According to the 1881 census, father James is a mariner - which might explain the family's absence on other censuses?
Reference is RG11, Piece 5600, Folio, 115, Page 5
12 Dale Street, Maughold, Isle of Man
Greenwood, James Head M 42 Mariner b England
Greenwood, Mary Wife F 40 b England
Greenwood, Arthur Son M 16 Scholar b England
Greenwood, George Son M 14 Scholar b England
Greenwood, Anna Daughter F 11 Scholar b England