A standard normal deviate is a normally distributed deviate. It is a realization of a standard normal random variable, defined as a random variable with expected value 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 numbe ...

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 Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...

. 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 Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...

and

time series analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

. 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 and

random variate In probability and statistics, a random variate or simply variate is a particular outcome of a ''random variable'': the random variates which are other outcomes of the same random variable might have different values ( random numbers). A random ...

. 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 A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...

to be made for the validity of the model. :*In the computer generation of a

pseudorandom number sequence A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...

, 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 ...

: 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 varia ...

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