# Cumulative Binomial for Test Design and Analysis

[Please note that the following article — while it has been updated from our newsletter archives — may not reflect the latest software interface and plot graphics, but the original methodology and analysis steps remain applicable.]

This article presents a summary of the techniques available to use the cumulative binomial distribution to assist in the design of effective reliability demonstration tests and also to determine the reliability that has been demonstrated in a test with few or no failures.

The Need for Effective Design and Analysis of Demonstration Tests

The need to design a test that proves that a product has met a certain reliability goal is a fairly common occurrence in industry. In many cases, the manufacturers desire to prove that their product designs meet a certain reliability specification, but do not have the resources to conduct a full-scale reliability/life test. The result is not a rigorous life test in which failure information is eagerly sought, but rather a demonstration test in which failures are hoped to be avoided. These demonstration tests are often of a limited scope and are frequently conducted by product engineers with little or no knowledge of life data analysis principles. These demonstration tests often occur toward the end of a product development cycle, when the product design has been improved to the point at which failures are, it is hoped, relatively infrequent. There is usually a great deal of pressure to maintain the development schedule and as a result, tests that merely demonstrate an acceptable minimum reliability level are required.

Although these tests yield minimal meaningful information about the product's life characteristics, they are a common requirement for many engineers in the design and manufacturing arena. Therefore, these engineers need to be able to design and allocate resources for these tests without having a great deal of detailed information beforehand. Fortunately, the cumulative binomial distribution can be put to use to help develop a rough estimate of the test design, which includes test duration and the number of units to be tested, without having to develop a complete life test. Otherwise, a large quantity of failures must be achieved before any conclusions can be drawn about the reliability of the product. The cumulative binomial distribution can also be used to analyze the results of tests in which there were few or no failures.

Using the Cumulative Binomial Distribution

The cumulative binomial distribution takes the form:

where:

• C.L. is the confidence level for the test. This can be thought of as the probability of more than the maximum number of allowable failures occurring on test.
• N is the number of units on test.
• r is the maximum allowable number of failures on test.
• R is the reliability of the product for the duration of the test.

Essentially, the test design process involves solving the cumulative binomial equation for one variable, given that the other variables are known or can be assumed. This is particularly important for the variable R, the reliability. An estimate of the reliability value for the duration of the test is necessary when using the cumulative binomial for test design. In some cases, it may be necessary to provide values for the parameter estimates of the product's life distribution for more detailed calculations. The next sections describe how test design information can be obtained by solving the cumulative binomial equation. Solving the cumulative binomial equation for certain variables can be difficult and in some cases almost impossible without the use of a computer. The test design utility included with ReliaSoft Weibull++ software performs these estimations automatically.

 Figure 1: Weibull++ Reliability Demonstration Test Utility

Number of Test Units Based on Allotted Test Time and Required Reliability or MTTF

In this case, the engineer has an allotted test time and needs to know how many units need to be tested for that time in order to demonstrate a required reliability. It is assumed that the engineer either has information regarding the life distribution of the product or can supply an estimate of the reliability of the product at the specified test time. If the engineer has the distribution parameters, he or she can calculate the reliability at the test duration time, otherwise the estimate of the product's reliability can be substituted directly for R. Then, given the desired confidence level (C.L.), the maximum allowable failures (r) and the reliability value (R), the cumulative binomial equation can be solved for the number of units (N). Similarly, if the test is to demonstrate a required MTTF (mean time to failure), at least one of the parameters of the product's life distribution must be known in order to solve for R. The exception to this is for the one-parameter exponential distribution, where the parameter is assumed to be the specified MTTF. Then, the MTTF and the parameter are used to calculate the value of R for substitution in the cumulative binomial equation and the procedure is the same as that described above.

Test Duration Based on Number of Test Units and Reliability Characteristics

In order to determine the proper test duration, the distribution of the product's life data must be known. The engineer knows the number of test units (N), the number of allowable failures (r) and the desired confidence level (C.L.). With this information, the cumulative binomial equation can be solved for R. Once the reliability value (R) has been determined, it can be substituted into the appropriate reliability equation and the test time can be solved for given the distribution parameters and the reliability value.

Demonstrated Reliability Based on Test Results

The cumulative binomial can also be used to determine the reliability that has been demonstrated on a test with few or no failures. There does not have to be any prior information about the life distribution of the products on test in order to make this assessment. However, if enough failures are encountered on the test, it may be advisable to perform a reliability analysis on the data. In order to determine the demonstrated reliability, you need the number of units on test (N), the number of failures on the test (r) and the desired confidence level (C.L.). With this information, the cumulative binomial equation can be solved for R, which is the reliability demonstrated on the test. However, the reliability value is associated only with the test duration.

Demonstrated Confidence Level Based on Test Results and Reliability Characteristics

The confidence level demonstrated on a test can be calculated based on the test results and reliability characteristics. Given the number of units on test (N), the number of failures (r) and the calculated or estimated reliability (R), the value of the confidence level (C.L.) can easily be calculated with the cumulative binomial equation.