Normality was the state of being normal or usual.
The Romulan commander and Leonard McCoy discussed James T. Kirk's behavior and the commander said "Then by your own standards of normality, this man is not fully competent." (TOS: "The Enterprise Incident")
In 2371, Tolian Soran stated that Geordi La Forge's VISOR was marvelous technology and a remarkable piece of equipment, but not very stylish. Soran asked La Forge if he ever considered using a prosthesis to make himself look a little more normal, and explained that "normal" was what everyone else was, which La Forge wasn't. (Star Trek Generations)