*r*(i.e., further away from zero) are needed to attain statistical significance, than is the case with larger samples. In other words, with smaller samples, it takes a stronger correlation (in a positive or negative direction) to reject the null hypothesis of no true correlation in the full population (rho = 0) and rule out (to the degree of certainty indicated by the

*p*level) that the correlation in your sample (

*r*) has arisen purely from chance.

As you can see, statisticians sometimes talk about sample sizes in terms of

*degrees of freedom*(

*df*). We'll discuss df more thoroughly later in the course in connection with other statistical techniques. For now, though, suffice it to say that for ordinary correlations,

*df*and sample size (

*N*) are very similar, with

*df*=

*N*- 2 (i.e., the sample size, minus the number of variables in the correlation).

For a partial correlation that controls (holds constant) one variable beyond the two main variables being correlated (a first-order partial),

*df*=

*N*- 3; for one that controls for two variables beyond the two main ones (a second-order partial),

*df*=

*N*- 4, etc.

This web document also has some useful information.