Yes, I'm not exactly clear on what the results are showing. 

1) I did a correlation matrix of the following variables: BirthRate, Agriculture, Service, GDPPerCap, Population, InfantMort. I did NOT click on "ensure uniform sample size".

The top of the results box says

      Correlation Coefficients, using the observations 4 - 229
      (missing values were skipped)
      Two-tailed critical values for n = 221: 5% 0.1320, 1% 0.1729

A couple of results:

BirthRate and Agriculture = 0.7021

...

3) Next, I did a correlation matrix of JUST BirthRate and Agriculture.Here are the results:

corr(BirthRate, Agriculture) = 0.68261942
Under the null hypothesis of no correlation:
t(220) = 13.855, with two-tailed p-value 0.0000

The correlation here is different from steps 1 or 2, 

Yes, I can confirm that, and indeed the difference between 1 and 3 (values 0.702 and 0.683) is unexpected, I'd say. Perhaps there's something wrong when a subset of variables are selected and missing values are all over the place, but I don't have time right now to do further testing or investigation.

We'll definitely look at this further.

thanks

sven