Analysis of “One-Shot” Devices

Introduction

A "one-shot" device is defined as a product, system, weapon, or equipment that can be used only once. After use, the device is destroyed or must undergo extensive rebuild. "Oneshot" devices typically spend their life in dormant storage or stand-by readiness. The device may end its useful life without ever being called upon to provide the function for which it was designed, limiting the availability of failure data during its life cycle.

Determining the reliability of a "one-shot" device poses a unique challenge to the manufacturers and users of these devices. Due to the destructive nature and costs of the testing, the current trend is to minimize testing. But the expectations are to have a high level of system reliability. Therefore, the test planner must have the knowledge necessary to determine the minimum sample size that must be tested to demonstrate a desired reliability of the population at some acceptable level of confidence.

This START sheet addresses the steps necessary to statistically establish the reliability, or probability of success, of "one-shot" devices.

Background and Concepts

Statistical tools are designed to analyze the distribution characteristics of some population based on a sample drawn from the population. For "one-shot" devices, acceptance sampling is a statistical method used to predict the probability of success, or reliability, by estimating an attribute of the population through a sample. An attribute is an inherent characteristic that is evaluated in terms of whether or not the product performs as designed. Test results are measured by determining if the product was good or bad, passed or failed, etc. Non-conformance of the product characteristic is generally expressed as a proportion defective. Proportion defective is the number of failures that occurred in a sample size divided by the sample size.

Attribute sampling uses the binomial equation to test a hypothesis that a product has an acceptable defective rate at some acceptable level of risk. For "one-shot" devices, the object is to verify that the probability of success, when the device is called upon to function, is satisfactory at some desired level of confidence.

Binomial Distribution

The binomial distribution is based on the work of Jacob Bernoulli (1654 ­ 1705). The distribution is based on "Bernoulli trials", where each trial will result in only of two possible outcomes, i.e., passed or failed. To use the binomial distribution to predict the probability of success for "oneshot" devices, the trials in the sample must meet the following conditions:
• Each trial must be independent. The outcome of one trial cannot influence an outcome of another trial.
• For each trial, there is only one of two possible outcomes.
• The number of trials in a sample must be fixed in advance and be a positive integer number.
• The probability of success must be the same for all trials.
The binomial equation to predict the probability of a specific number of r defects or failures in n samples is:
 (1) where: p = proportion defective n = sample size r = number defective P(r) = probability of getting exactly r defective or failed units in a sample size of n units

The desired proportion defective is the Lot Tolerance Percent Defective (LTPD), which is the poorest quality in an individual lot that one is willing to accept.

To calculate the probability of k or fewer failures occurring in a test of n units, the probability of each failure occurring must be summed, as shown in Equation 2.

 (2)

The Confidence Level that the population is only p defective based on r k defects from a sample of n is:

 (3)

For example, assume that the population of a part can be no more than 10% defective (p = 0.1). The plan is to test twenty parts and allow only one failure (pass-fail criterion). Using Equation 1, the probabilities of exactly one and exactly zero failures occurring are:

P(r = 0) = 0.122
P(r = 1) = 0.270

Using Equation 2, the probability of one failure or less is the sum of these probabilities; i.e.,

P(r ≤ 1) = 0.122 + 0.270 = 0.392

Using Equation 3, if the sample passes the test, one would only be 60.8% confident (CL = 1 - 0.392 = 0.608, or 60.8%) that the proportion defective in the population is 10% or less.

The sampling plan is inadequate for us to be 90% confident that the population is no more than 10% defective. The sample size must be increased, the number of allowable failures decreased, or both. Using Table 1, when one failure is allowed, the sample size must be at least 38 for us to be 90% confident that the population is no more than 10% defective.

If no failures are allowed in the 20 tests, P(r = 0) = 0.122, the confidence level increases to 87.8% that the proportion defective in the population is 10% or less. To reach a 90% CL, the sample size would have to be 22 with no failures allowed. Table 1 shows the relationship between the number of failures and sample size for tests to demonstrate that the proportion defective in the population is 10% or less at Confidence Levels of 90% and 95%.

Estimating p

Assume a sample of 38 units was tested and two failures occurred. The proportion defective of the population can be estimated by calculating the upper and lower confidence limits of the true p for the population from which the sample was drawn. To do this, we use the F distribution.

