Damage Dealt = round(Power * Weapon Strength * Skill Coefficient * Multipliers / 2597)
Multipliers: critical damage, slaying potions, force sigil, % trait mods, etc

multi-hitting skills that display "Damage(x#):" in their tooltip are calculated by rounding each hit and multiplying by the total # of hits.
dmg = [# of hits] * round(Power * WS * [coefficient per hit] * Multipliers / 2597)

the damage numbers on the wiki are taken from the skill tooltips with base (1000) power, no modifiers active, and an exotic level 80 weapon (1 handed mainhand weapons have 952.5 WS)

the wiki has a good table showing midpoints of level 80 exotic and ascended weapons
https://wiki.guildwars2.com/wiki/Weapon_strength

PvP uses level 78 exotic weapons in their calculations, the avg is 922.5

while using an ability without a weapon equipped, the game will use 690.5 WS (656-725, 5% variance) in the calculation. this is the same value used for all (i think) utility/trait/rune/sigil damage calculations

hovering over a skill in the chatbox will generally use 690.5 in the displayed damage calculation. a bug introduced with HoT sometimes offsets this number by a percentage based on the weapon involved.
example: vault (thief staff 5) displayed chatbox damage will be
[690.5 * 2.25 (coef) * power / 2597] * 1.1
this is the result of staves having 10% more damage than an "average" weapon (any 1 hand mainhand weapon)
this bug does not apply to every skill and as such is incredibly frustrating when working off of chat tooltips.

engineer kits (and other bundles) use the pvp base 922.5 weapon strength in all game modes. this results in them having 968.5 WS when wielding an ascended weapon

engineer toolbelt abilities use the WS of your currently equipped weapon (all profession f1-f5 skills are like this). e.g. grenade barrage with a rifle equipped will do 15% more damage than it would with a pistol equipped. (note that currently equipped does not mean currently wielding, it doesn't matter if you're using a kit when you cast the abilities.) if you don't have a weapon equipped, it will use the unequipped 690.5 value

engineer turret detonations do steady damage, resulting in them always hitting for 1270 against a 2597 armor target (make sure not to have the explosives line slotted). this can be very useful for calculating an enemy mob's armor if you're careful about avoiding other modifiers (vuln, etc.)

thief dual-wield skill coefficients are clean numbers multiplied by 1.05
example: shadow shot uses 1.3125 in the tooltip calculation, the result of 1.25 * 1.05

mesmer mind spike (third hit of sword auto) has incorrect damage. the tooltip uses a 1.32 coefficient for no-boon targets but the actual damage dealt is 10% less at a 1.188 coefficient. originally this skill was a 1.0 coef to all targets, then it got a 20% hit (0.8) and split to do 50% more to targets with no boons (1.2). it received another buff of 10% damage to (0.88 and 1.32), but the actual tooltip coefficients are 0.9 and 1.32. damage dealt (to no-boon targets) is wrong.

warriors received a 10% increase in damage to their longbow skills in the specializations update. there was previously a trait that gave longbow skills 10% extra damage against burning targets, and when they removed it they decided to make the 10% baseline (but they forgot to mention it in the patch notes).

mesmer is fucked, sorry.

symbol of energy is not affected by symbolic power

symbol of punishment only pulses damage 4 times, inconsistent with every other symbol having 5 pulses

spear of justice is inconsistent and will provide between 5 and 7 cripple/burn pulses (probably due to server clock. on a side note, server clock is maybe 20hz)

transformation skills have different scaling based on pve and pvp, they do significantly less in pve

tooltip damage and actual damage use separate variables in the game code. this can lead to inconsistencies if the devs don't input the numbers properly

some examples:

engi pistol auto (fragmentation shot) has a 0.4 power scaling, but the tooltip uses 0.4005
druid solar beam (staff auto) has 0.36 power scaling, but the tooltip uses 0.3