*This blog is translated from German with DeepL.*

###### 1. Why are tolerances needed?

While a CAD model represents the ideal and perfect geometry of a part, in reality there are disturbing factors that influence the effective part geometry. One of the most important influencing factors is the manufacturing process as such, but also environmental influences such as temperature and humidity variations have an impact on our part geometry.

The following deviations from the ideal geometry can occur in a production part:

- Dimensional deviations
- Shape deviations
- Surface deviations

Tolerances or tolerance ranges are used to define the permissible deviations, i.e. a tolerance range, in order to ensure fault-free function and assembly at all times.

###### 2 Arithmetic tolerance analysis (worst case)

Arithmetic tolerance analysis, also known as the extreme value method, minimum-maximum principle or worst case analysis, is probably the most widely used and most frequently applied method for tolerance analysis.

The main advantage of this method is certainly the 100% interchangeability of components, which is guaranteed at all times. However, long dimensional chains result in a large tolerance field of the closing dimension (Z) due to the summation of the individual tolerances. This could, for example, jeopardize the fault-free function of the assembly. To avoid this, the tolerance zones of the individual dimensions would have to be reduced, which in turn would result in higher manufacturing costs. Mostly, of course, the “both/and” is strived for. In this context, this means keeping production costs as low as possible while still guaranteeing perfect function. For this reason, other methods of tolerance analysis are being used more and more frequently, especially for systems that are manufactured in large quantities.

###### 3 Statistical tolerance analysis

Statistical methods for dimensional tolerance analysis are based on probability calculation, as well as the following assumptions:

- During production, the center of the tolerance field is always aimed at
- The manufacturing process is stable
- The probability of exclusively assembling boundary-layered components together in an assembly is low (and decreases further as the number of components increases)

In contrast to arithmetic tolerance analysis, only partial interchangeability of components is guaranteed here, while a low percentage of unfavorable cases (rejects) can occur. However, the individual dimensions of a dimensional chain can be provided with larger tolerance zones, which leads to a reduction in production costs.

In the case of a tolerance chain of at least four or more individual dimensions (rule of thumb), a statistical consideration can be useful.

###### 3.1 Distributions and process capability

So-called distributions play an important role in connection with the statistical consideration of a dimensional or tolerance chain. With the help of the density function of a distribution, for example, a (stable) manufacturing process can be described mathematically.

The table below shows some distributions with density function and their application:

In a stable manufacturing process, the dispersion of the individual mass can be described in most cases by the normal distribution. This distribution is described by the density function of the Gaussian bell curve.

The shape of this curve is determined by the following two parameters:

- Expected value µ (mean value) -> Determines the position of the maximum.
- Variance σ2 or standard deviation σ (sigma) -> Measure for the dispersion (width of the curve).

The following table shows the process yield (good parts) for a normally distributed manufacturing process depending on the position of the limiting mass:

###### 3.2 Root Sum Squares (RSS) Method

This method of statistical tolerance analysis is probably the most widely used. It assumes that all part offsets are produced according to the 3σ process quality. Linear tolerance chains can be statistically analyzed in this way (with usefully at least four or more individual masses).

Calculation of the standard deviation σ for single masses:

Expected value µ (mean value) of the closing measure (Z):

Standard deviation σ of the closing dimension (Z):

(Attention! Observe the directional dependence of the signs).

Results for the closing dimension:

With a probability of 99.74 %, the closing dimension (Z) will be within the tolerance range of ±3σZ.

###### 4. Conclusion

Of course, there are many other more accurate, but usually more complex, methods for tolerance analysis. As is certainly the case in many other companies, we still use arithmetic tolerance calculation for the most part. Nevertheless, in some cases the RSS method offers a good alternative for estimating the probability of occurrence of the “worst case”. Which method is ultimately used for tolerance analysis varies, of course, from situation to situation. Is it a single device or a series product that is to be manufactured in quantities of millions? Our developers and designers are challenged every day in the interest of our customers to select and apply the appropriate method in each case.