Hi Sue,
I am not convinced that the William ARMSTRONG that I found in Fulford, Yorks is the correct one :0(
As following him through census does not provide the proof.
Found my family on the 1911 Census on IOM
William though is missing,but his family are there:
Household ARMSTRONG, Florence - 1887 - 24 - F - Isle of Man - Isle of Man
Household ARMSTRONG, Ivy May - 1906 - 5 - F - Isle of Man - Isle of Man
Household ARMSTRONG, John Norman - 1904 - 7 - M - Isle of Man - Isle of Man ( My Granddad b. 1903)
Household ARMSTRONG, L . E - 1874 - 37 - F - Isle of Man - Isle of Man
Household ARMSTRONG, William Arthur - 1909 - 2 - M - Isle of Man - Isle of Man.

My family told me that William ARMSTRONG Sr came from either Galway, Ireland OR Galway, Scotland.
I do know from his marriage cert copy that his father was called Samuel ARMSTRONG, Occ: Farmer.
So, I am afraid that I am still searching:0(

Can I ask, how do you connect to this family?
Sorry I am not much help.
Kind Regards,
Carol