Log transformation and correlation I am studying the price of fish in a rural African market. I used Spearman's to test for the degree of correlation between weight and price, which gave a value of .773. I then took the log10 of price and re-tested correlation, and it gave the same Spearman's value of .773. This seems odd, since I thought a transformation would make it less or more correlated (e.g., transforming bacterial colony growth over time from exponential to linear). 
I understand that every variable is affected the same way by the transformation, hence the correlation doesn't change, however the scatterplots are qualitatively different-so shouldn't the correlation coefficients be as well?
 A: The reason you aren't seeing any difference is because you're calculating Spearman's rather than Pearson's correlation.  The latter is a measure of linear association, but Spearman's correlation measures the strength of any monotone relationship, which should be invariant to monotone transformations.
The way we calculate Spearman's correlation is by first converting the observations into their ranks and then applying Pearson's correlation.  Since any monotone increasing transformation (such as the logarithm) does not change the order of the observations, you will get exactly the same ranks as before you applying the transformation, and so you get the same value for Spearman's correlation.
A: Spearman's correlation coefficient uses rank, rather than the actual data values. Using Spearman's correlation is actually therefore already a transformation, as you are transforming the data values into ranks.
A log transformation will change the values of the variable, but it won't change the ranking of the values relative to one another. Thus, the Spearman correlation coefficient will remain unchanged.
A: Spearman correlation tests for monotonic association (tendency to increase together and decrease together); it's unaffected by monotonic-increasing transformation (like taking logs, square roots or squaring positive values).
To Spearman correlation, these are all perfectly correlated:

... since each variable increases (though by varying amounts) as the other one does.
If you expect the correlation to change when you transform one or the other, you're probably thinking of something more like Pearson correlation, which measures linear association and is affected by monotonic transformation.
(Incidentally, if you do want to transform for a Pearson correlation, I'd suggest considering transforming both variables by taking logs.)
