Studentization
Encyclopedia
In statistics
, Studentization, named after William Sealy Gosset
, who wrote under the pseudonym
Student, is the adjustment consisting of division of a first-degree statistic derived from a sample, by a sample-based estimate of a population standard deviation
. The term is also used for the standardisation of a higher-degree statistic by another statistic of the same degree: for example, an estimate of the third central moment would be standardised by dividing by the cube of the sample standard deviation.
A simple example is the process of dividing a sample mean by the sample standard deviation when data arise from a location-scale family
. The consequence of "Studentization" is that the complication of treating the probability distribution
of the mean, which depends on both the location and scale parameters, has been reduced to considering a distribution which depends only on the location parameter. However, the fact that a sample standard deviation is used, rather than the unknown population standard deviation, complicates the mathematics of finding the probability distribution
of a Studentized statistic.
In computational statistics
, the idea of using Studentized statistics is of some importance in the development of confidence intervals with improved properties in the context of resampling
and, in particular, bootstrapping
.
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....
, Studentization, named after William Sealy Gosset
William Sealy Gosset
William Sealy Gosset is famous as a statistician, best known by his pen name Student and for his work on Student's t-distribution....
, who wrote under the pseudonym
Pseudonym
A pseudonym is a name that a person assumes for a particular purpose and that differs from his or her original orthonym...
Student, is the adjustment consisting of division of a first-degree statistic derived from a sample, by a sample-based estimate of a population standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
. The term is also used for the standardisation of a higher-degree statistic by another statistic of the same degree: for example, an estimate of the third central moment would be standardised by dividing by the cube of the sample standard deviation.
A simple example is the process of dividing a sample mean by the sample standard deviation when data arise from a location-scale family
Location-scale family
In probability theory, especially as that field is used in statistics, a location-scale family is a family of univariate probability distributions parametrized by a location parameter and a non-negative scale parameter; if X is any random variable whose probability distribution belongs to such a...
. The consequence of "Studentization" is that the complication of treating the 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....
of the mean, which depends on both the location and scale parameters, has been reduced to considering a distribution which depends only on the location parameter. However, the fact that a sample standard deviation is used, rather than the unknown population standard deviation, complicates the mathematics of finding the 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....
of a Studentized statistic.
In computational statistics
Computational statistics
Computational statistics, or statistical computing, is the interface between statistics and computer science. It is the area of computational science specific to the mathematical science of statistics....
, the idea of using Studentized statistics is of some importance in the development of confidence intervals with improved properties in the context of resampling
Resampling (statistics)
In statistics, resampling is any of a variety of methods for doing one of the following:# Estimating the precision of sample statistics by using subsets of available data or drawing randomly with replacement from a set of data points # Exchanging labels on data points when performing significance...
and, in particular, bootstrapping
Bootstrapping (statistics)
In statistics, bootstrapping is a computer-based method for assigning measures of accuracy to sample estimates . This technique allows estimation of the sample distribution of almost any statistic using only very simple methods...
.