The Correlation coefficient

Example:

The observation pairs (x, y)

Characteristics:
A correlation describes the strenght of an association between variables. An association between variables means that the value of one variable can be predicted, to some extent, by the value of the other. A correlation is a special kind of association: there is a linear relation between the values of the variables. A non-linear relation can be transformed into a linear one before the correlation is calculated.
For a set of variable pairs, the correlation coefficient gives the strength of the association. The square of the size of the correlation coefficient is the fraction of the variance of the one variable that can be explained from the variance of the other variable. The relation between the variables is called the regression line. The regression line is defined as the best fitting straight line through all value pairs, i.e., the one explaining the largest part of the variance.
The correlation coefficient is calculated with the assumption that both variables are stochastic (i.e., bivariate Gaussian). If one of the variables is deterministic, e.g., a time series or a series of doses, this is called regression analysis. In regression analysis, the interpretation of the correlation coefficient is different from that of correlation analysis. In regression analysis, tests on statistical significance can only be used when the conditional probability distribution of the other variable is known or can be guessed. However, the regression line can still be used.
If the aim is only to prove a monotonic relation, i.e., if one variable increases the other either always increases or decreases, then the >Rank Correlation test is a better test.

H0:
The values of the members of the pairs are uncorrelated, i.e., there are no linear dependencies.

Assumptions:
The values of both members of the pairs are Normal (bivariate) distributed.

Scale:
Interval

Procedure:
The correlation coefficient R of the pairs ( x , y ) is calculated as:
R = { Sum( x * y ) - Sum(x) * Sum(y) / N } /
sqrt( {Sum( x**2 ) - Sum( x )**2 / N} * {Sum( y**2 ) - Sum( y )**2 / N} )
The regression line y = a * x + b is calculated as:
a = { Sum( x * y ) - Sum(x) * Sum(y) / N } / {Sum( x**2 ) - Sum(x)**2 / N}
b = Sum( y )/ N - a * Sum( x ) / N

Level of Significance:
The value of t = R * sqrt( ( N - 2 ) / ( 1 - R**2 ) ) has a &dof=> Student-t distribution with Degrees of Freedom = N - 2.

Approximation:
If the Degrees of Freedom > 30, the distribution of t can be approximated by a > Standard Normal Distribution.

Remarks:
This could be called the most mis-used of statistical procedures. It is able to show whether two variables are connected. It is not able to show that the variables are not connected. If one variable depends on another, i.e., there is a causal relation, then it is always possible to find some kind of correlation between the two variables. However, if both variables depend on a third, they can show a sizable correlation without any causal dependency between them. A famous example is the fact that the position of the hands of all clocks are correlated, without one clock being the cause of the position of the others. Another example is the significant correlation between human birth rates and stork population sizes.
WARNING: the level of significance given here is only an approximation, take care when using it! (use a table if necessary)