Correlation coefficient

A correlation coefficient is a numerical measure of some type of correlation, meaning a linear function between two variables. 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. Several types of correlation coefficient exist, each with their own definition and 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. 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).

A correlation coefficient is a numerical measure of some type of correlation, meaning a linear function between two variables. 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. Several types of correlation coefficient exist, each with their own definition and 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. 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). ==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. r = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} where r is the Pearson correlation coefficient; cov(X,Y) denotes the covariance between variables X and Y; and \sigma_X and \sigma_Y denote the standard deviations of X and Y, respectively. r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} where n is the number of data pairs; \sum xy is the sum of the product of paired scores; \sum x and \sum y are the sums of x-scores and y-scores; \sum x^2 and \sum y^2 are the sums of squared x-scores and squared y-scores. ===Interpreting correlation coefficient values=== The correlation between two variables have different associations that are measured in values such as or . Correlation values range from −1 to +1, where ±1 indicates the strongest possible correlation and 0 indicates no correlation between variables. ==See also== 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 ==Notes== ==References==

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