In applied 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....

, a variance-stabilizing transformation is a data transformation

Data transformation (statistics)

In statistics, data transformation refers to the application of a deterministic mathematical function to each point in a data set — that is, each data point zi is replaced with the transformed value yi = f, where f is a function...

 that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or 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...

 techniques.

The aim behind the choice of a variance-stabilizing transformation is to find a simple function ƒ to apply to values x in a data set to create new values y = ƒ(x) such that the variability of the values y is not related to their mean value. For example, suppose that the values x are realizations from different Poisson distribution

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

s: i.e. the distributions each have different mean values μ. Then, because for the Poisson distribution the variance is identical to the mean, the variance varies with the mean. However, if the simple variance-stabilizing transformation


is applied, the sampling variance associated with observation will be nearly constant: see Anscombe transform

Anscombe transform

In statistics, the Anscombe transform, named after Francis Anscombe, is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately standard Gaussian distribution. The Anscombe transform is widely used in photon-limited imaging ...

 for details and some alternative transformations.

While variance-stabilizing transformations are well known for certain parametric families of distributions, such as the Poisson and the binomial distribution, some types of data analysis proceed more empirically: for example by searching among power transformation

Power transform

In statistics, the power transform is from a family of functions that are applied to create a rank-preserving transformation of data using power functions. This is a useful data processing technique used to stabilize variance, make the data more normal distribution-like, improve the correlation...

s to find a suitable fixed transformation. Alternatively, if data analysis suggests a functional form for the relation between variance and mean, this can be used to deduce a variance-stabilizing transformation. Thus if, for a mean μ,


a suitable basis for a variance stabilizing transformation would be


where the arbitrary constant of integration can be chosen for convenience.