A standard normal deviate is a
normally distributed deviate. It is a
realization of a standard normal random variable, defined as a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
0 and
variance 1.
[Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms. OUP. ] Where collections of such random variables are used, there is often an associated (possibly unstated) assumption that members of such collections are
statistically independent.
Standard normal variables play a major role in theoretical statistics in the description of many types of models, particularly in
regression analysis, the
analysis of variance and
time series analysis.
When the term "deviate" is used, rather than "variable", there is a connotation that the value concerned is treated as the no-longer-random outcome of a standard normal random variable. The terminology here is the same as that for
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
and
random variate. Standard normal deviates arise in practical
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 ...
in two ways.
:*Given a model for a set of observed data, a set of manipulations of the data can result in a derived quantity which, assuming that the model is a true representation of reality, is a standard normal deviate (perhaps in an approximate sense). This enables a
significance test to be made for the validity of the model.
:*In the computer generation of a
pseudorandom number sequence, the aim may be to generate random numbers having a
normal distribution: these can be obtained from standard normal deviates (themselves the output of a pseudorandom number sequence) by multiplying by the scale parameter and adding the location parameter. More generally, the generation of pseudorandom number sequence having other
marginal distributions may involve manipulating sequences of standard normal deviates: an example here is the
chi-squared distribution, random values of which can be obtained by adding the squares of standard normal deviates (although this would seldom be the fastest method of generating such values).
See also
*
Standard normal table A standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of Φ, which are the values of the cumulative distribution function of the normal distribution. It is used to find the probability that ...
References
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Normal distribution