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

## Statistical Tolerance Analysis

Statistical adding is always preferred. It gives good results and does not lead to unnecessarily high demands on tolerances. But even more refinement is possible. So that your tolerance analysis becomes an even better reflection of the actual produced assemblies. In short: you make a distinction between statistical contributions and worst-case contributions.

## Worst-case Contributions in Tolerance Analysis

Suppose, for example, that there is a moving part in your tolerance chain. Consider, for example, the position of a robot arm or a carriage that moves over a guide. It is also possible that there are different machine states. For example, a machine that comes back into operation after some inactivity. If the machine temperature rises, the expansion of parts may affect your tolerance chain.

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

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

If you make the distinction 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 looks as follows:

Ttot = W1 + W2 +… Wn + √(T12 + T22 +… + Tn2)

Where Wi are all the contributions that you add worst-case and Ti are all the contributions that you add statistically.

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

Maybe it gets complicated now. Which contributions in your tolerance chain should you add statistically and which one worst case? Can’t you bend the result too much to your liking and get ‘every’ outcome you want? Of course you can strongly influence the result by your choice. But you want the most accurate reflection of reality. So here’s a rule of thumb for the choice between statistic and worst-case:

If the contribution in your chain (almost) always occurs in its extreme values, then you add that item worst-case. Because they don’t average out. Think of:

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

The 3rd item ‘handling many products’ needs some explanation. Suppose you want to analyze tolerance chain in a production process and the size of a product is part in your chain. Of course, not one product passes through the production process, but many. The dimensions of the products will probably be normally distributed. But you can’t include the size as a static contribution in your tolerance chain! You also want the products with the greatest deviation to pass through the production process. This means that you take into account the largest deviation and thus allow it to contribute as a worst-case item in your tolerance chain.
The latter two, poorly managed processes and play, are not necessarily worst-case but certainly not normally distributed. You could also choose to add these random (as a uniform distribution). If so, you apply a correction factor √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 calculation of the statistical addition is given at the end of the article ‘Statistical Tolerance Stack-up Analysis‘. The result, with all items added statistically, is +/-0.62 mm. Now suppose you decide that item 2 should be a worst-case contribution. If you make that adjustment, in the same table, then the outcome has increased by 0.17 mm to +/-0.79 mm. See calculation below. If the decision that item 2 should be a worst-case contribution is right, then your tolerance analysis has now become more accurate. And better reflects the practice.

If you assign the individual contributions as good as possible, your stack-up analysis will be a better reflection of the real produced assemblies.

## Increasingly Used Method

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