Normal began as a woodworking term. Latin norma was a carpenter's square — the L-shaped tool used to check that an angle is exactly ninety degrees. Something normālis was 'made according to the square': straight, right-angled, correct.
The mathematical sense preserves the original meaning most faithfully. In geometry, a normal line is one perpendicular to a surface — literally at right angles, just as a carpenter's norma would verify.
The expansion from 'right-angled' to 'conforming to a standard' happened in Late Latin, as norma shifted from a physical tool to an abstract rule. By the time French normal entered English in the 17th century, the carpenter's square had been mostly forgotten.
The social sense — 'typical, ordinary, what most people do' — is surprisingly recent. It emerged in the 19th century alongside statistics. When Adolphe Quetelet described the 'normal distribution' in the 1830s, he mathematised the concept: normal became what the average person does. Before statistics, there was no concept of 'normal' as a population average.
Enormous is normal's dark twin. Latin ē-normis meant 'out of the square' — deviating from the pattern, departing from the norm. In early usage, enormous meant not just large but aberrant. An enormous crime was monstrous because it broke the rules, not because it was big.