Hi Beverly, maybe you can check into some of this to verify if they're your family.
1881 Census

Dwelling: Wood Street
Census Place: Sorn, Ayr, Scotland
Source: FHL Film 0203608 GRO Ref Volume 613 EnumDist 9 Page 30
Marr Age Sex Birthplace
James BLACK M 43 M Ramshorn, Lanark, Scotland
Rel: Head
Occ: Coal Miner
Agnes BLACK M 42 F Gatehead, Ayr, Scotland
Rel: Wife
Andrew BLACK U 19 M Dreghorn, Ayr, Scotland
Rel: Son
Occ: Coal Miner
Archibald BLACK U 17 M Craigbank, Ayr, Scotland
Rel: Son
Occ: Coal Miner
Elizabeth BLACK U 15 F Kilmaurs, Ayr, Scotland
Rel: Daur
Occ: Cotton Weaver
Jeanie BLACK 13 F Craigmack, Ayr, Scotland
Rel: Daur
Occ: Scholar
John BLACK 12 M Craigmack, Ayr, Scotland
Rel: Son
Occ: Scholar
William BLACK 7 M Craigmack, Ayr, Scotland
Rel: Son
Occ: Scholar
David BLACK 5 M Dalmellington, Ayr, Scotland
Rel: Son
Robert BLACK 2 M Rakinston, Ayr, Scotland
Rel: Son
Mary BLACK M 20 F Harthill, Linlithgow, Scotland




Vital Records Index: search on James Black born around 1838 Lanarkshire, (with possible parents Andrew and Jean according to naming customs)

BLACK, James Christening
Gender: Male
Birth Date: 10 Sep 1839 Birthplace: , Biggar, Lanark, Scotland
Recorded in: Biggar, Lanarkshire, Scotland
Father: Andrew BLACK
Mother: Jean HOGG
Source: FHL Film 1042963 Dates: 1728 - 1854

Search for chn of Andrew Black and Jean Hogg:

BLACK, Anna Bir 1835 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG
BLACK, Jane Bir 1837 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG
BLACK, James Bir 1839 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG
BLACK, Alexander Bir 1841 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG
BLACK, Margaret Bir 1843 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG
BLACK, Mary Bir 1845 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG
BLACK, Andrew Bir 1847 Scot Lana Biggar
Fa: Andrew BLACK
Mo: Jean HOGG

Search for possible parents of Andrew born around 1810-15 to James and Jane (again following naming customs):
BLACK, Andrew Christening
Gender: Male
Birth Date: 14 Feb 1806
Christening Date: 26 Feb 1806 Recorded in: Carnwath, Lanarkshire, Scotland
Father: James BLACK
Mother: Janet SCOTT
Source: FHL Film 1042971 Dates: 1792 - 1855

other chn with same parents:
BLACK, Andrew Bir 1806 Scot Lana Carnwa
Fa: James BLACK
Mo: Janet SCOTT
BLACK, Margaret Bir 1807 Scot Lana Carnwa
Fa: James BLACK
Mo: Janet SCOTT
BLACK, James Bir 1810 Scot Lana Carnwa
Fa: James BLACK
Mo: Janet SCOTT


These are some things to go on, maybe there is something more in FreeCen.