Correlation coefficient

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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]


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.


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.


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

Tetrachoric and polychoric[edit]

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.

See also[edit]


  1. ^ 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]


  1. ^ "correlation coefficient". National Council on Measurement in Education. Archived from the original on July 22, 2017. Retrieved April 17, 2014.
  2. ^ 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.
  3. ^ 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.
  4. ^ Weisstein, Eric W. "Statistical Correlation". Retrieved 2020-08-22.