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ANY type of damage. As long it is dealt on target.
For example, engies and medics can have crazy melee crits because the engie uses his sentry to fill 100000 idiots with bullets, and medic gets assist damage.
The chance scales based on damage dealt in the past 30 seconds of so, 0 damage for minimun chance to 800 damage for max chance. Healing done and sentry/pocketed player damage counts as your own damage. It doesn't depend on server, except for forced crits server and nocrits server.
Current health doesn't affect it, and all the weapons have the same crit chance.
The base crit chance for standard weapons is 2%. This is modified by a bonus percentage based on how much damage the player has done in the last 20 seconds, which scales from 0% at 0 damage to 10% at 800 damage.[1] For example, if the player has dealt 200 damage in the past 20 seconds, the additional bonus percentage is 2.5% and so the overall crit chance is 4.5%. As a result, the highest natural crit chance attainable is 12%.
Melee weapons are designed to crit far more often than ranged ones in order to encourage their usage at close range. Their base crit chance starts at 15% - higher than attainable for ranged weapons with the recent damage bonus maxed out - and their recent damage bonus scales from 0% to 45%, meaning a player that has done 800 damage or more in the last 20 seconds has a melee crit rate of 60%.
Among those weapons that trigger randomly, critical hit probability is determined differently based on the type of weapon. Single-shot weapons, such as the Shotgun, Revolver, and Rocket Launcher, determine the chance to crit with every shot. Thus, every shot of the Shotgun will have a 2% chance to launch a flurry of critical pellets, assuming no previous damage done. Rapid-fire weapons, however, which include the Minigun, Syringe Gun, Flamethrower, and Pistol check for critical hits once a second. With weapons like these, a positive crit determination results in all shots for the next two seconds being critical. This only applies to the active weapon; getting two seconds of crits on one's Pistol will not cause their Shotgun to fire crits during this time.
Are you doing good? Then everyone around you gets crits.
Are you sucking? No crits for you!