What is the best correlation that I can use in order to determine the relationship of these 2 variables?

How can I correlate 2 dichotomous variables?

Var1: Passed:1 Failed: 19

Var2: Passed:3 Failed:13

I tried the $$\phi$$ coefficient $$X$$: Var1 Var2

$$Y$$: Passed Failed

Everything went well but I realized that I'm correlating the failing and passing of each var1 and var2 separately.

What statistical tool is the best to correlate this?

• Have you tried tetrachoric correlations? It's intended for use with binary information. jstor.org/stable/3701350 Feb 8 at 2:19
• Galen, OP noted they were using Excel only. So, I thought of applying the tag. Feb 8 at 2:49
• How have you correlated $20$ observations with $16?$
– Dave
Feb 8 at 2:50

In that case, your Var1 and Var2 are really one variable that indicates the state, if the subject is in the control group (typically coded with a zero) or the treatment group (typically coded as a one). Call this variable $$X,$$ and it has $$36 = (1 + 19) + (3 + 13)$$ observations.
Then your $$Y$$ variable is another indicator of passing or failing. It would be typical to code these as $$0$$s and $$1$$s, leaving you with another set of $$36$$ observations, paired with the $$X$$-values (for each treatment state, you know the pass/fail state).
You now have two equal-length variables that are paired. The usual Pearson correlation makes sense here (though the sign will depend on how you code the categorical states of exam pass/fail and control/treatment). This is the $$\phi$$ coefficient, which (discussed in the link) has a relationship to a $$\chi^2$$ test of the corresponding contingency table.