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]

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:

• 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 are measured at the ordinal level.

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.

The correlation between two variables have different associations that are measured in values such as r or R. Correlation values range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation between variables.^[5]

• Correlation disattenuation
• 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