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For example, let's say I have a population of I have 5 doctors and have to determine how many times they correctly diagnose cancer per patient. How would I go about finding the variance and determining statistical significance of the variance?
I see you asking two different questions here. First is to test the null hypothesis that the five doctors' conclusions have similar variances. The second is to test if the five doctors' conclusions are concordant with a standard of truth or in other words whether they correctly identify cancer. Ultimately, it seems you're interested in the latter, not the former. In this, I'm not sure if it would apply to your situation. You imply your doctors are giving a binary response: Cancer or No Cancer.
I don't think that finding the variances of this binary response will tell you whether the doctors are correctly identifying it or not. To extend that to many groups or many doctors , Bartlett's Test can be used, which tests the null hypothesis that all variances are equal; the alternative is when even 1 variance deviates from the others. These are very sensitive to the normal assumption, and binary responses are not normal.
If the doctors give a response that can be reasonably assumed to be normal say reporting tumor size , then these tests can be used. If you only have the binary response, this wouldn't be a good test. I put this here to directly answer your question about comparing variances. What I think you're more interested in testing whether the doctors are concordant with some standard of known truth Cancer or Not. Sadly, oftentimes with cancer, the standard of truth is whether they died from said cancer or not.