Parametric statistics is a branch of statistics which assumes that sample data comes from a population that can be adequately modeled by a
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 ...
that has a fixed set of
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
.
Conversely a non-parametric model does not assume an explicit (finite-parametric) mathematical form for the distribution when modeling the data. However, it may make some assumptions about that distribution, such as continuity or symmetry.
Most well-known statistical methods are parametric.
Regarding nonparametric (and semiparametric) models,
Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".
Example
The
normal family of distributions all have the same general shape and are ''parameterized'' by
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
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 ...
. That means that if the mean and standard deviation are known and if the distribution is normal, the probability of any future observation lying in a given range is known.
Suppose that we have a sample of 99 test scores with a mean of 100 and a standard deviation of 1. If we assume all 99 test scores are random observations from a normal distribution, then we predict there is a 1% chance that the 100th test score will be higher than 102.33 (that is, the mean plus 2.33 standard deviations), assuming that the 100th test score comes from the same distribution as the others. Parametric statistical methods are used to compute the 2.33 value above, given 99
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
observations from the same normal distribution.
A
non-parametric
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distri ...
estimate of the same thing is the maximum of the first 99 scores. We don't need to assume anything about the distribution of test scores to reason that before we gave the test it was equally likely that the highest score would be any of the first 100. Thus there is a 1% chance that the 100th score is higher than any of the 99 that preceded it.
History
Parametric statistics was mentioned by
R. A. Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who ...
in his work ''
Statistical Methods for Research Workers
''Statistical Methods for Research Workers'' is a classic book on statistics, written by the statistician R. A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods, together with his ''The ...
'' in 1925, which created the foundation for modern statistics.
See also
*
Aggregated distribution
An aggregated distribution, commonly found among predators and parasites, is a highly uneven (skewed) statistical distribution pattern in which they collect or aggregate in regions, which may be widely separated, where their prey or hosts are at ...
*
All models are wrong
All or ALL may refer to:
Language
* All, an indefinite pronoun in English
* All, one of the English determiners
* Allar language (ISO 639-3 code)
* Allative case (abbreviated ALL)
Music
* All (band), an American punk rock band
* ''All'' (All al ...
*
Inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
*
Parametric model
In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.
Def ...
*
Sufficient statistic
In statistics, a statistic is ''sufficient'' with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the pa ...
References
{{DEFAULTSORT:Parametric Statistics
Statistical inference