Parametric statistics

# Parametric statistics

Discussion

Encyclopedia
Parametric statistics is a branch of statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

that assumes that the data has come from a type of probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

and makes inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...

s about the parameters of the distribution. Most well-known elementary statistical methods are parametric.

Generally speaking parametric methods make more assumptions than non-parametric methods
Non-parametric statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...

. If those extra assumptions are correct, parametric methods can produce more accurate and precise estimates. They are said to have more statistical power
Statistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...

. However, if those assumptions are incorrect, parametric methods can be very misleading. For that reason they are often not considered robust
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

. On the other hand, parametric formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

e are often simpler to write down and faster to compute. In some, but definitely not all cases, their simplicity makes up for their non-robustness
Robust statistics
Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

, especially if care is taken to examine diagnostic statistics.

Because parametric statistics require a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

, they are not distribution-free
Non-parametric statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...

.

## Example

Suppose we have a sample of 99 test scores with a mean of 100 and a standard deviation of 10. If we assume all 99 test scores are random samples from a normal distribution we predict there is a 1% chance that the 100th test score will be higher than 123.65 (that is the mean plus 2.365 standard deviations) assuming that the 100th test score comes from the same distribution as the others. The normal family of distributions all have the same shape and are parameterized by mean and standard deviation. That means if you know the mean and standard deviation, and that the distribution is normal, you know the probability of any future observation. Parametric statistical methods are used to compute the 2.365 value above, given 99 independent observations from the same normal distribution.

A non-parametric
Non-parametric statistics
In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...

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 is higher than any of the 99 that preceded it.

## History

Statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...

Jacob Wolfowitz
Jacob Wolfowitz
Jacob Wolfowitz was a Polish-born American statistician and Shannon Award-winning information theorist. He was the father of former Deputy Secretary of Defense and World Bank Group President Paul Wolfowitz....

coined the statistical term "parametric" in order to define its opposite in 1942:

"Most of these developments have this feature in common, that the distribution functions of the various stochastic variables which enter into their problems are assumed to be of known functional form, and the theories of estimation and of testing hypotheses are theories of estimation of and of testing hypotheses about, one or more parameters. . ., the knowledge of which would completely determine the various distribution functions involved. We shall refer to this situation. . .as the parametric case, and denote the opposite case, where the functional forms of the distributions are unknown, as the non-parametric case."