# Significance of (mean of?) many Spearman rank correlation coefficients

I have data for a school class of grades and family income(*). To see whether the richest kids do better, I find the spearman rank correlation coefficient for the grades and income of the kids in the class. To figure out if the result is statistically significant I can consult a table.

Now suppose I have this data for K classes of different sizes (ranging from 10 kids to 40). K might be in the hundreds or thousands. I have the spearman coefficient for each class. How shall I proceed?

1. I take the mean of these coefficients - is there a way to decide if it's significant? Say the mean is 0.03, too low to be significant for any particular class - but averaged over so many might it be significant?

2. Is it much more sensible to lump the data together and find just one spearman coefficient for all the data?

3. What if I can't lump the together (e.g. they're graded on different scales so it only makes sense to compare kids in the same class as each other)? How can I use all this data to check whether there's a statistically significant relationship between household income and grades?

(*) these details are fake but I'm facing an equivalent problem.

• What hypothesis are you trying to test? Income causes grades to be higher? If you are interested in the association between grades and income, you should use a linear model predicting grades from income. Calculating correlation for each grade is inferior to regression with adjustment or stratification for grade level. Commented Jun 15, 2017 at 17:49
• @Todd: Computing the correlation between grade and income is equivalent to running a linear model. He has chosen to correlate on ranks rather than values, which is equivalent to running a linear model on the ranks. Commented Jun 15, 2017 at 18:05

• @Todd: Certainly $\rho$ has an interpretation as a correlation coefficient independent of $n$. But it only has a probability interpretation, as the chance of observing such a large value under the null hypothesis of no correlation, for a specific $n$. Another way to think of this is that for your measurement to be useful, you need to not only report not on a value for $\rho$, but also an error bar, and how to compute that error bar by combining different $\rho$ with different $n$ is not clear. Commented Jun 19, 2017 at 18:46