458 
2083 
137500 
Pressure vessels 
0.5 
1.5 
6 
52083 
83333 
1375000 
Filters strainers 
0.5 
1 
3 
208333 
208333 
8333333 
Check valves 
0.5 
1 
3 
4167 
4167 
52083 
Relief valves 
0.5 
1 
3 
4167 
4167 
41667 
Service Liquids 
Low 
Typical 
High 
Low (days) 
Typical (days) 
High (days) 
Coolants 
0.5 
1.1 
2 
458 
625 
1375 
Lubricants screw compr. 
0.5 
1.1 
3 
458 
625 
1667 
Lube oils mineral 
0.5 
1.1 
3 
125 
417 
1042 
Lube oils synthetic 
0.5 
1.1 
3 
1375 
2083 
10417 
Greases 
0.5 
1.1 
3 
292 
417 
1375 
Weibull Results Interpretation
GE Digital APM Reliability shows the failure pattern of a single piece of equipment or groups of similar equipment using Weibull analysis methods. This helps you determine the appropriate repair strategy to improve reliability.
Is the Probability Plot a good fit?
Follow these steps to determine whether or not the plot is a good fit:
 Identify Beta (slope) and its associated failure pattern.
 Compare Eta (characteristic life) to standard values.
 Check goodness of fit, compare with Weibull database.
 Make a decision about the nature of the failure and its prevention.
The following chart demonstrates how to interpret the Weibull analysis data using the Beta parameter, Eta parameter, and typical failure mode to determine a failure cause.
Weibull Results  Interpretation  

Beta 
Eta 
Typical Failure Mode 
Failure Cause 
Greater than 4 
Low compared with standard values for failed parts (less than 20%) 
Old age, rapid wear out (systematic, regular) 
Poor machine design 
Greater than 4 
Low compared with standard values for failed parts (less than 20%) 
Old age, rapid wear out (systematic, regular) 
Poor material selection 
Between 1 and 4 
Low compared with standard values for failed parts (less than 20%) 
Early wear out 
Poor system design 
Between 1 and 4 
Low 
Early wear out 
Construction problem 
Less than 1 
Low 
Infant Mortality 
Inadequate maintenance procedure 
Between 1 and 4 
Between 1 and 4 
Less than manufacturer recommended PM cycle 
Inadequate PM schedule

Around 1 
Much less than 
Random failures with definable causes 
Inadequate operating procedure 
Goodness of Fit (GOF) Tests for a Weibull Distribution
A Goodness of Fit test is a statistical test that determines whether the analysis data follows the distribution model.
 If the data passes the Goodness of Fit test, it means that it follows the model pattern closely enough that predictions can be made based on that model.
 If the data fails the Goodness of Fit test, it means that the data does not follow the model closely enough to confidently make predictions and that the data does not appear to follow a specific pattern.
Weibull results are valid if Goodness of Fit (GOF) tests are satisfied. Goodness of Fit tests for a Weibull distribution include the following types:
 RSquared Linear regression (least squares): An RSquared test statistic greater than 0.9 is considered a good fit for linear regression.
 KolmogorovSmirnov: The GE Digital APM system uses confidence level and PValue to determine if the data is considered a good fit. If the PValue is greater than 1 minus the confidence level, the test passes.
About Exponential Distribution
An Exponential Distribution is a mathematical distribution that describes a purely random process. It is a single parameter distribution where the mean value describes MTBF (Mean Time Between Failures). It is simulated by the Weibull distribution for value of Beta = 1. When applied to failure data, the Exponential distribution exhibits a constant failure rate, independent of time in service. The Exponential Distribution is often used in reliability modeling, when the failure rate is known but the failure pattern is not.
An Exponential Distribution uses the following parameter:
 MTBF: Mean time between failures calculated for the analysis.
About Lognormal Distribution
In Lognormal Distributions of failure data, two parameters are calculated: Mu and Sigma. These do not represent mean and standard deviation, but they are used to calculate MTBF. In Lognormal analysis, the median (antilog of mu) is often used as the MTBF. The standard deviation factor (antilog of sigma) gives the degree of variance in the data.
A Lognormal Distribution uses the following parameters:
 Mu: The logarithmic average for the Distribution function.
 Sigma: The scatter.
 Gamma: A location parameter.
About Triangular Distribution
Triangular Distribution is typically used as a subjective description of a population for which there is only limited sample data, and especially in cases where the relationship between variables is known, but data is scarce (possibly because of the high cost of collection). It is based on a knowledge of the minimum (a) and maximum (b) and an inspired guess as to the modal value (c).
 Lower limit a
 Upper limit b
 Mode c
…where a < b and a ≤ c ≤ b.
About Gumbel Distribution
The Gumbel Distribution is a continuous probability distribution. Gumbel distributions are a family of distributions of the same general form. These distributions differ in their location and scale parameters: the mean of the distribution defines its location, and the standard deviation, or variability, defines the scale.
The Gumbel Distribution is a probability distribution of extreme values.
In probability theory and statistics, the Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
About Generalized Extreme Value Distribution
In probability theory and statistics, the Generalized Extreme Value (GEV) Distribution is a family of continuous probability distributions developed within extreme value theory.
By the Extreme Value Theorem, the GEV Distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables.