Statistical interference

When two probability distributions overlap, statistical interference exists. Knowledge of the distributions can be used to determine the likelihood that one parameter exceeds another, and by how much.

This technique can be used for dimensioning of mechanical parts, determining when an applied load exceeds the strength of a structure, and in many other situations. This type of analysis can also be used to estimate the probability of failure or the frequency of failure.

Dimensional interference

Interference of measurement distributions to determine fit of parts

When two mechanical parts are designed to fit together, traditional tolerances may suggest to some that all actual dimensions fall within those tolerances. A process capability study, however, may show normal distributions with long tails. For example, it there is a shaft that must have a sliding fit in a hole, both the shaft and hole may have measurements which have distributions with an average (mean) and a standard deviation.

With two such normal distributions, a distribution of interference can be calculated. The new distribution would also be normal and have a average (mean): the difference between the means of the two base distributions. The variance would be the sum of the variances of the two base distributions. This distribution can be used to determine when the difference in dimensions is less than zero (the shaft cannot fit in the hole) or when the difference is less than the required sliding gap needed.

Physical property interference

Interference of distributions of applied load and strength

Physical properties and the conditions of use also have inherent variability. For example, the applied load (stress) on a mechanical part may have some variation. The measured strength of that part (tensile strength, etc) may also have variation. The part will break when the stress exceeds the strength.

With two normal distributions, the interference may be calculated as above. (This also is workable for transformed units such as the Log-normal distribution). With other distributions or combinations of different distributions, a Monte Carlo method or simulation is often best.

See also

• Tolerance (engineering)
• Specification
• Process capability
• Normal distribution
• Probabilistic design

References

• Paul H. Garthwaite, Byron Jones, Ian T. Jolliffe ‘’Statistical Inference’’, 2002, ISBN: 0198572263
• Haugen, "Probabilistic mechanical design", (1980) Wiley. ISBN 0471058475