The below table contains detailed information on the damage inflicted by each weapon. The continuous firing rate is approximate as it was obtained by timing each weapon with a stopwatch. A missed shot inflicts zero damage (except for rocket blast damage). Once a shot is calculated to hit a monster (or other player), the amount of damage is calculated by multiplying a random number (or several in the case of the shotguns), and multiplying by some constant. For example, the shotgun does 7d3 x 5 damage (assuming all 7 pellets hit), which means you throw seven 3-sided dice, add the results and multiplying by 5. In the game of Dungeons and Dragons, this is similar to the way in which the Dungeon Master calculated damages (although they didn't have 3 sided dice!).
The Min, Max and Mean Damage Columns result directly from the Damage column. The Damage in HP/sec is simply the Mean Damage times the firing rate. The table is sorted in order of increasing Damage in HP/sec. The berserk fist in these terms is actually more powerful than the SSG, although practically speaking you can't punch any monster continuously unless you are invulnerable or it is busy fighting something else.
Notes:

(*) These damage calculations are valid firing at monsters at ANY skill level. In deathmatch, the raw numbers are multiplied by a constant that depends on the skill level.
(**) This number is very rough, as I got it by estimating the HP/sec (<2 seconds to kill a Demon) and working backwards. Effectively, the chainsaw appears to be a quadruple speed fist.

(***) The Super Shotgun does 3 times the damage of the Shotgun, even though it only uses twice the ammo.

(#) Cyberdemons and Spider Masterminds take only the direct hit damage from a rocket, not the blast damage. This explains why it takes on average 44 rockets to kill a Cyberdemon, 33 for a Spider Mastermind.

(##) BFG damage calculations are quite complex, the figures listed show the damage taken by the direct hit plus all 40 traces. Each trace does a random amount of damage between 49 and 85 HP, with a mean of 67 HP. At point blank range the total damage of a BFG impact can exceed 3000 HP! It is not all that uncommon to kill a Spider Mastermind with one BFG shot. See The BFG FAQ for more info on the BFG (although their trace damage figures are wrong).

The following table contains the number of hit points for each monster and how much damage it's attacks do. This table is for skill levels 2-5, damage at skill 1 is half this (rounded down to the nearest integer). The hit point figures may differ from those in some FAQ's, however these were obtained from Dehacked and are believed to be correct. If you don't believe some HP figure, check it in Dehacked first.
Note that this pistol carried by the Trooper does different damage from the one you carry, as does the Sargeant's shotgun.
The player at 100% health and no armor has 100 hit points. At 200% health and armor, though, you effectively have 400 hit points.

If you know probability theory, you can use the damage and monster HP information to calculate the probability distribution of the damage done by a weapon, or the probability distribution of the number of shots required to kill a type of monster. It is important to realize that all the below calculations assume zero misses. This is particularly significant for the shotguns - some of the 21 SSG pellets will often miss a monster unless it is very close.

For example, you should be able to kill the Cyberdemon with the SSG in 19 shots over half the time. You can demonstrate this by going invulnerable and also going into no clipping mode so that you can fire the SSG when you are actually inside him, ensuring that not a single pellet misses. However in real game situations 20-22 shots are usually required, because it is very hard to make sure every pellet is hitting him.

The table below also shows why it often takes 3 shots with the normal shotgun to kill a demon (150 HP) but almost always only 1 shot with the SSG.

Two ways: Dehacked and using actual gameplay. Dehacked contains monster HP values plus damage multipliers for all of the missiles in the game. If you look at "Things" in Dehacked, you will see "Rocket (in air)" and "Plasma Bullet". Under the attributes of these things you will find a "Missile damage", which is the damage multiplier (20 in the case of the rocket). It appears that all missiles are based on a d8 random damage. Bullets or shotgun pellets do instantaneous damage and are not "Things" - their damage must be obtained by the second technique below.
The second, more primitive technique is just to observe health decreases during a controlled attack by a monster, or to create a "voodoo doll". If you create a level with two starts for Player 1, when you start a single player game the second start will be your voodoo doll. If you shoot him with a pistol, for example, your own health will go down at the same time. If you do this, you will notice your health always goes down by 5%, 10% or 15% with equal probability (when you have no armor).

The BFG damage calculations are very complex and it even has it's own BFG FAQ. As far as I can tell, the BFG FAQ is perfectly correct, except their trace damage numbers are too low, and they claim there are only 20 traces. We believe there are actually 40 traces and each does a random amount of damage approximately between 49 and 85 HP. The latter was obtained by doing voodoo experiments set up so that only a single trace hits the voodoo doll.

I'm sure there are some mistakes but most of this is believed to be very accurate. It is certainly more accurate than the information I have seem in all the FAQ's. Many of the monster damage figures are rather uncertain and more experiments are needed. If you believe anything here is wrong, please e-mail me, but first you should check the following:
Are your shots always hitting?? In real gameplay it is hard to ensure every shotgun pellet is hitting some monster that you aim at, so it is virtually impossible to take out a Cyberdemon in 19 SSG shots.
Can you make an lmp to illustrate your point? If so, send it to me.
The odds calculations get pretty complex, especially when calculating the odds of taking out a Cyberdemon. Please feel free to try to duplicate them on your own, and let me know if you get something different.

What do you correct in my tier list?