Northwich is in the county of Cheshire, Norwich is in the county of Norfolk, on the opposite side of the country.

1841 census, HO107/120/12 folio 6 page 7, Fleming Street, Witton-cum-Twambrooks, Northwich, Cheshire
(All surname SMITH, all born in Cheshire)
Joseph 49 Stone mason
Mary 41
Fanny 17
Charly 15
Henry 12
Mary 11
John 10
George 9
Sarah 7
William 5
Margret 3

The ages of those 15+ should have been rounded down to the nearest 5 years by the enumerator, but in this case they were not.
There is the birth listed on FreeBMD of a Margaret Smith, March quarter 1838, Northwich registration district, volume 19 page 157.
I also have an ancestor from Northwich at that time & her baptism is not listed on the IGI, although it is in the church records.