The Concept of MTTF Used in Arrhenius: The Role of MTTF in Reliability Testing #31

In reliability testing and material evaluation, the Arrhenius model is a very commonly used analysis method for accelerated testing. To predict lifespan, one must first define "
what point is considered a failure" and then statistically process those failure times.

If failure times can be defined for each temperature condition, you can extrapolate to room temperature environments to estimate "how long it will take to break in the market." In this article, we will organize
MTTF (Mean Time To Failure), which is a representative value, from a practical perspective.

1. MTTF and MTBF: Why is MTTF necessary?

There are two very similar indicators in reliability engineering.

• MTTF (Mean Time To Failure)
  The average time until a non-repairable product (semiconductor, LED, etc.) fails for the first time.

• MTBF (Mean Time Between Failures)
  The average interval for a repairable system from one failure to the next.

Since the subjects for which lifespan is predicted using the Arrhenius model are generally non-repairable parts, the indicator that should be used is MTTF. We calculate the MTTF at each temperature in accelerated testing and extrapolate it to estimate the "lifespan at room temperature."

2. Definition of MTTF

MTTF (Mean Time To Failure) is defined as the expected value (mean value) of the time until a system or part fails for the first time.

If failure time T is a random variable, MTTF is expressed as follows.

$$
MTTF = E[T]
$$

$$
MTTF = \int_{0}^{\infty} t f(t), dt
$$

Here, f(t) is the probability density function of failure time. In other words, MTTF is not just "average failure time," but has a
strict definition based on probability theory.

In practice, it is treated as the "average of failure times of multiple samples," but its reliability depends heavily on the number of samples.

3. Sample size and MTTF stability

Although MTTF is practically treated as the "average value of failure times of multiple samples," its reliability depends strongly on the number of samples N.

(1) When the number of samples is large (N ≥ 11)

• Because the central limit theorem applies, the sample mean is likely to approach the population mean (the true MTTF).

• If fitted to a Weibull distribution or log-normal distribution, highly accurate MTTF estimation including confidence intervals is possible.

• It is also useful for understanding degradation modes.

(2) When the sample size is small (N < 11)

• Statistical stability is insufficient, making it difficult to estimate distribution parameters.

• Since only a few samples are available in the early stages of R&D, it is practical to average the non-outlier data, assuming they all failed in the same way, to obtain a provisional MTTF.

• In this case, it is important to explicitly state that it is a "provisional value."

Supplementary Note

The guideline of N=11 shown here is merely an indicator based on standards and empirical knowledge. In reality, the required sample size varies depending on organizational policy and the purpose of the test. Since the handling also differs depending on whether it is the evaluation stage or the mass production stage, it is important to check the scope of application before use.

4. Why is "N=11" a guideline?

The criterion of "11 or more" is not just a rule of thumb; it has international standards and statistical backing.

• JEDEC (Semiconductors/Electronic Components)
  In JESD22-A108 (High Temperature Operating Life, HTOL), 22 to 77 units is standard. In the evaluation stage, there are cases where
  about 11 units are used as a reduced version.

• IEC (Batteries/Environmental Testing)
  In IEC 62660 (Secondary batteries for vehicles), 10 cells or more is required. In the IEC 60068 series (Environmental testing), "10 or more" is also the minimum line.
  

In other words, around 11 samples is positioned as the "minimum line capable of withstanding distribution estimation" both in terms of standards and experience.

5. Practical Compromise

There is a gap between the ideal and reality in calculating MTTF for Arrhenius analysis.

• Ideal: Ensure N ≥ 11 and present distribution estimation and confidence intervals. The more, the better.

• Reality (Research Stage): Since samples are limited, use the average excluding outliers as the provisional MTTF.

As a researcher, you are required to understand this "ideal vs. reality" and present data with appropriate caveats.

6. Summary

• MTTF is used as the representative value for lifetime in Arrhenius analysis.

• If N ≥ 11, distribution approximation and calculation of confidence intervals are possible.

• N < 11, the average should be treated as a provisional value, and its limitations must be clearly stated.

• JEDEC and IEC standards also set '10 to 22 samples' as the minimum requirement.

Treating 'ideals' and 'reality' separately is a practical approach for researchers and engineers.

References (Standards/Literature)

• JEDEC Standard JESD22-A108: High Temperature Operating Life (HTOL)

• IEC 62660-2: Secondary lithium-ion cells for the propulsion of electric road vehicles – Part 2: Reliability and abuse testing

• IEC 60068-2: Environmental testing – Basic environmental test procedures

Reference Articles

• Arrhenius Model Memorandum | How to Interpret Failure Modes and Activation Energy