In previous articles about tolerance analysis, I discussed the step-by-step plan and the way of adding up. The latter 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.

Add Statistically

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 practice.

Worst-case Contributions

Now 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 Analysis

If you make that distinction, you get a more accurate tolerance analysis: a table divided into contributions that you add statistically and contributions that you add worst case. The general formula for this 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.

What to Choose

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 this way 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;
  • wear;
  • poorly managed processes;
  • play.

The latter two 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 calculation of 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. If you now add one item worst-case, for example item 2, the result is +/-0.79 mm. See the calculation below.

Increasingly Used

This sophisticated 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.