**The (Pearson) correlation coefficient
{From the Institute of Phonetic Sciences (IFA):
http://www.fon.hum.uva.nl/}**

*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

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

The correlation coefficient is calculated with the assumption that

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
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).

You can compute this correlation by clicking HERE.