There is no standard rule about the height to install a bathroom mirror. Instead, hang the mirror so the top of it sits a few inches higher than eye level for the tallest person in the home.
General Installation Guidelines
Take the heights of all household members into consideration when hanging a bathroom mirror; ideally, everyone should be able to see the reflection of his entire face when standing in front of the mirror. At very least, place the mirror at a height at which all teens and adults easily can see themselves without squatting down or requiring a stool. Placement also depends on the height of the mirror itself. If it is the large, frameless style that secures to a wall with an adhesive, you may wish to position the bottom edge so it rests upon the top of the tile or backsplash edge behind the sink; otherwise, a small gap may prove difficult to clean. Ideally, adjust the height so the mirror is centered between the top of the vanity or backsplash and the ceiling, unless the ceiling is too high to do so.