In previous articles about tolerance analysis, I discussed the step-by-step plan and how to add up tolerances. The sum of tolerances can be a worst-case sum or a statistical sum. In ‘The Distribution of Tolerances‘ I also discussed the types of distributions that can exist: a normal distribution or not. But there is an even more advanced method of tolerance analysis, a combination of the worst-case and statistical methods.

## Statistical Tolerance Analysis

Statistical summation is **always preferred**. It gives good results and does not lead to unnecessarily high tolerance requirements. But even more refinement is possible. To make your tolerance analysis a better reflection of the actual assemblies being produced. In short, you **distinguish between** statistical contributions and worst-case contributions.

## Worst-case Contributions in Tolerance Analysis

For **example**, suppose there is a **moving part** in your tolerance chain. Consider the position of a robot arm or a carriage moving over a guide. It is also possible that there are different **machine states**. For instance, a machine that comes back into operation after a period of inactivity. As the machine temperature rises, the **expansion** of parts can affect your chain of tolerance.

All these and similar cases have in common that the **extremes** (amplitude of movement, difference between hot and cold state, etc.) **will actually always** occur. You cannot add them statistically because there is **no averaging**. You have to add these contributions in your worst case tolerance chain!

## A Better Tolerance Analysis, Combining Worst-Case and Statistics

When you distinguish between worst-case contributions and statistical contributions, you get a more accurate tolerance analysis. Your table will contain contributions that you add statistically and contributions that you add worst case. The** general formula** for calculating the sum is as follows:

T_{tot} = W_{1} + W_{2} +… W_{n} + √(T_{1}^{2} + T_{2}^{2} +… + T_{n}^{2})

Where W_{i} are all the worst-case contributions you add and T_{i} are all the contributions you add statistically.

## How to Assign a Worst-Case or Statistical Contribution

Maybe this is where it gets complicated. Which contributions in your tolerance chain **should you statistically add**, and **which are worst-case**? Can’t you bend the result too much to your liking and **get “any” result you want**? Of course, you can strongly influence the result by your choices. But you want the **most accurate reflection** of reality. So here’s a rule of thumb for choosing between statistics and worst case:

*If the contribution in your chain (almost) always occurs in its extreme values, then add that item as a worst case. Because they don’t average out. Some examples worst-case contributions are*:

- vibrations, movements;
- expansions and shrinkage;
- handling of many products;
- wear;
- poorly managed processes;
- play.

The 3rd point, ‘handling many products’, needs some explanation. Suppose you want to analyze a tolerance chain **in a production process**, and the size of a product is a part of your chain. Consider a robot that is constantly moving sheets of glass, and the thickness of the glass is part of the tolerance chain. Of course, not just one product (glass plate) goes through the production process, but many day in day out. The dimensions of the products (glass plates) are likely to be normally distributed. But you **can’t include the size as a statistical contribution** in your tolerance chain! Because you also want the products with the largest deviation to pass through the production process (instead of rejecting them). This means that you include the largest variation and allow it to contribute as a worst-case item in your tolerance chain.

The last two points, poorly managed processes and randomness, are not necessarily worst-case, but they are certainly not normally distributed. An alternative could be to add them as a uniform distribution (a.k.a. randomly). If so, apply a correction factor of √3 to the statistical contribution. See also the article ‘The Distribution of Tolerances‘. Contributions that are (reasonably well) normally distributed **do average out** and you add them statistically.

## An Example with Numbers

An example of the statistical addition calculation is given at the end of the ‘Statistical Tolerance Stack-up Analysis‘ article. The result, with all items statistically added, is +/-0.62 mm. Now suppose you decide that Item 2 should be a worst-case contribution. When you make this adjustment in the same spreadsheet, the result **has increased** by 0.17 mm to +/-0.79 mm. See the calculation below. If the decision that Item 2 should be a worst-case contribution is correct, then your tolerance analysis **is now more accurate**. And better reflects the real world.

Your stack-up analysis **will better reflect** the actual produced assemblies if you allocate each contribution as well as possible..

## Increasingly Used Method

This sophisticated and refined method is used by more and more companies and is for example the standard way of working with ASML. It is also part of Mikrocentrum’s tolerance analysis course. And it is available by default in the TolStackUp Excel spreadsheet template offered on this site.