In a previous article the statistical tolerance stack-up analysis is discussed. It is usually profitable to do a statistical tolerance stack-up analysis instead of a worst-case analysis. A worst-case analysis often leads to low tolerance specifications and therefore high(er) costs.
Gaussian Distribution or Not?
In a statistical tolerance stack-up analysis, a Gaussian distribution of tolerances is often assumed. But is this a realistic assumption? That depends on the actual variation and the used tolerances. The question that needs to be answered is: is the process to realize the dimensional tolerance specification, balanced (in control, perfectly adapted to the required accuracy) or not. There are basically three possibilities.
- Yes, it is a well balanced process step. The Gaussian distribution is an adequate assumption.
- No, the process step is not balanced and the tolerance specification is much wider (or narrower) than the capability of the actual process. An example: the tolerance specification is +/-0.5 mm while it is a simple milling process with an actual variation of +/-0.1 mm.
- No, the process step is not balanced and the required accuracy is quite difficult to realize. The Gaussian distribution is too optimistic.
Normal Process with Gaussian Distribution
Process with Narrow Gaussian Distribution
Distribution of an Uncontrolled Process
What to Choose, Statistical or Something Else?
In case 1 one has also to realize that the average value of the dimension can be off centre with respect to the required dimension. Also, the actual distribution is probably not 100% perfect Gaussian. All these deviations can be corrected for. Many companies use a correction factor of 1.5x. This is called Benderizing. It is not too scientific but it works wonderfully well.
In case 2 you can still work with the Gaussian distribution but the analysis will give you a too pessimistic (or too optimistic) result. Maybe your analysis shows you that non-fitting assemblies will occur from time to time. But in practice this never happens because the actual variation is much lower than in your analysis. A solution can be:
- Adapt the tolerance specification(s) on drawings;
- Use a correction factor for the difference between tolerance specification and the expected variation. Explain this in your analysis.
In case 3 you can try to guess the distribution of the realized tolerances. Than use a correction factor for the differences with a Gaussian distribution. Examples of distributions are:
- uniform (random, 1.73x);
- triangular (1.225x);
- trapezoid (1.369x);
- elliptical (1.5x).
Correction Factor Corrects for Non-Gaussian Distribution
For each distribution type a correction factor can be used. For the uniform distribution this factor is exactly √3. In the article ‘Tolerance Stack Analysis Methods‘ of Fritz Scholtz, the theoretical background of tolerance stack-up methods is described and gives correction factors for many distributions.
It will be clear, as long as you don’t have knowledge about the actual tolerance distribution, you will have to make a (educated) guess. In many cases you’ll end up using a correction factor.
Field of tolerance analysis
Tolerance analysis is one of the fields of expertise of Jaap Vink of Vink System Design & Analysis. I have done several (precision) mechanical analyses in this field. As a guest lecturer for Mikrocentrum I have given a three-day course on tolerance analysis for several years.
