SOLUTION EXISTS!

1. RELIABLITY:
NOT
ENOUGH
SAMPLES
SOLUTION:
Predicting
Product
Lifetime
with Small
Sample
Sizes
Allyson Hartzell
VERYST ENGINEERING

2. 1
Predicting Product Lifetime with Small Sample Sizes
How 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 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.

3. 2
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 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.

4. 3
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.

5. 4
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.
dte
t
tF
T
o
Tt








 


2
2
50
2
))ln()(ln(
2
1
)(


Figure 4. CDF of the Lognormal distribution

6. 5
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.

7. 6
Figure 6. Weibull CDF plot—failure rate versus time with 60% confidence limits
The 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.

8. 7
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

9. 8
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 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 Hartzell
Veryst Engineering, LLC
47A Kearney Road
Needham, MA 02494
www.veryst.com
ahartzell@veryst.com
781-433-0433 x330