Statistics

 You did a great job of demonstrating partial correlation in your post. The researcher usually only conducts partial correlation when the third variable has some relationship to one or both of the primary variables. In other words, correlational analysis on all variables would be done first so that you can see whether there are significant relationships among all the variables. It's also important to keep in mind that the researcher also typically wants some theoretical reason why the third variable would be impacting the results.  Can you give an example of this?