Standard normal deviate
Encyclopedia
A standard normal deviate (or standard normal variable) is a normally distributed random variable
with expected value
0 and variance
1. A fuller term is standard normal random variable. 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 model, 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 norm random variable. The terminology here is the same as that for random variable
and random variate
. Standard normal deviates arise in practical statistics
in two ways.
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
with expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
0 and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
1. A fuller term is standard normal random variable. 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 model, particularly in regression analysis
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
, the analysis of variance
Analysis of variance
In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...
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 norm random variable. The terminology here is the same as that for random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
and random variate
Random variate
A random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable would have different values. Random variates are used when simulating processes driven by random influences...
. Standard normal deviates arise in practical 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....
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 distributionMarginal distributionIn probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. The term marginal variable is used to refer to those variables in the subset of variables being retained...
s 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).