Test for No Correlation I'm wondering: Is there a statistical test for determining "no correlation"? I would not want to use Fisher's transformation.
 A: "No correlation" means r=0.00000000000000000... that is almost never the case and never of practical interest. You could define, how small an r is "as small as to be zero for practical purposes". Computing correlation indices in statistics software will give you a confidence interval for the correlation coefficient. 
If you say, that abs(r)<.10 you can compute the confidence interval and see, if it includes only values that meet that requirement.
A: R values range from -1 to 1. A value of -1 indicates a completely negative correlation, and 1 indicates a completely positive correlation. The distribution for the correlation coefficient is not normally distributed, and its variance is not constant. Fisher's transformation attempts to solve this problem, which allows us to compute the confidence interval. Determining correlation and no correlation would depend on your confidence interval. This interval defines a range that estimates that values of r that are likely to contain unknown. This is useful because it provides a range of potential values for r given an unknown. Otherwise you are simply left with a number from -1 to 1, and you have to make a personal decision if it correlated. Is .5 a correlation, what about .4? I would consider that something interesting when examining data, but its not enough to definitely say yes this is the golden relationship. Confidence interval gives us a mathematical way to say yes or no under given criteria which is what is lacking in the correlation equation. 
The CI is calculated using the mean and standard deviation. For variance uses size and sample size and for this, CI cannot be calculated directly. Fisher's transformation provides an indirect calculation that can be be made for CI. This is why most of the Fischer problems from your stats books solve for or give CI. 
