Statistical Tolerance Intervals

Explained in a simple and practical way based on ISO 16269-6

Statistical tolerance intervals are frequently used during process validation and design verification, maybe because the approach is straightforward and useful. The approach of statistical tolerance intervals assumes normally distributed data. However, assuming it is not enough, we have to prove it, but slowly let's do it step-by-step.

This post answers the following questions:

• What are Statistical Tolerance Intervals?
• Why is it useful?
• Proofing Normal Distribution
• How to apply it?

What are Statistical Tolerance Intervals?

The standard ISO 16269-6 writes about the determination of statistical tolerance intervals and explains it as follows:

?"A statistical tolerance interval is an estimated interval, based on a sample, which can be asserted with confidence level 1 ? α, for example, 0,95, to contain at least a specified proportion p of the items in the population."

Well, that explains it very well if you already know what this is all about. So, let’s take a few steps back.?Figure?2 (left)?shows a normal distribution curve. This curve can is characterized by two values, the mean?m, and the standard deviation?s. Both values can be calculated relatively quickly and usually are, but no further calculation is often made with these values. Assuming a single dimension would be measured on every product in a batch. We would observe that this dimension will most likely be normally distributed, meaning most values will be close to the mean?and others are lower or higher.

You might already know that some 68,3% of these values fall within?±?1s; this correlates to the green shaded area in?Figure?1. Quick reminder: 1s?is at the inflection point of the curve, the point where you would have to turn the steering wheel in the other direction if you were driving along the curve. Based on the mean?m?and the standard deviation?s, one can calculate how many products of a batch (population) fall within a specific interval.

Why is it useful?

In the medical device industry, we deal with risk all the time. Our entire industry is risk-based, and even ISO 13485, the standard for quality management systems, wants us to use risk-based approaches (ISO 13485:2016, 4.1.2, b). Risk management is also built on probabilities; therefore, we say how likely each individual risk is or not and assign it an occurrence. A sequence of events (each with its own probability) can lead to harm, i.e., death, burn, discomfort. A couple of these events originate in production, and we must know how frequently possible problems occur. Therefore, knowing the mean?m?and standard deviation?s?of a critical measure and the associated probability that the measure is nonconforming is literally a lifesaver.

Proofing Normal Distribution (or failing to do so)

Unfortunately, the concept of statistical tolerance intervals only works for normally distributed data, which means we must prove it before doing further calculations. Due to the multiple ways of proving the normal distribution, we will not look at it in detail (Check out various statistic software packages such as Minitab, R, JMP, SPSS, PSPP, MATLAB, etc.). We will talk about the situation when we must reject the null hypothesis (H0?means data is normally distributed), i.e., the p-value is less than the chosen alpha level (1%, 5%, or 10%), and the data fails the normality test. Common reasons include [1]:

• The underlying distribution is not normal
• Outliers or mixed distributions are present
• A low discriminant measure is used (“Resolution”)
• Skewness is present in the data
• Large sample size (ideally: 15 to 100 values [2])

A failed normality test is not necessarily a big issue. We just need to do some more math and address the violated assumption. Common ways to address it are [2]:

• Show outlier is a measurement error
• Determine if high capability acceptance criteria are applicable
• Acceptance criteria are met, and data passes the skewness-kurtosis specific normality test
• Acceptance criteria are met, and other batches routinely pass normality tests
• Use an attribute sampling plan instead

Let’s take a closer look at two common ways to address it; higher capability acceptance criteria and skewness-kurtosis-specific normality test.

A sufficiently high capability does not need the data to be normally distributed; it already proves with high confidence that the product is conforming [2]. What is sufficiently high? Based on the sample size and the RQL (rejectable quality level), a multiplier for Ppk?and Pp?is used to calculate the new acceptance criteria.

The skewness-kurtosis-specific normality test will reject data that have tails towards a specification limit that are longer than a normal distribution.?Figure?3?shows a positive skewed distribution, which would be better if the specification only has a lower limit.??Figure?3?also shows that the mean?m?is no longer the "highest point" on the curve. Skewed curves need to be assessed based on average (mean), median, and mode. The order of these values defines whether the distribution is positive or negative skewed, i.e., positive skews have a higher mean than mode.

In contrast, negative skews have a lower mean than mode. The median is always between the two. If the data is normally distributed, these three values are consistent with each other.

How to apply the concept of statistical tolerance intervals

The application of statistical tolerance intervals is straightforward. Once we have the mean?m, the standard deviation?s, and proved normality, we only need to know one formula:

xL, U=m±k*s?(E.1)

(- for lower and + for upper specification limit)

The test passes if:

• xL greater or equal SL?(if there is only a lower specification limit)
• xU smaller or equal SU?(if there is only an upper specification limit)
• xL greater or equal SL?and xU smaller or equal SU?(if there is a lower and an upper specification limit)

Let us consider an example with which most of us are familiar. Seal strength of a sterile barrier system with an assumed confidence and probability level of 95% each. NOTE: some notified bodies want to see 95%/99%; 95%/95% has only been chosen for the sake of this example; an organization has to assess their needs in accordance to their intended use and risk management.?

Only a lower specification limit at 1.2N/15mm is given (SL?in?Figure?4). The mean of the normally distributed sample is 10.28N/15mm, and the standard deviation is 0.76N/15mm. Using ISO 16269-6, table C.2 (one-sided) and a sample size of 30 gives us a k-value of 2.2199. The lower limit of the statistical tolerance interval (xL) was calculated using E.1. Since xL?is greater than SL, the test passes. We can conclude with 95% confidence that 95% of the population (green shaded area in?Figure?4) is greater than xL. In the case of SL=xL, not more than 95% of the population is conforming. However, the greater the distance between xL?and SL, the higher the proportion of the population conforming - if SL<xL?- of course.

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References

[1] https://blog.minitab.com/en/understanding-statistics-and-its-application/what-should-i-do-if-my-data-is-not-normal-v2

[2] Taylor, Wayne (2017). Statistical Procedures for the Medical Device Industry. Taylor Enterprises, Inc., www.variation.com