The 1850 census states that Ballard was age 1, although he reports his own birthday in the 1900 census as being October 1851.
Given his parents report in July (24th) of 1850 and his report in 1900 of an October birthday, he was most likely born in October of 1849. He was unable to read or write in 1900, which could account for the birth reporting error.
I also can't dismiss the idea that there might be two children with the name, one born in 49, and another in 51.
US Census 1850 District 32, Logan, Virginia
US Census 1860 Buffalo, Logan, Virginia
US Census 1879 Rich Creek, Logan, Virginia
US Census 1880 West Virginia Wyoming Oceana 158
US Census 1900 West Virginia Logan Skin Creek District 0065
US Census 1910 West Virginia Boone Washington District 0006
West Virginia, Deaths Index, 1853-1973 VIEW RECORD Name Ballard Burgess Death Date 7 Oct 1925 Death Place Logan, West Virginia FHL Film Number 1953239
WV Dept of Vital Statistics
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On 9 Feb 2016 at 19:22 GMT Maggie Hedrick wrote:
Ballard is 17 degrees from Louisa Alcott, 15 degrees from David Douglass, 20 degrees from AJ Jacobs and 17 degrees from Queen Elizabeth II Windsor on our single family tree. Login to find your connection.