In the 1881 Census for the Isle there are no matching Armstrongs. I only found one William Armstrong in England in 1881 whose father was Samuel.:

Household:

Name Relation Marital Status Gender Age Birthplace Occupation Disability
Samuel W. ARMSTRONG Head M Male 26 Fulford, Lincoln, England Farmer 83 Acres 2 Boys & 1 Labourer
Elizabeth J. ARMSTRONG Wife M Female 26 Scotland
William ARMSTRONG Son Male 2 Fulford, York, England
Elizabeth M. ARMSTRONG Daur Female 1 Osbaldwick, York, England
Anthony T. HUMBLE Serv U Male 19 Newcastle Farm Servant (Indoor)
William WOODHALL Serv U Male 23 York, York, England Farm Servant (Indoor)
Alice SLEIGHT Serv U Female 25 York, York, England General Servant Domestic

Source Information:

Census Place Osbaldwick, York, England
Family History Library Film 1342143
Public Records Office Reference RG11
Piece / Folio 4727 / 22
Page Number 4
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Same family 1891:
Source information: RG12/3895
Registration district: York
Sub registration district: Walmgate
ED, institution, or vessel: 29
Folio: 7

Armstrong, Samuel Wilson 36 Yorkshire Head Fulford Yorkshire
Armstrong, Elizabeth Jane 36 Scotland Wife Fulford Yorkshire
Armstrong, Elizabeth Mary 11 Yorkshire Daughter Fulford Yorkshire
Armstrong, Elsie 7 Yorkshire Daughter Fulford Yorkshire
Armstrong, Lillian Davidson 6/12 Yorkshire Daughter Fulford Yorkshire
Armstrong, William 12 Yorkshire Son Fulford Yorkshire
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For what it's worth...

Barbara