Yup. we taught that way in classes, if you don't SEE it go down or HEAR it go down, be still and quiet. A deer hit too far back generally has a pass through or no huge trauma (bone contact) and will normally run away for 20-80 yards and stop in an available cover, turn and watch for something to follow.
If nothing follows, the deer likely will bed right there. If you quietly leave your stand, after sitting for a time (do not follow), and leave the area, that very well may be where you will find the critter later.
Immediately after it runs, watch it, get sight landmarks, but also take a compass reading from your tree, not just E, W, but degrees. That can come in very handy later, especially if you come back out after dark.
Give the animal plenty of time to expire, then come back out and find it.
Lots of strategies are out there for the best way to trail such a shot. Another discussion
ChuckC