One idea runs through these words: looking closely to judge or tell apart. To examine something is to test it; to test sharply is to distinguish one thing from another, and that same act, applied to a patient, becomes a diagnosis. From the verb "to examine" the root radiates into the quiz, the exam, the diagnosis, and the abstract noun for examination. Notice the pair בֹּחַן / מִבְחָן: two different moulds for the same idea "a test". Every word here is verified against pealim, niqqud included.
The same root runs through the verb system. Each binyan is a voice: it bends the one idea into active, passive, causative or reciprocal. Tap to hear.
The pa'al livchon is the plain act of examining; the hif'il lehavchin sharpens it into discernment, telling things apart. The nif'al, pi'el and hitpa'el are not lexicalized for this root in everyday Hebrew, so the family runs through the pa'al/hif'il pair plus a rich set of nouns.
Same root letters, different noun moulds (mishkalim). The mould tells you the function: who does it, the act of doing it, or the thing produced.
Strip the prefixes, suffixes and vowels. Three consonants remain, and the meaning lives there.