# Correlation coefficient

A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables.[a] The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution.[citation needed]

Several types of correlation coefficient exist, each with their own definition and own range of usability and characteristics. They all assume values in the range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation.[2] As tools of analysis, correlation coefficients present certain problems, including the propensity of some types to be distorted by outliers and the possibility of incorrectly being used to infer a causal relationship between the variables (for more, see Correlation does not imply causation).[3]

## Types

There are several different measures for the degree of correlation in data, depending on the kind of data: principally whether the data is a measurement, ordinal, or categorical.

### Pearson

The Pearson product-moment correlation coefficient, also known as r, R, or Pearson's r, is a measure of the strength and direction of the linear relationship between two variables that is defined as the covariance of the variables divided by the product of their standard deviations.[4] This is the best-known and most commonly used type of correlation coefficient. When the term "correlation coefficient" is used without further qualification, it usually refers to the Pearson product-moment correlation coefficient.

### Intra-class

Intraclass correlation (ICC) is a descriptive statistic that can be used, when quantitative measurements are made on units that are organized into groups; it describes how strongly units in the same group resemble each other.

### Rank

Rank correlation is a measure of the relationship between the rankings of two variables, or two rankings of the same variable:

### Tetrachoric and polychoric

The polychoric correlation coefficient measures association between two ordered-categorical variables. It's technically defined as the estimate of the Pearson correlation coefficient one would obtain if:

1. The two variables were measured on a continuous scale, instead of as ordered-category variables.
2. The two continuous variables followed a bivariate normal distribution.

When both variables are dichotomous instead of ordered-categorical, the polychoric correlation coefficient is called the tetrachoric correlation coefficient.