A **correlation coefficient** is a numerical measure of some type of 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 agreement and 0 the strongest possible disagreement.^{[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]}^{[5]} 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:

- Spearman's rank correlation coefficient is a measure of how well the relationship between two variables can be described by a monotonic function.
- The Kendall tau rank correlation coefficient is a measure of the portion of ranks that match between two data sets.
- Goodman and Kruskal's gamma is a measure of the strength of association of the cross tabulated data when both variables ae measured at the ordinal level.

### 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:

- The two variables were measured on a continuous scale, instead of as ordered-category variables.
- 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.

## See also

- Coefficient of determination
- Correlation and dependence
- Correlation ratio
- Distance correlation
- Goodness of fit, any of several measures that measure how well a statistical model fits observations by summarizing the discrepancy between observed values and the values expected under the model
- Multiple correlation
- Partial correlation

## Footnotes

**^**Correlation coefficient: A statistic used to show how the scores from one measure relate to scores on a second measure for the same group of individuals. A high value (approaching +1.00) is a strong direct relationship, values near 0.50 are considered moderate and values below 0.30 are considered to show weak relationship. A low negative value (approaching -1.00) is similarly a strong inverse relationship, and values near 0.00 indicate little, if any, relationship.^{[1]}

## References

**^**"correlation coefficient".*NCME.org*. National Council on Measurement in Education. Archived from the original on July 22, 2017. Retrieved April 17, 2014.**^**Taylor, John R. (1997).*An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements*(PDF) (2nd ed.). Sausalito, CA: University Science Books. p. 217. ISBN 0-935702-75-X. Archived from the original (PDF) on 15 February 2019. Retrieved 14 February 2019.**^**Boddy, Richard; Smith, Gordon (2009).*Statistical Methods in Practice: For scientists and technologists*. Chichester, U.K.: Wiley. pp. 95–96. ISBN 978-0-470-74664-6.**^**"List of Probability and Statistics Symbols".*Math Vault*. 2020-04-26. Retrieved 2020-08-22.**^**Weisstein, Eric W. "Statistical Correlation".*mathworld.wolfram.com*. Retrieved 2020-08-22.