Is this correct in terms of subject and verb agreement; for some strange reason, it seems like the verb should be "depends" (a singular) even though the subject is plural, i.e. compound:

"The quality and accuracy of the results of any statistical data analysis for large amounts of data collected from different sources depend on many factors."

Yes, it is correct. You are right that it does not sound correct however it is.

Reason being is that you have amounts of data from different sources. Therefore you have more than one piece of data and more than one source, therefore you use the word depend because it is plural. If you had one piece of data and one source, it would be singular and you would use depends.

Simplified:

They (many pieces of data) depend on many factors (plural)

Or

It (one piece of data) depends on many factors (singular)

Is this correct in terms of subject and verb agreement; for some strange reason, it seems like the verb should be "depends" (a singular) even though the subject is plural, i.e., compound:

"The quality and accuracy of the results of any statistical data analysis for large amounts of data collected from different sources depend on many factors."
Forgive me, it's the right answer but wrong subject identified, I think?

"The quality and accuracy" is a plural subject, not the data and that is what determines the verb.

Ex: "The quality and accuracy (they) ...... depend on many factors."

If it had been a singular subject it would have been depends.
Ex: "The quality or accuracy (it)...... depends on many factors."

This part is simply subject description and doesn't really matter... "of the results of any statistical data analysis for large amounts of data collected from different sources"

Sorry Val, had to spread the love. But I do believe you are right. It is plural because there is quality AND accuracy.

I humbly stand corrected.