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Small Sample Size Lifetime Prediction

Technical Challenge:
How long will a product last? Virtually all reliability engineers must address this question during product development. Yet prediction of accurate product end of life can be hampered by limited data. 

Veryst Solution:
This study provides a method for the reliability engineer to predict end of life with a small sample size. Data from ten samples tested to failure were compared using two cumulative probability distributions. Veryst shows how the proper lifetime prediction method can eliminate unexpected field failures.

Lifetime and Bathtub Curve
The 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.

The bathtub curve is an instantaneous failure rate curve versus time.  It consists of three distinct failure regimes:  early lifetime failure, steady state failure, and end of life failure.

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

Lifetime Distributions
Now that we understand the instantaneous life curve, how does one pick the proper statistical distribution to predict product end of life using a cumulative distribution failure (CDF) model? Veryst’s 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.

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.

The mathematical expression for the 2-parameter Weibull cumulative failure distribution (CDF) is shown.

Figure 2. CDF of Weibull distribution, 2-Parameter  


The graphical depiction of the 2-parameter Weibull cumulative failure distribution (CDF) is shown. 
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 while Figure 5 depicts the CDF graphically. The time at which 50% of the population has failed is termed T50 and σ is the standard deviation.

The mathematical expression for the Lognormal cumulative failure distribution (CDF) is shown.

Figure 4. CDF of the Lognormal distribution

 

The graphical depiction of the Lognormal cumulative failure distribution (CDF) is shown.

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 (prior to end of life) can have deleterious effects on product marketplace acceptance. 

 The Weibull CDF plot with 60% confidence limits is shown for the data set of ten samples.
Figure 6. Weibull CDF plot—failure rate versus time with 60% confidence limits

The r² fit 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.

The Lognormal CDF plot with 60% confidence limits is shown for the data set of ten samples. 
Figure 7. Lognormal CDF plot—failure rate versus time with 60% confidence limits

 

Distribution

Lognormal

Weibull 2P

1% Failure Rate

22,398 hours

11,050 hours

80% Lower Confidence Limit at 1%

14,337 hours

4812 hours

 

Table 1. Data summary at 1% failure rate

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.  

Summary
Technologies require lifetime prediction, often with small sample sizes. The prediction of end of life 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 during operation. This case study highlights that the Weibull 2P prediction is more conservative when compared to the Lognormal distribution when predicting end of life.

 

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