According to the the 1881 Wales Census she lives at:
Street address: Brynhyfryd 3
Registration district: Bangor
Sub registration district: Llanllechid
"England and Wales Census, 1881," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:Q2WS-VT91 : 1 September 2017), Elizabeth Williams in household of John Williams, Llanfair-Fechan, Caernarvonshire, Wales; from "1881 England, Scotland and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing p. 13, Piece/Folio 5580/81, The National Archives, Kew, Surrey; FHL microfilm 101,775,520.
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It may be possible to confirm family relationships with Elizabeth by comparing test results with other carriers of her ancestors' mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share some percentage (beta) of DNA with Elizabeth: