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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 ...
In both examples, as the number of comparisons increases, it becomes more likely that the groups being compared will appear to differ in terms of at least one attribute. Our confidence that a result will generalize to independent data should generally be weaker if it is observed as part of an analysis that involves multiple comparisons, rather ...
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.
Correlations between the two variables are determined as strong or weak correlations and are rated on a scale of –1 to 1, where 1 is a perfect direct correlation, –1 is a perfect inverse correlation, and 0 is no correlation. In the case of long legs and long strides, there would be a strong direct correlation. [6]
Diametrically opposed points represent a correlation of –1 = cos(π), called anti-correlation. Any two points not in the same hemisphere have negative correlation. An example would be a negative cross-sectional relationship between illness and vaccination, if it is observed that where the incidence of one is higher than average, the incidence ...
A specific case of biserial correlation occurs where X is the sum of a number of dichotomous variables of which Y is one. An example of this is where X is a person's total score on a test composed of n dichotomously scored items. A statistic of interest (which is a discrimination index) is the correlation between responses to a given item and ...