**The objective of the tolerance is to identify the limits of production variability that ensure the functional quality of a product.**

## Introduction

The objective of tolerance is to identify limits of production variability to ensure the functional quality of a product, while ensuring the economic reality. As any system generates variability, it is necessary to accept differences in relation to an ideal situation.

Based on a functional specification to guarantee customer satisfaction, the statistical tolerance determines that they are the tolerances on the parts composing my assembly.

Let’s take the example of the base of a connecting rod. We have the following assembly:

Our JC play is our rated condition, necessary for the proper operation of our Assembly. The calculation formula is as follows :

**Play C = C3 – C1 – C4 – C5 – C6**

We know that :

**Game C :**We want a play of 2 ± 0.5 mm**C5 and C6 :**Bolt length = 10 mm**C4 :**width of washer = 3 mm**C1 :**width of the connecting rod = 40 mm**C3 :**length of our threaded stem = 66 mm

## 1 : The traditional * “worst case* ” technique

Our conditional tolerance is equal to the sum of the tolerances of the parts that compose it. We guarantee that in all cases our parts can be assembled. This is ” *Ideal* “.

Taking our example, we get the following result:

We have a tolerance on each of our 5 partss of ± 0.1 mm.

This condition is very good. The problem is that it does not consider that the probability of having 2 parts in their extreme dimension is statistically rare. Very restrictive, it will either cost us expensive to hold it or we generate waste.

## 2 : Statistical tolerance

The statistical tolerance proposes to identify our tolerance limits by studying the probability distribution of the parts of our ensemble.

Under the assumption of independence of the dimension of our parts (we will be able to do a test of **Durbin Watson** To check this condition) and under the fact that we have at least one assembly with 5 parts, the probability of distribution of our set is the sum of the probabilities of distribution of the parts of our subset. By the sum/subtraction, using the Monte Carlo method, we obtain the distribution of our resultant following a **Normal** law :

- The average of our play is equal to the sum/subtract of the averages.
- The variance of our play is equal to the sum of the variances.

**This then amounts to multiplying tolerance to the worst case by √ n** (the demonstration of this theorem is quite complicated, because it uses complex numbers and Fourier analysis. It was made in 1920 by Levy and Lindeberg).

In our example, we have a tolerance on our 5 parts of ± 0.23 mm, more than double the tolerance in the worst case.