He appears to have generally written his given name as Frederic, without the final letter k.
In the 1880 US Census for District 31, Enfield; Hartford County; Connecticut Frederick Lee Barnum was enumerated with his parents and siblings, as follows:
Dwelling #285; Family #538
Barnum, Frederick; W.; M; 18; Son; Single; Student; Attended school within the year; b. Massachusetts; Father b. Connecticut; Mother b. New York
In the 1900 US Census for District 72, Brooklyn Ward 7, Kings County, New York the family of Frederick Lee Barnum was enumerated as follows:
Dwelling #44; Family #88
Barnum, Frederick L.; Head; W; M; b. Jun 1862; 37; Married 11 years; b. Connecticut; Father b. Connecticut; Mother b. New York; Physician; Can read, write and speak English; Owns non-farm home with mortgage
Barnum, May; Wife; W; F; b. 1864; 35; Married 11 years; Mother of 2 children; 1 child living; b. New York; Both parents b. New York; Can read, write and speak English
Barnum, Lee F.; Son; W; M; b. May 1892, 8, Single, b. Pennsylvania, Father b. Connecticut; Mother b. New York; At school; Attended school 10 months during the year
Barnum, Isaac, W.; Uncle; W; M; b. Dec. 1824; 76; Widowed; b. Connecticut; Both parents b. Connecticut; Retired; Can read, write and speak English
Frage, Edna; Servant; W; F; b. Jan 1886; 14; Single; b. New York; Both parents b. New York; Servant; Can read, write and speak English
[Note: Isaac W. Barnum, shown here as Frederick's uncle, was a son of Frederick's grandfather Daniel E. Barnum by his first wife, not by Frederick's grandmother who was his second wife].