"England and Wales Census, 1901," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XSV2-5SP : 21 May 2019), William Brown in household of Thomas Brown, Lamplugh, Cumberland, England, United Kingdom; from "1901 England, Scotland and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing Harrington subdistrict, PRO RG 13, The National Archives, Kew, Surrey.
"England and Wales Census, 1911," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:X727-DFK : 22 December 2018), William Brown in household of Thomas Brown, Salter And Eskett, Near Frizington, Cumberland, England; from "1911 England and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing PRO RG 14, The National Archives of the UK, Kew, Surrey.
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It may be possible to confirm family relationships with William by comparing test results with other carriers of his ancestors' Y-chromosome or 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 DNA with William: