Is binning data valid prior to Pearson correlation? Is it acceptable to bin data, calculate the mean of the bins, and then derive the Pearson correlation coefficient on the basis of these means? It seems a somewhat fishy procedure to me in that (if you think of the data as a population sample) the scatter of these means will be the standard error of the mean and hence very tight if $n$ is large.  So you will probably get a much better correlation coefficient than from the primary data, and that seems wrong.
On the other hand, people often average replicate measurements before a correlation calculation which isn't very different.
 A: Not exactly the same as your question, but on a related note, I remember reading an article a while back (either The American Statistician or Chance magazine, sometime between 2000 and 2003) that showed that for any dataset of 2 variables where they are pretty much uncorrelated you can find a way to bin the "predictor" variable, then take the average of the response variable within each bin and depending on how you do the binning show either a positive relationship or a negative relationship in a table or simple plot.
A: Correlation coefficient is a measure of uncertainty in predicting value Y from value X measured for an individual sampled object, such as a patient. In focusing on this prediction you do not 'bin' individual readouts: your blood sample is your blood sample not other's. If however the nature of your data permits binning / averaging it normally means that you do not care about individual readouts (happy to mix them together)  but want to see if there is a TREND in your sample. Here, you look for a (linear) regression and its significance instead because the correlation coeffcient would depend  directly on the way you bin your data. Somehow, most biological papers ignore this.
A: The main reason to bin data is to allow for the possibility of a nonlinear relationship between the variables. The Pearson correlation measures strength of linear association, so it doesn't work well when the relationship is nonlinear.
There are obviously much better ways to handle this issue than binning. For example, you might fit a nonlinear or local regression model and correlate the predicted and actual response values (although this assumes that a predictor-response approach is valid, whereas correlation is symmetric). Binning is just a way of solving the problem of nonlinearity that people without a statistics background or statistical tools might use.
