Critical hit

As you get to some of the higher experience levels, you will begin to score critical hits more and more often. There are five levels of critical hit, ranging from:

Critical hit types table

Type	Magnitude
CRITICAL	2x damage
CRUSHING CRITICAL	4x damage
OBLITERATING CRITICAL	8x damage
ANNIHILATINGLY POWERFUL CRITICAL	16x damage
WORLD-SHATTERING CRITICAL	32x damage

Note: if you have a 10% chance to score a standard critical hit, then you have a 10% chance for a standard critical to become a crushing critical hit, and a 10% chance for a crushing critical to become an obliterating critical hit, etc.

In aethership vs. denizen combat, crew members can score critical hits while manning a battle turret. The chance of critical hits there is determined by one's skill in Aethercraft rather than level, however.

Note that certain denizens are immune to critical hits - notably city and commune guards and certain extraordinarily powerful denizens (i.e., the Demon Lords of Nil).

Maths

Expected Value

To calculate the expected value (aka the mean) of some attack a:
Let d = the base damage of the attack
Let c = the percentage critical hit chance

Expected Value Calculation

Type	Damage	Chance	Expected Value
Normal hit	d	1 - c	d(1-c)
CRITICAL	2 * d	c - c^2	2d(c-c^2)
CRUSHING CRITICAL	4 * d	c^2 - c^4	4d(c^2-c^4)
OBLITERATING CRITICAL	8 * d	c^4 - c^8	8d(c^4-c^8)
ANNIHILATINGLY POWERFUL CRITICAL	16 * d	c^8 - c^16	16d(c^8-c^16)
WORLD-SHATTERING CRITICAL	32 * d	c^16	32dc^16

So our expected value for this attack is the sum of rightmost column above
d(1-c) + 2d(c-c^2) + 4d(c^2-c^4) + 8d(c^4-c^8) + 16d(c^8-c^16) + 32dc^16
Which expands to
16dc^16 + 8dc^8 + 4dc^4 + 2dc^2 + dc + d
and we can factor this into
d((4c^8+1)^2 + 4c^4 + 2c^2 + c)