RELIABLITY: NOT ENOUGH SAMPLES

SOLUTION:Predicting Product Lifetime with Small Sample SizesAllyson HartzellVERYST ENGINEERING

Predicting Product Lifetime with Small Sample SizesHow long will a product last? Virtually all reliability engineers must address this question during product development. Yet prediction of accurate product lifetime can be hampered by limited data. This case study will provide a method for the reliability engineer to predict lifetime with a small sample size. Data from ten samples tested to failure will be compared using two cumulative probability distributions. We will show how the proper lifetime prediction method will eliminate unexpected field failures.Lifetime and Bathtub CurveThe lifetime of a product falls into three failure categories. Figure 1 is the bathtub curve, which depicts the failure over the life of the product. The plot is the instantaneous failure rate versus product operating time. There are three distinct time ranges that exhibit different failure rate behavior in this plot:

1) Early life failure, also called infant mortality (parts with defects will fail early in the product lifetime).

2) Low steady state failure, also called useful life (this defect-free population has a low failure rate; the failures that occur in this timeframe are due to external random events).

3) End of life failure, also called wearout (the population failure rate increases after the useful life as the product intrinsically wears out).

Product operational lifetime is targeted for the useful life time range, as this is the lowest observed failure rate over time.

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Figure 1. The bathtub curve is an instantaneous failure rate curve versus time.

Lifetime DistributionsNow that we understand the instantaneous life curve, how does one pick the proper statistical distribution to predict product lifetime using a cumulative distribution failure (CDF) model? Our answer: start with the most popular distributions for reliability lifetime prediction. These are the Weibull, Lognormal and Exponential, and their definitions are summarized here:

The Weibull distribution is also a continuous probability distribution and was empirically determined to model particle size distribution. Here we work with the Weibull 2-parameter (2P) version.

The Lognormal distribution is a continuous probability distribution of a random variable with a normally distributed logarithm.

The Exponential distribution is a Poisson-based probability distribution that describes the time between events which occur independently at a constant average rate.

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The two parameters in the Weibull 2P distribution are β, the shape parameter, and α, the characteristic life. The characteristic life is the time at which 63.2% of the population has failed. For Weibull distributions, β > 1 is in wearout and β < 1 is termed early life failure (both of these terms are described via the bathtub curve). Figure 2 is the mathematical expression of the CDF (Cumulative Distribution Function) for the Weibull 2P distribution. Figure 3 is the plot of the Weibull 2-parameter CDF with time as a function of 1/α, while varying β.

Figure 2. CDF of Weibull distribution, 2-Parameter.

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Figure 3. Weibull CDF curve as function of 1/α, varying β.

Figure 4 is the mathematical expression of the CDF for the Lognormal distribution, where σ is the standard deviation. The time at which 50% of the population has failed is termed T50.

F ( t )=∫o

T1

σt √2πe(−(ln (t )−ln(T50))

2

2 σ2 )dt

Figure 4. CDF of the Lognormal distribution

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Figure 5. Lognormal CDF curve, varying σ.Use of the 2-parameter Weibull for small sample sizes is the best choice for lifetime prediction. The same set of data will be plotted via both the Lognormal and the Weibull 2P distributions, and this will help us discover why.Small Sample Size Consider the case of having lifetime data for 10 parts that all failed during reliability testing. Why is 10 a small sample size? Because in this case study we follow the advice of reliability pioneer Dr. Bob Abernethy who identified <21 as a small sample size. We will derive the 1% predicted failure rate assuming both a Weibull and Lognormal distribution to illustrate the degree to which the predictions can vary. Targeting 1% failure rate instead of the typical 50% failure rate is more important to the producer as early failure can have deleterious effects on product marketplace acceptance.

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Figure 6. Weibull CDF plot—failure rate versus time with 60% confidence limitsThe r2 fit is for the Weibull prediction in Figure 6 is good at 0.908. This dataset has 60% confidence limits calculated. The 60% confidence limits means that 20% of the time the product will fail earlier than the lower confidence limit (see Table 1). The confidence limits are split percentage-wise around the prediction, which explains why Table 1 has 80% lower confidence limits.The same data are also plotted in a Lognormal CDF plot (Figure 7). The r2 fit of 0.0966 is slightly better than the Weibull. The 60% confidence limits are again plotted.

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Figure 7. Lognormal CDF plot—failure rate versus time with 60% confidence limits

Distribution 1% Failure Rate 80% Lower Confidence Limit at

1% Lognormal 22,398 hours 14,337 hours Weibull 2P 11,050 hours 4812 hours

Table 1. Data summary at 1% failure rate

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Table 1 illustrates how the 2-parameter Weibull prediction can avoid an overly optimistic lifetime prediction. The Weibull predicts that 1% failure will occur 11,000 hours earlier than the Lognormal prediction. The Lognormal 80% lower confidence limit at 1% failure rate is also very optimistic in its prediction versus the Weibull. SummaryTechnologies require lifetime prediction, often with small sample sizes. The prediction of lifetime with limited samples using different predictive methodologies results in a large range of lifetimes. It is also important to target a low failure rate instead of the typical 50% failure rate during prediction. Early population failure is important to the producer as knowledge of good reliability is critical to product introduction. Use of the wrong distribution can result in an overly optimistic prediction and unhappy customers who experience early product failure. This case study highlights that the Weibull 2P prediction is more conservative when compared to the Lognormal distribution.

Allyson HartzellVeryst Engineering, LLC47A Kearney RoadNeedham, MA 02494www.veryst.comahartzell@veryst.com

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781-433-0433 x330

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