# Fisher transformation

**Fisher's ''z'' transformationFisher’s Z-transformation**

In statistics, hypotheses about the value of the population correlation coefficient ρ between variables X and Y can be tested using the Fisher transformation (aka Fisher z-transformation) applied to the sample correlation coefficient.wikipedia

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### Pearson correlation coefficient

**correlation coefficientcorrelationPearson correlation**

In statistics, hypotheses about the value of the population correlation coefficient ρ between variables X and Y can be tested using the Fisher transformation (aka Fisher z-transformation) applied to the sample correlation coefficient. While the Fisher transformation is mainly associated with the Pearson product-moment correlation coefficient for bivariate normal observations, it can also be applied to Spearman's rank correlation coefficient in more general cases.

In practice, confidence intervals and hypothesis tests relating to ρ are usually carried out using the Fisher transformation, the inverse hyperbolic function (artanh) of r:

### Spearman's rank correlation coefficient

**rank correlation coefficientSpearmanSpearman's rho**

While the Fisher transformation is mainly associated with the Pearson product-moment correlation coefficient for bivariate normal observations, it can also be applied to Spearman's rank correlation coefficient in more general cases.

Another approach parallels the use of the Fisher transformation in the case of the Pearson product-moment correlation coefficient.

### Data transformation (statistics)

**data transformationtransformationData Transformations**

Data transformation (statistics)

Examples of variance-stabilizing transformations are the Fisher transformation for the sample correlation coefficient, the square root transformation or Anscombe transform for Poisson data (count data), the Box–Cox transformation for regression analysis and the arcsine square root transformation or angular transformation for proportions (binomial data).

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, hypotheses about the value of the population correlation coefficient ρ between variables X and Y can be tested using the Fisher transformation (aka Fisher z-transformation) applied to the sample correlation coefficient.

### Covariance

**covariantcovariationcovary**

Here stands for the covariance between the variables X and Y and \sigma stands for the standard deviation of the respective variable.

### Standard deviation

**standard deviationssample standard deviationsigma**

Here stands for the covariance between the variables X and Y and \sigma stands for the standard deviation of the respective variable.

### Natural logarithm

**lnnatural logarithmsnatural log**

where "ln" is the natural logarithm function and "arctanh" is the inverse hyperbolic tangent function.

### Inverse hyperbolic functions

**inverse hyperbolic functioninverse hyperbolic tangentinverse hyperbolic cosine**

where "ln" is the natural logarithm function and "arctanh" is the inverse hyperbolic tangent function.

### Multivariate normal distribution

**multivariate normalbivariate normal distributionjointly normally distributed**

If (X, Y) has a bivariate normal distribution with correlation ρ and the pairs (X i, Y i ) are independent and identically distributed, then z is approximately normally distributed with mean

### Independent and identically distributed random variables

**independent and identically distributedi.i.d.iid**

If (X, Y) has a bivariate normal distribution with correlation ρ and the pairs (X i, Y i ) are independent and identically distributed, then z is approximately normally distributed with mean

### Normal distribution

**normally distributednormalGaussian**

If (X, Y) has a bivariate normal distribution with correlation ρ and the pairs (X i, Y i ) are independent and identically distributed, then z is approximately normally distributed with mean

### Standard error

**SEstandard errorsstandard error of the mean**

and standard error

### Confidence interval

**confidence intervalsconfidence levelconfidence**

can be used to construct a large-sample confidence interval for r using standard normal theory and derivations.

### Partial correlation

See also application to partial correlation.

### Variance-stabilizing transformation

**variance stabilizedvariance stabilizing transformation**

The Fisher transformation is an approximate variance-stabilizing transformation for r when X and Y follow a bivariate normal distribution.

### Ronald Fisher

**FisherR.A. FisherR. A. Fisher**

The behavior of this transform has been extensively studied since Fisher introduced it in 1915.

### Harold Hotelling

**HotellingHotelling, HaroldH. Hotelling**

Hotelling in 1953 calculated the Taylor series expressions for the moments of z and several related statistics and Hawkins in 1989 discovered the asymptotic distribution of z for data from a distribution with bounded fourth moments.

### Asymptotic distribution

**asymptotically normalasymptotic normalitylimiting distribution**

A similar result for the asymptotic distribution applies, but with a minor adjustment factor: see the latter article for details.

### Meta-analysis

**meta-analysesmeta analysismeta-analytic**

Meta-analysis (this transformation is used in meta analysis for stabilizing the variance)

### R (programming language)

**RR programming languageCRAN**

R implementation

### List of statistics articles

**list of statistical topicslist of statistics topics**

Fisher transformation

### Denoising Algorithm based on Relevance network Topology

The correlation coefficient then underwent a Fisher's transform:

### Gene co-expression network

**coexpression**

Another approach is to use Fisher’s Z-transformation which calculates a z-score for each correlation based on the number of samples.

### Confidence distribution

It is well known that Fisher's z defined by the Fisher transformation:

### Fisher

Fisher transformation, a transformation in statistics used to test some hypotheses