You are lucky because it should reflect the arrest and conviction. There must be some sort of error. (Did you send in your fingerprints to the FBI for the extensive background check? If so, you are lucky. If not, then the fingerprint background check should certainly reveal the conviction. The FBI is the central repository for all fingerprints.)
UNLESS you were a juvenile. Juvenile records are not public to anyone but law enforcement.
All records depend on human action and decisions and apparently, something has allowed your record to escape detection this time. This doesn't mean that the record is not in the system and certainly the best possible scenario is that it has possibly been lost from the county that you were convicted, but most likely, it wasn't reported properly years ago, which means a more through search would discover it. Records do get destroyed and maybe this happened and it is gone forever, but probably. It is out there to be discovered, so don't make any misrepresentation by commission or omission that could prove to be a big problem for you later. If you must know, a search of the county's records is possible, but sleeping dogs are often better dogs.
These records are entered on the computer by human beings and we know how often improper entrys are made on computers. Count your lucky stars and say nothing about this to anyone. I would run a criminal history every couple of years just to be sure.