Weibull Distribution

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Distribution - Weibull


References: Dr. Mikel J. Harry, Juran's Quality Control Handbook

Purpose

The Weibull distribution is applicable to make population predictions around a wide variety of patterns of variation, including departures from the normal and exponential distributions.

It covers many shapes of distributions thus reducing the problem of deciding which of the common distributions (e.g. normal or exponential) best fits a set of data.

Weibull Distribution Analysis

Weibull analysis can make predictions about a product's life, compare the reliability of competing product designs, statistically establish warranty policies or proactively manage spare parts inventories, to name a few common industrial applications. 

It is one of the most widely-used solutions for modeling lifetime data (time-to-failure data) and reliability. 

With reliability analysis we make predictions about the life of products by fitting a distribution to the data from a sample of the products. The distribution from the data is then used to estimate life of the product such as reliability or probability of failure at a specific time, the average life and the failure rate. 

Anatomy

The Weibull DistributionWeibull Distribution

Terminology

A. Vertical axis - Scale to measure the value of the Weibull function.

B. Horizontal axis - Scale of measure of the independent variable (X).

C. Curves representing the Weibull Distribution for different values of its parameters beta, alpha and gamma

  • (beta) Shape parameter - Reflects the pattern of the curve. When Beta = 1.0 the Weibull function reduces to the exponential function and when Beta is about 3.5 (and Alpha = 1 and Gamma = 0), it closely approximates the normal distribution.
  • (alpha) Scale parameter - As changes the curve becomes flatter or more peaked.
  • (gamma) Location parameter - Smallest possible value of X (often assumed to be zero, thereby simplifying the equation.

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Major Considerations

The location parameter is usually assumed zero to simplify calculations.

Application Cookbook

An analytical approach for the Weibull distribution (even with tables) is cumbersome, and predictions are usually made with Weibull probability paper.

1. To create a plot in Minitab, enter the values in a column and choose the menu Graph > Probability Plot specifying the column where data is entered.

2. Observe if the points fall approximately in a straight line, and if so, read the probability predictions from the graph. For example, based on a sample taken on the life of a component we want to predict the percentage failure of the population.

The failure data (expressed in hours) is 10 263, 12 187, 16 908, 18 042 and 23 271. Applying steps 1 and 2 we obtain the following graph.

Weibull Probability Plot

Reading the graph we can see that about 80% of the population will fail in less than 20,000 hours.

From Weibull Distribution to Six Sigma Tools.

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