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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]
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect inverse (decreasing) linear relationship (anti-correlation), [5] and some value in the open interval (,) in all other cases, indicating the degree of linear dependence between the variables. As it ...
The correlation coefficient ρ, expressed as an autocorrelation function or cross-correlation function, depends on the lag-time between the times being considered.Typically such functions, ρ(t), decay to zero with increasing lag-time, but they can assume values across all levels of correlations: strong and weak, and positive and negative as in the table.
The correlation ratio was introduced by Karl Pearson as part of analysis of variance. Ronald Fisher commented: "As a descriptive statistic the utility of the correlation ratio is extremely limited. It will be noticed that the number of degrees of freedom in the numerator of depends on the number of the arrays" [1]
The Spearman correlation coefficient is often described as being "nonparametric". This can have two meanings. First, a perfect Spearman correlation results when X and Y are related by any monotonic function. Contrast this with the Pearson correlation, which only gives a perfect value when X and Y are related by a linear function.
Examples are Spearman’s correlation coefficient, Kendall’s tau, Biserial correlation, and Chi-square analysis. Pearson correlation coefficient. Three important notes should be highlighted with regard to correlation: The presence of outliers can severely bias the correlation coefficient.
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