Table 1. Failures Allowed vs. Sample Size vs. Confidence Level (CL) for 90% Reliability (10% defective rate)
No. of Failures Sample Size
90% Confidence 95% Confidence
0 22 29
1 38 47
2 52 63
3 65 77
4 78 92
5 91 104
6 104 116
7 116 129
8 128 143
9 140 156
10 152 168

Equations 4 and 5 show how the lower (PL) and upper (PU) limits on p are calculated.

 (4) where: r is the number of failures observed n is the sample size FL corresponds to the F distribution for the following degrees of freedom and associated required CL (5) where: r is the number of failures observed n is the sample size FU corresponds to the F distribution for the following degrees of freedom and associated required CL v1= 2 (r + 1) v2= 2 (n - r)

Tables of the values of the F distribution can be found in statistics textbooks. Using the example where 38 parts were tested and two failures occurred, the proportion defective of the population can be estimated with 90% confidence by using Equations 4 and 5. We find that FL = 3.79 and pL = 0.014, and that FU = 1.86 and pU = 0.134. We can, therefore, state with 90% confidence that the true p of the population lies between 1.4% and 13.4%. To narrow the range, we must test a larger sample or accept a lower Confidence Level. Of course, if we observed fewer failures, the range would also be smaller.

Tables Developed by the RIAC

Given a desired confidence level, Equation 2 can be used to determine the sample size n given r defects via a trial and error approach. The value of n is varied until the desired Confidence Level is reached. Computer spreadsheets, e.g., Excel, can be used to make the calculations. Note that Excel has a limitation in that the largest factorial that it can use is 170!. If you need to calculate factorials of numbers larger than 170, RIAC recommends you solve the binomial equation using logarithms.

RIAC recognized that calculations using the Binomial distribution can be tedious, time consuming, and easily result in mistakes. Therefore, we developed a series of tables for different proportion defective LTPD: p = 0.01, 0.05, 0.10, 0.15, and 0.20 and a calculator.

Table 2 shows the sample size required for a given number of failures to achieve a desired confidence level for p = 0.10. To use the table, assume the plan is to test 45 units and allow no failures as the criterion of success. Go to the row of the table with "0" under the column "Number of Failures" and read across to the right to the last column. The value is 45. One would be 99% confident that zero failures in a test of a sample of 45 items indicates that the population is no more than 10% defective. If two failures were allowed, you would go to the left column and find the value 2. A sample size of 45 lies between a 80% and 90% Confidence Level. By interpolation, the Confidence Level that the population is 10% defective or less is 83%.

The tables available from the RIAC provides the user with a quick method of approximating:
• The sample size required to achieve a desired CL given an expected or allowable number of failures.
• The CL given the allowable number of failures and sample size.
• The allowable number of failures given the CL and the sample size.
Table 2. Sample Size Required for p = 0.1 To Achieve a Desired Confidence Level
No.
of
Failures
Confidence Levels
60% 80% 90% 95% 99%
Sample Size
0 9 16 22 29 45
1 20 29 38 47 65
2 31 42 52 63 83
3 41 55 65 77 98
4 52 67 78 92 113
5 63 78 91 104 128
6 73 90 104 116 142
7 84 101 116 129 158
8 95 112 128 143 170
9 105 124 140 156 184
10 115 135 152 168 197
11 125 146 164 179 210
12 135 157 176 191 223
13 146 169 187 203 236
14 156 178 198 217 250
15 167 189 210 228 264
16 177 200 223 239 278
17 188 211 234 252 289
18 198 223 245 264 301
19 208 233 256 276 315
20 218 244 267 288 327
22 241 266 290 313 342
24 262 286 312 340 378
26 282 308 330 364 395
28 303 331 354 385 430
30 319 354 377 408 448
35 374 403 430 462 505
40 414 432 490 512 565
45 478 510 550 580 620
50 513 534 595 628 675

For Further Study
1. O'Connor, P. D. T., "Practical Reliability Engineering," John Wiley & Sons, 1995.
2. John, P. W. M., "Statistical Methods in Engineering and Quality Assurance," John Wiley & Sons, 1990.
3. Juran, J. M. & F. M. Gryna, Jr., "Quality Planning & Analysis," McGraw-Hill, 1980.
4. Reliability Information Analysis Center, "Practical Statistical Tools for the Reliability Engineer," September 1999.
5. http://faculty.vassar.edu/lowry/binomialX.html (online Exact Binomial Probability Calculator)