At 81 Cumberland Street Area in the 1881 census:
Andrew MILLAR (50), Mary (50), William (17) - all born St Cuthberts, Edinburgh
George (15), Christina (13), Emil (5), John C (2) - all born St Andrews, Edinburgh
In this context, I would take area to mean basement flat accessed via external stairway, rather than district. (I am not sure if that definition is unique to Edinburgh) Basement area flats are commonly found in Edinburgh's Georgian New Town district, which would include Cumberland Street.
ADP