# How Are Manufacture Specifications Distributed: Gaussian or Uniform

Recently, I encountered several individuals (technical experts) who believed that a manufacturer’s published accuracy specification is characterized by a uniform or rectangular probability distribution. For several years, I have observed the opinions of others who seem to share the same consensus. However, I have never accepted this assumption; so, I thought I would share my perspective.

Based on my observations and experience, the majority of the time, a manufacturer’s accuracy specification is actually characterized by a Gaussian distribution. Why do I believe this to be true, and how could so many people have it wrong?

Well, let’s think about it; what is a manufacturer’s accuracy specification? A quantified boundary which the majority of manufactured items are expected to conform to? First, all items can not conform; it is not practically possible. Second, each item tested for conformance will not perform equally; the group’s behavior will have variability.

Assuming that the manufacturer’s accuracy specification is uniformly distributed, we expect that all outcomes are equally probable. If we test 100 items for conformance, where will they lie in the probability distribution and will all outcomes be equally probable? I, do not believe so.

Therefore, let’s evaluate two realistic scenarios;

Example 1:
A manufacturer’s accuracy specification is developed based on the combination of various error sources (Error Analysis). In accordance with the Central Limit Theorem, the convolution of the independent probability densities will result in a Gaussian distribution that exhibits central tendency.

Example 2:
A manufacture produces 100 similar items, anticipating that 95% of the yield will successfully perform within the established specifications. The behavior of the group will most likely resemble a Gaussian distribution with central tendency, where 68.27% of the yield will be distributed 1-sigma about the mean and 27.18% will be distributed 2-sigma about the mean (68.27% + 27.18% = 95.45%). The remaining 4.55% will lie outside the 2-sigma range.

The two examples exhibit realistic scenarios where the use of the Gaussian distribution to describe a manufacturer’s accuracy specification is practical and probable.

What do you think? Share your opinion.

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### Richard Hogan

Richard Hogan is the CEO of ISO Budgets, L.L.C., a U.S.-based consulting and data analysis firm. Services include measurement consulting, data analysis, uncertainty budgets, and control charts. Richard is a systems engineer who has laboratory management and quality control experience in the Metrology industry. He specializes in uncertainty analysis, industrial statistics, and process optimization. Richard holds a Masters degree in Engineering from Old Dominion University in Norfolk, VA. Connect with Richard on LinkedIn.