First of all to answer this we need to think about what a knockout is.
A Knockout is not a medical term. It is a competition term. The term's use varies between competition styles, but basically it amounts to this:
A Knockout is when your opponent can no longer continue to fight you.
So while your opponent may be unconscious they don't have to be.
For example in Thai Boxing: lower leg knockouts
Now that we have established what we mean by a "knockout" let's think about your question.
We can achieve a knockout by disabling the leg or legs, we can achieve a knockout by causing a concussion (Which involves hitting the head)... but would you expect that hitting someone's arm could cause them to give fighting?
Probably not right? Even if you are able to disable an arm they might be able to fight on with the other. Some people might fight on even with a broken arm!
So we have an answer, if you can hit someone on the head or leg and possibly get a knockout but never on the arm then it definitely matters 'where you hit' more than 'how hard you hit'.
Now although the question is answered you probably want to know where you can hit someone to expect a knockout. Let's cover these roughly in order of how conventional they are.
Concussion. This is probably the most common type of knockout. Your opponent suffers brain damage from concussion because their brain hits the side of their skull. You usually achieve this by hitting the jaw and causing the head to rotate. The referee will check your opponent's eyes to make sure you have not caused:
Compression. This is where you actually damage the skull enough to cause damage to the brain. It's much easier than you might think and will typically cause opponents eyes to have pupils of different sizes.
Winding. This is where you paralyze your opponent's diaphragm so they can't breath. If this lasts long enough you will be awarded a win. You strike to the solar plexus to achieve this.
Leg paralysation. Strikes to the thigh can cause the leg to be paralyzed. You may need to strike one or both legs before your opponent will give up or be unable to continue. The exact target is about two thirds of the way up the thigh.
Carotid artery damage. We are getting on to methods now that are less conventional. A strike to the carotid artery or better still the vegus nerve on the carotid artery will cause extreme light-headedness or unconsciousness similar to a strangle.
Spinal compression. Not legal in most competition but included for completeness the easiest way to cause this is to strike to the back of the head. It causes a similar symptom to a concussion but is much more dangerous as you run the risk of breaking the neck.
Neural/Acupressure/Appo. No everyone accepts this one as existing and I have never personally seen it done. But some martial arts traditions hold that damage to certain points on the head will be painful enough to cause a knockout. The one typically given as an example is to hit under both ears at once with the knuckles on each of the thumbs. Note that the angle of attack would have to be correct for this to work. In this case it is towards the centre of the forehead.
Also worth noting of course but omitted here because we are talking about knockouts and not submissions:
- The strangle and the choke. I don't think I need to explain these. One blocks the air to the lungs, the other the blood to the brain but since no-one really agrees which is which it gets confusing.