Weight has many settings as this can be shown in numerous locations ...

Under Configuration:
Shipping/Packaging ...
Display Number of Boxes and Weight Status
Display Shipping Weight and Number of Boxes?
0= off
1= Boxes Only
2= Weight Only
3= Both Boxes and Weight
Layout Settings ...
Shopping Cart - Show Totals
Show Totals Above Shopping Cart?
0= off
1= on: Items Weight Amount
2= on: Items Weight Amount, but no weight when 0
3= on: Items Amount
Product Listing ...
New Products ...
Featured Products ...
All Products ...

Under Catalog ... Product Types ...
Select a Product type ...
Hit EDIT LAYOUT not the EDIT button ...
Change the setting for weight ...