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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 .
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 ...
Intuitively, the Spearman correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully opposed for a ...
A negative correlation between variables is also called inverse correlation. Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the circular arc of separation of the points on a great circle of the sphere. [ 1 ]
The instrument must be correlated with the endogenous explanatory variables, conditionally on the other covariates. If this correlation is strong, then the instrument is said to have a strong first stage. A weak correlation may provide misleading inferences about parameter estimates and standard errors. [3] [4]
Uncorrelated random variables have a Pearson correlation coefficient, when it exists, of zero, except in the trivial case when either variable has zero variance (is a constant). In this case the correlation is undefined.
A bivariate correlation is a measure of whether and how two variables covary linearly, that is, whether the variance of one changes in a linear fashion as the variance of the other changes. Covariance can be difficult to interpret across studies because it depends on the scale or level of measurement used.