also -Samuel the son of Robert Burton and Sarah -had more than 1 brother and more sisters than you mentioned
as already posted
1841 children at home were Sarah 5, Samuel 4, Jane 2, Henry 6mths
1851 Children at home were Sarah Ann 15, Jane 12, Henry 10, Mary 7, Elizabeth 4, Hannah 1
then 1861 shows
20 bath St Derby
Robert Burton Head m 50 Lace maker Castle Donnington Leics
Sarah wife 43 Disworth Leics
Jane daughter 22 Castle Donington
Henry son 20 Castle Donington
Elizabeth daughter 14 Castle Donington
hannah daughter 11 CastleDonington
William son 9 Castle Donington
Stephen son 6 Derby
Mary daughter 3 Derby
RG9 2495 77 22
1871
20 Bath St Derby
Robert Burton head M 59 Lab Railway stores Castle Donington leics
Sarah wife 54 Diseworth Leics
Hannah daughter 21 Silk Mill Engine Hand Castle Donington
William son 19 Painter's lab Castle Donington
Mary h daughter 12 Silk Mill Engine hand Derby
Emily daughter 9 Derby (I assume this is "Amelia")
Thomas Twigger Lodger M 44 Elastic Weaver Bedworth Warwicks
Mary A Twigger U 18 Bedworth Warwicks
RG10 3560 89 24
The other coincidence is -the "other" Samuel (who I think is the son of the above family) -moved to Derby -his wife Elizabeth was b Derby -then they moved to York

??
Suz