Putting up installation in walls is an easy job that most homeowners can do themselves. However, some homeowners find themselves in trouble when they go to hire someone to put up the walls because they have stapled the insulation in a way they are told is wrong. Wrong depends on your perspective, though.
Wall insulation can come in different types. It can be blown in, polystyrene or fiberglass batts. Fiberglass batts are the most common. These batts can come faced or unfaced. Faced means with built in Kraft paper on one side that is used to form a vapor barrier. The paper should face the inside of the home. Many states require a vapor barrier in their building codes. A vapor barrier can also be put in as one big plastic sheet and stapled in place. In this case you would use unfaced batts because you are adding your own vapor barrier.
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When installing fiberglass insulation batts with the built in Kraft paper, you will want to staple the insulation in place so that it doesn't fall. The paper covers the insulation plus enough left over on each side to staple it onto wall studs. The built in overlap is made in a way that you can get a consistent barrier between the inside of the house and the wall cavities. However, sheet rock installers do not like this because it makes it harder for them to easily see where the studs are when they put up the walls. So the sheet rock installers prefer insulators staple the paper inside the cavity so that the stud is clearly visible. Some instructions will tell you to staple it inside also but it's up to you. As long as you don't compress the insulation you should be fine. If you are going to hire someone to put up your sheet rock then you should probably staple the paper to the inside of the cavity and not on the outside of the stud.