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This living paper developed and maintained by Kim Niles, is co-published at San Diego SAMPE (Soc.for Adv.of Material & Process Engineering), and was featured under "Quality Web Watch" of Quality Progress Magazine, March 2001; page 22.  

This is a living paper in that any related comments, definitions, or stories you submit to us using the on-line form below will be added as appropriate and you will be featured in the reference section at the bottom.     

 Jump to any of the 28 distributions below as follows (* indicates most common types):

The population of a continuous variable has an underlying distribution. Throwing enough rocks at a spot on the earth will soon form a pile that would then represent the distribution of the process of the particular rock thrower throwing rocks at that spot from a fixed distance.  If the rocks are magnetic the pile would be more peaked (higher kurtosis).  If the distance to the point is increased, the pile will be less peaked and likely have a more skewed, less bell shape (normal/gaussian) distribution.  If half the rocks are thrown by one person and the other half by another, the pile is likely to have two peaks (bi-modal).

The normal distribution is the most used distribution for process improvement statistics.  Other distributions such as Binomial and Poisson are very common in QC inspection.   The Weibull, Gamma, and Exponential distributions are very common in Reliability Engineering.

We've had numerous requests for formulas which is beyond the scope of this site. However, a great source for these formulas as well as other distribution information is Mike Mclaughlin's  -  "A Compendium of Common Probability Distributions" (pdf format) which lists 56  continuous symmetric, continuous skewed, continuous mixture, discrete, and discrete mixture distributions.  

Bernoulli Distribution:  A distribution whose outcome has only two possibilities: Success or failure.  Also known as Binomial distribution. 

Flipping a non-biased coin should approximate the 0.5 event probability.

Binomial Distribution:  A distribution that gives the probability of observing successes in a fixed number of independent or Bernoulli trials. 

Pass or fail situations with only two outcomes and the probability remains the same for all trials.   Also called Bernoulli distribution. 

Flipping a non-biased coin should approximate the 0.5 event probability.  Used in part sampling with pass or fail specification criteria.

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Geometric Distribution:  A distribution that characterizes the number of failures before the first success in a series of Bernoulli trials, a special case of Negative Binomial distributions, where k = 1. 

Flipping a non-biased coin should approximate the 0.5 event probability.

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Example:  Binomial sampling where a sample of 50 is randomly chosen from a population of size 100.

Negative Binomial Distribution:  A distribution that characterizes the number of failures before the kth success in a series of Bernoulli trials.

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Poisson Distribution:  A distribution that expresses probabilities that concern the number of events per unit time.  Often used to describe situations in which the probability of an event is small and the number of opportunities for the event is large. 

Good for applicable inspection sampling plans.  Events occur at a constant average rate, with only one of two outcomes countable.  The Poisson is an extension of the binomial distribution in which the number of samples is infinite.

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Beta Distribution:  A distribution that is useful for random variables constrained to lie between 0 and 1.   Characterized by two parameters:  Shape 1 and Shape 2. 

Possible use with project task times.

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Chi-Square Distribution:  A distribution that is useful for random variables constrained to be greater than 0.   Characterized by one parameter:  Degrees of Freedom.

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Erlang Distribution:  A distribution useful for random variables constrained to be greater than 0.   Characterized by the Shape and Scale parameters.

 

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Exponential Distribution:  A continuous distribution useful for characterizing random variables that can take only positive values.   Completely determined by its mean. 

A distribution that fits time-series data, such as arrival times, where you expect arrivals at a constant rate. 

This distribution is a special case of both the Gamma and the Weibull distributions. The Expo(b) density function is (1/b)e-x/b for x = 0, and it has mean = b and variance = b2. This is the only continuous distribution having a "memoryless" property.

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Extreme Value Distribution:  A distribution useful for random variables constrained to be greater than 0.   Characterized by two parameters:  Mode and Scale. 

A distribution for fitting the limiting distribution for the maximum number of samples collected from a process.  This distribution applies when extremes rather than means are collected from samples from an unknown or complex underlying distribution.

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F (Variance Ratio) Distribution:  A distribution useful for random variables constrained to be greater than 0.   Characterized by two parameters: Numerator Degrees of Freedom and Denominator Degrees of Freedom.

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Gamma Distribution: A distribution useful for random variables constrained to be greater than 0.   Characterized by the Shape and Scale parameters.

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Laplace Distribution:  A distribution useful for random variables from a distribution that is more peaked than a Normal distribution.   Characterized by two parameters: Mean and Scale.

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Logistic Distribution:  A distribution useful for random variables that are not constrained to be greater than or equal to 0.   Characterized by two parameters: Mean and Standard Deviation.

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Lognormal Distribution:  A distribution useful for random variables constrained to be greater than 0.   Characterized by two parameters: Mean and Standard Deviation. 

A positively skewed distribution that can take on various shapes.

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Normal Distribution:  A continuous probability distribution that is useful in characterizing a large variety and type of data. 

It is a symmetric, bell-shaped distribution, completely determined by its Mean and Standard Deviation. 

A common bell-shaped curve used to calculate probabilities of events that tend to occur around a mean value and trail off with decreasing likelihood.  Also called Gaussian Distribution.  Values are grouped about a central value with symmetrical variance (bell curve).

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Pareto Distribution:  A distribution with a decreasing density function.  One parameter, Shape, is necessary to specify the distribution.

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Student's t Distribution:  A distribution useful in forming confidence intervals for the mean when the variance is unknown, testing to determine if two sample means are significantly different, or testing to determine the significance of coefficients in a regression. 

The distribution is similar in shape to a Normal distribution.

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Triangular Distribution:  A distribution useful for random variables constrained to lie between two fixed limits. 

This distribution peaks at some value between two limits and characterized by three parameters: Lower Limit, Central Value (Mode), and Upper Limit.

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A distribution that is an appropriate model for product failures because its failure rate curves can take various shapes.   This distribution is a generalization of the exponential distribution. 

Other Interesting Weibull distribution/analysis links:
1- Using MS Excell for Weibull Analysis
2- See animated distribution under "W" in this Stat Glossary

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