In statistics, dependence is any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence, though it most in common usage often refers to the extent to which two variables have a linear relationship with each other.
Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring, and the correlation between the demand for a product and its price.
Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling; however, statistical dependence is not sufficient to demonstrate the presence of such a causal relationship (i.e., correlation does not imply causation).
In projective geometry, a correlation is a transformation of a d-dimensional projective space that transforms objects of dimension k into objects of dimension d − k − 1, preserving incidence. Correlations are also called reciprocities or reciprocal transformations.
Given a line m and P a point not on m, an elementary correlation is obtained as follows: for every Q on m form the line PQ. The inverse correlation starts with the pencil on P: for any line q in this pencil take the point m ∩ q. The composition of two correlations that share the same pencil is a perspectivity.
In three dimensions
In a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook:
Three-dimensional correlations also transform lines into lines, so they may be considered to be collineations of the two spaces.