# The Evolution of Correlations

This is the evolution of a bivariate correlation between two questionnaire scales, “hope of power” and “fear of losing control”. Both scales were administered in an open online study. The video shows how the correlation evolves from *r* = .69*** (n=20) to *r* = .26*** (*n*=271). It does not stabilize until *n* = 150.

Data has not been rearranged – it is the random order how participants dropped into the study. This had been a rather extreme case of an unstable correlation – other scales in this study were stable right from the beginning. Maybe this video could help as an anecdotal caveat for a careful interpretation of correlations with small n’s (and with ‘small’ I mean *n* < 100) â€¦

The evolution of correlations from Felix SchÃ¶nbrodt on Vimeo.

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Nicely done, although this could have been demonstrated analytically. The approximate standard error of the correlation is given through SE[r] = (1-rho^2)/sqrt(n-1). So let us assume that the population correlation (rho) equals .30.

With n=20 we get SE=.21.

With n=90 the SE is still .096 (so the “typical” sampling deviation from the true correlation is about +/- .10), and with n=270 it is .056.

So, even in this simple case of a bivariate correlation small n means n<100.

Hi Psychometrix,

both approaches answer a similar (but not identical) question. Both are concerned with the accuracy of an estimate (although in your example of rho=.3 and n=90, I’d rather calculate the 95% CI which is [.10;.48]. Now the deviation is rather +/- .18).

The “stability approach” in contrast asks “Does the correlation estimate stay close to the true value with increasing sample size”.

But much more details on this approach can be found in a new blog post and in a new paper: SchÃ¶nbrodt, F. D., & Perugini, M. (in press). At what sample size do correlations stabilize? Journal of Research in Personality. doi:10.1016/j.jrp.2013.05.009 [PDF]