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 p ...

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 ...

0 and

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

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 In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...

, the analysis of variance and

time series analysis In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...

. 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

and random variate. Standard normal deviates arise in practical statistics 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 In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...

: 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 distribution In 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. It gives the probabilities of various values of the variables ...

s may involve manipulating sequences of standard normal deviates: an example here is the

chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...

, 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).

References

{{statistics-stub Normal distribution

See also

References