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At the top of the staircase, use a pencil to make a tick mark on the wall 36" (or the required height in your area) above the front tread of the second step. That's where the top of the handrail will rest.
Figure out where to place the handrail brackets by measuring the height of the handrail, top to bottom. Then subtract that distance from the 36", and make another mark below the 36" mark. For example, if the handrail is 2" deep, you'd pencil a mark on the wall above the front tread of the second step on the staircase. That's where the top of the bracket should rest.
Take the same measurements and make the same type of marks at the bottom of the staircase, using the step that's second from the bottom to measure.
Run a string from the top of the staircase wall to the bottom, starting with the mark you made to indicate where the top of the bracket should rest. Attach the string at the top with a thumbtack or dry-wall screw. Then pull it taut, at an angle, until it reaches the corresponding mark on the bottom, and tack it onto the wall there too. The string represents the bottom of the handrail and will give you a line to follow when you space the handrail brackets out along the stairwell wall.
Use a stud finder to locate studs to attach the brackets to. Find one at the top first, as near to the second tread as possible, then one at the bottom, as close to the second step from the bottom as possible. Then find studs between those two, so that you can evenly space the two remaining brackets. The brackets cannot be attached more than 48" apart, and they need to be an equal distance from each other and the end bracket.
Working smarter: Don't install brackets in places that don't have studs, or they'll pull right out of the wall, taking the handrail with them.
Predrill the holes before screwing in the brackets.
Screw in the top bracket first, using an electric. Next attach the bottom bracket, then the middle two.
Lift the handrail onto the brackets, placing the ends wherever is aesthetically pleasing.
Safety alert: The end of the handrail or banister should never rest more than 24" from the end bracket on either side.
It's also up to you whether to make a plumb cut on the two ends of the handrail or to leave them at a 90-degree angle, the way they came.
Predrill, and then use an electric screwdriver to attach the U-brackets that hold the handrail in place.