His death certificate has Feb. 8, 1876 for birth date. His headstone and SS application have Feb. 1, 1877.

He is not listed on the 1880 census with his parents:
Home in 1880: Burkesville, Cumberland, Kentucky
Calvin Shaw 36
Missouri Shaw 35
Mary E. Shaw 14
Martha E. Shaw 12
Herchel P. Shaw 9
Arminta Shaw 8
Julia Shaw 9m

Home in 1900: Hiseville, Barren, Kentucky
John K Shaw 53
Missouri Shaw 53 [Transcribed as Mirrisa.]
John B Shaw 22 [Feb 1878]
Enner B Shaw 12
Martha Anderson 29
Effie E Anderson 6
Ova Anderson 5

Is this him? If so, he is listed twice in the 1900 census.
Home in 1900: Hiseville, Barren, Kentucky
Robert Wood 27
Mollie H Wood 64
Ann Thompson 66
Willie Redford 15
Alice Douly 21
Barton Shaw 24 [Feb 1876; Laborer]

Home in 1910: Leslie, Cumberland, Kentucky
Barton Shaw 27 [Age would be incorrect.]
Elizabeth Shaw 21 [Married 1 year; 0 children]

Home in 1920: Leslie, Cumberland, Kentucky
John B Shaw 43
Lizzie Shaw 33
May Shaw 9
Pearl Shaw 3 8/12

Home in 1930: District 7, Cumberland, Kentucky
John B Shaw 53
Elizabeth Shaw 43
Pearl Shaw 13
Catherine Shaw 8

Home in 1940: Metcalfe, Kentucky
Map of Home in 1940: View Map
Street: Gaimon Avenue
Inferred Residence in 1935: Sh, Metcalfe, Kentucky
Residence in 1935: Same House
Resident on farm in 1935: Yes
Sheet Number: 12B
Attended School or College: No
Highest Grade Completed: Elementary school, 2nd grade
Neighbors: View others on page
Household Members: Name Age
Barton Shaw 63
Lisa Shaw 54
Johnny Collins 9 [Grandson]

U.S., Social Security Applications and Claims Index, 1936-2007
Name: John Barton Shaw
Gender: Male
Race: White
Birth Date: 1 Feb 1877
Birth Place: Burkesville, Kentucky
Father: John C Shaw
Mother: Missouri D Anderson
Type of Claim: Original SSN.
Notes: Jul 1944: Name listed as JOHN BARTON SHAW

Kentucky, Death Records, 1852-1953
Name: John Barton Shaw
Death Date: 28 Apr 1950
Death Location: Barren, Kentucky, USA
Age: 74
Gender: Male
Race: White
Birth Date: 8 Feb 1876
Father's name: Calvin Shaw
Mother's name: Katherine Anderson

Son-in-law:
Melvin John Reisinger