Alexander was born about 1805. He passed away about 1873.
Sources
Kentucky marriages 1802-1850
"United States Census, 1850," database with images, FamilySearch (https://www.familysearch.org/ark:/61903/1:1:M85R-46N : 20 December 2020), Alexander Shaw, Aux Sable, Grundy, Illinois, United States; citing family , NARA microfilm publication (Washington, D.C.: National Archives and Records Administration, n.d.).
"United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MF5Q-W6C : 19 February 2021), Alexander Shaw in household of P Stevenson, Osage, Labette, Kansas, United States; citing enumeration district ED 117, sheet 388A, NARA microfilm publication T9 (Washington, D.C.: National Archives and Records Administration, n.d.), FHL microfilm 1,254,385.
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DNA Connections
It may be possible to confirm family relationships with Alexander by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Alexander: