When two
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s 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
Mechanical parts are usually designed to fit precisely together. For example, if a shaft is designed to have a "sliding fit" in a hole, the shaft must be a little smaller than the hole. (Traditional
tolerances
Engineering tolerance is the permissible limit or limits of variation in:
# a physical dimension;
# a measured value or physical property of a material, manufacturing, manufactured object, system, or service;
# other measured values (such as t ...
may suggest that all dimensions fall within those intended tolerances. A
process capability
The process capability is a measurable property of a process to the specification, expressed as a process capability index (e.g., Cpk or Cpm) or as a process performance index (e.g., Ppk or Ppm). The output of this measurement is often illustrated ...
study of actual production, however, may reveal
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
s with long tails.) Both the shaft and hole sizes will usually form normal distributions with some average (
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
) and
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
.
With two such normal distributions, a distribution of interference can be calculated. The derived distribution will also be normal, and its average will be equal to the difference between the means of the two base distributions. The
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of the derived distribution will be the sum of the variances of the two base distributions.
This derived distribution can be used to determine how often the difference in dimensions will be less than zero (i.e., the shaft cannot fit in the hole), how often the difference will be less than the required sliding gap (the shaft fits, but too tightly), and how often the difference will be greater than the maximum acceptable gap (the shaft fits, but not tightly enough).
Physical property interference
Physical properties and the conditions of use are also inherently variable. For example, the applied load (stress) on a mechanical part may vary. The measured strength of that part (tensile strength, etc.) may also be variable. The part will break when the stress exceeds the strength.
With two normal distributions, the statistical interference may be calculated as above. (This problem is also workable for transformed units such as the
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
). With other distributions, or combinations of different distributions, a
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
or simulation is often the most practical way to quantify the effects of statistical interference.
See also
*
Interference fit
An interference fit, also known as a pressed fit or friction fit is a form of fastening between two ''tight'' fitting mating parts that produces a joint which is held together by friction after the parts are pushed together.
Depending on the am ...
*
Joint probability distribution
Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
*
Probabilistic design
Probabilistic design is a discipline within engineering design. It deals primarily with the consideration of the effects of random variability upon the performance of an engineering system during the design phase. Typically, these effects are rela ...
*
Process capability
The process capability is a measurable property of a process to the specification, expressed as a process capability index (e.g., Cpk or Cpm) or as a process performance index (e.g., Ppk or Ppm). The output of this measurement is often illustrated ...
*
Reliability engineering
Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability describes the ability of a system or component to function under stated conditions for a specifie ...
*
Specification
A specification often refers to a set of documented requirements to be satisfied by a material, design, product, or service. A specification is often a type of technical standard.
There are different types of technical or engineering specificati ...
*
Tolerance (engineering)
Engineering tolerance is the permissible limit or limits of variation in:
# a physical dimension;
# a measured value or physical property of a material, manufactured object, system, or service;
# other measured values (such as temperature, hum ...
References
*Paul H. Garthwaite, Byron Jones, Ian T. Jolliffe (2002) ''Statistical Inference''.
*Haugen, (1980) ''Probabilistic mechanical design'', Wiley.
{{DEFAULTSORT:Statistical Interference
Statistical theory
Survival analysis
Reliability engineering
Probability theory
Applied probability