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 wear-out). 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, wear-out, 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.

Note: This is an advanced feature and should be used in the proper context and with a good understanding of how to apply a three-parameter Weibull distribution.

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
Electro-mechanical 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:

R-Squared Linear regression (least squares): An R-Squared test statistic greater than 0.9 is considered a good fit for linear regression.
Kolmogorov-Smirnov: The GE Digital APM system uses confidence level and P-Value to determine if the data is considered a good fit. If the P-Value is greater than 1 minus the confidence level, the test passes.

Note: The R-Squared test statistic is calculated only for reference. The GE Digital APM system uses the Kolmogorov-Smirnov test as the Goodness of Fit test.

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).

A Triangular Distribution is a continuous Probability Distribution with:

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.