When all the stair materials are ready/store bought or cut and ready to install, there is a few simple steps to take when building the stair stringers so that the stair case installs fast and easy.
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Most stair cases are built out of 3 stair stringers, you want to layout/build the stair stringers close to where they will be installed. To start if you didn't already remove a 1-1/2" off the top stair run cut a 1-1/2" off and nail a 2x12 cut to the width of the stairs flush to the top of the top stringers. Then cut out a 1-1/2"x5-1/2" box at the bottom edge of the bottom stringer step to nail a 2x6 in place. These two boards are what you nail the stair case in with when ready.
Add only the riser boards first [standard riser boards would be a 1x8 or a 1x?] fill in every riser besides the first and last step. After all the risers are installed you will want to set the stair case [stair stringers] up into place before adding the treads on. [adding the treads on first will make the stair case 3 times as heavy] Start from the floor where the top of the stairs are to be installed measure down your RISE measurement plus a 1-1/4" [the 1-1/4" is to make up for 1-1/2" treads to be installed] level the 2x12 nailed on the backside of the stair stringers and nail it.
After the top of the stringer is nailed in good square up the bottom of the stair stringers with the walls around it then nail the 2x6 to the floor then add in the last two riser boards. To finish add sub-floor glue to the top of the stair stringers where the treads are to be installed. Using 3" screws add 3 screws through the treads into every stair stringer.