In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Studentization, named after
William Sealy Gosset
William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician, chemist and brewer who served as Head Brewer of Guinness and Head Experimental Brewer of Guinness and was a pioneer of modern statistics. He pioneered small s ...
, who wrote under the
pseudonym
A pseudonym (; ) or alias () is a fictitious name that a person or group assumes for a particular purpose, which differs from their original or true name (orthonym). This also differs from a new name that entirely or legally replaces an individua ...
''
Student
A student is a person enrolled in a school or other educational institution.
In the United Kingdom and most commonwealth countries, a "student" attends a secondary school or higher (e.g., college or university); those in primary or elementar ...
'', 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
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 ...
. 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
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 ...
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
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 ...
of a Studentized statistic.
In
computational statistics
Computational statistics, or statistical computing, is the bond between statistics and computer science. It means statistical methods that are enabled by using computational methods. It is the area of computational science (or scientific computi ...
, the idea of using Studentized statistics is of some importance in the development of
confidence intervals
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
with improved properties in the context of
resampling and, in particular,
bootstrapping
In general, bootstrapping usually refers to a self-starting process that is supposed to continue or grow without external input.
Etymology
Tall boots may have a tab, loop or handle at the top known as a bootstrap, allowing one to use fingers ...
.
[Davison, A.C., Hinkley, D.V. (1997) ''Bootstrap Methods and their Application'', CUP. ]
Examples
*
Studentized range In statistics, the studentized range, denoted ''q'', is the difference between the largest and smallest data in a sample normalized by the sample standard deviation.
It is named after William Sealy Gosset (who wrote under the pseudonym "''Student'' ...
*
Studentized residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points.
This is ...
See also
*
Pivotal quantity
In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need ...
References
Statistical ratios
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