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.

Contents of this article

## 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.

## 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.