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League of Legends Formulas
Last Updated: January 24th 2015



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Understanding the math behind League of Legends will help you analyze the game better.
The following formulas were also used to create "LoL Item Optimizer" program in this website.


Basic League of Legends Formulas

  • GO   Damage Reduction of Armor and Magic Resist
  • GO   Attack Speed
  • GO   Attack Speed Reduction

Intermediate League of Legends Formulas

% Damage Increase Formulas
  • GO   Armor Pen. / Magic Pen.
  • GO   Cooldown Reduction
% Durability Increase Formulas
  • GO    Armor against physical damage only (Or Magic Resist against magic damage)

Advanced Derivation of League of Legends Formulas

% Damage Increase Formulas
  • GO   Critical Strikes
% Durability Increase Formulas
  • GO   Armor, Magic Resist, Damage Reduction, Attack Speed Slow






Basic League of Legends Formulas



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Damage Reduction of Armor and Magic Resist


      Armor       
  100 + Armor


Example:
If you have 50 armor,

      50       
100  +  50

= 0.3333
You take 33.3% less damage from physical attacks.
Magic Resist works the same way as this against magic attacks.


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Attack Speed


Base_Attack_Speed * (1 + Bonus _Attack_Speed)

Example:
Ashe has  0.658 base attack speed and gets 4% bonus attack speed per level after level 1.  At level 10, Ashe will have

0.658 * (1 + 0.04 * 9levelups)
= 0.658 * (1 + 0.36)

= 0.895

Therefore, Ashe will autoattack 0.895 times per second.




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Attack Speed Reduction


Enemy's_Total_Attack_Speed * (1 - Attack_Speed_Slow)

(It literally takes away a percentage of enemy's total attack speed.)

Example:
When you buy Frozen Heart, enemies with 1.00 attack speed will now have 0.85 attack speed

Note: This can be considered as reducing enemy's autoattack DPS by 15%.






Intermediate League of Legends Formulas


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% Increase in Damage by Armor Pen / Magic Pen


100 + Old Armor
100 + New Armor

Example:
If an enemy has 100 armor and you have 35% armor pen and 15 armor pen,

           100 + 100           
100 + 100* (1-0.35) - 15

= 1.33333

Therefore, you do 33.33% more damage with the armor penetration stats

Concept
You can confirm the above example with this very simple logic.  If you have 100 Attack Damage and autoattack the enemy,

Previously the enemy had 100 armor so he would take only 50 damage

With Armor Pen the enemy now effectively has only 50 armor so he takes 66.66 damage


Therefore, the relative difference is 66.66/50 = 1.3333



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Casting DPS increase by Cooldown Reduction


    1 - Old CDR     
    1 - New CDR


Example:
If you have 0% CDR and you get 40% CDR

   1 - 0       
   1 - 0.40

= 1.66666
Therefore, your casting DPS goes up by 66.66% with 40% CDR.

Concept:
You can confirm the above example with this very simple logic.  
Say there is a spell with 10s cooldown.  In 60 seconds, you can cast it 6 times.
Now with 40% cooldown reduction, the new cooldown of the spell becomes 6s which means you can cast it 10 times in 60 seconds.
Hence, the casting DPS is 10/6 = 1.6666




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% Increase in Durability by Armor Against Only Physical Damage


100 + New Armor
100 + Old Armor

Example:
If you have 50 armor and you purchase Thornmail (100 armor), you now have 150 armor.

100 + 150
100 + 50

= 1.66666

Therefore, you are now 1.6666 times more durable against physical attacks (66.7% tankier against physical attacks)
Or take the reciprocal of that (1/answer) and you are left with 0.60 which means you take 40% less damage relative to when you didn't have Thornmail

Comcept
You can confirm the above example with this very simple logic.  When an enemy champion with 100 Attack Damage autoattacks you,

With 50 armor, you take 66.66 damage
With 150 
armor, you take 40 damage

Therefore, the relative difference is 66.66/40 = 1.6666




Advanced Derivation of League of Legends Formulas


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% Increase in DPS by Critical Strike


1 + NewCritChance * (1 or 1.5)
                       
1 + OldCritChance * (1 or 1.5)

*1.5 with  Infinity Edge only

Example:
When you have 0% critical chance, the critical strike and damage portion of Infinity Edge will increase your DPS by,

1 + 0.20 * (1.5)
    1 + 0 * (1)

= 1.3
Therefore, your DPS has increased by 30% relative to before with only the critical strike stat portion of
Infinity Edge

Concept:
You can confirm the above example with this very simple logic.  When you auto-attack 5 times, you deal

Before: 100, 100, 100, 100, 100  = 500 damage
After:    100, 250, 100, 100, 100  = 650 damage      (since 20% chance of critical striking)

Therefore, the relative difference is 650/500 = 1.3

Additional Notes:
Yasuo receives 10% damage penalty on his critical strikes, which means the formula now changes to
1 + NewCritChance * (0.8 or 1.25)
1 + OldCritChance * (0.8 or 1.25)

Garen's Judgment (E) receives a flat 50% critical damage penalty.  
Convert critical strike stats into bonus damage
by multiplying the respective AD ratio with the modified critical strike bonus
(70/80/90/100/110% AD) * CritChance * (0.5 or 1)
= Bonus Average Damage for
Judgment (E)

Master Yi's Alpha Strike (Q)benefits only 60% from critical strikes.
Convert critical strike stats into bonus damage by multiplying the AD ratio with the modified critical strike bonus
(100% AD) * CritChance * (0.6 or 0.9)
= Bonus Average Damage for
Alpha Strike (Q)




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% Increase in Durability From Armor, Magic Resist, Damage Reduction, and Attack Speed Slow


Old:     (AA*P.Red * AA.Red. ) + (P*P.Red.) + (M*M.Red.) + T                  
New:      
(AA*P.Red * AA.Red. ) + (P*P.Red.) + (M*M.Red.) + T    


Variable Meanings:
AA =  Autoattack damage you receive before armor and other modifiers are taken into account
P = Non-autoattack physical damage you receive before armor is taken into account
M = Magic damage you receive before magic resist is taken into account
T = True damage
AA.Red. = (1 -
Ninja Tabi[0 or 0.10] ) * (1 - %AttackSpeedSlow)
P.Red. = 100 / (100+Armor)
M.Red. =
100 / (100+Magic Resist)


Example:
You used to have 100 Armor and 100 Magic Resist and now you buy
Frozen Heart and Negatron Cloak which give you additional 100 armor, 40 magic resist and 15% attack speed slow.
  • You are autoattacked by a champion with 300 AD 5 times (AA = 1500)
  • You are hit by an ability which does 1000 raw physical damage (P = 1000)
  • You are hit by magic attacks which do 1500 raw magic damage (M = 1500)
  • You are hit by Ignite spell which deals 500 true damage (T = 500)
New: 200 Armor, 140 Magic Resist, and 15% attack speed slow yield the following.
  • AA.Red. = (1 - 0) * (1 - 0.15) = 0.85
  • P.Red. = 100 / (100 + 200) = 0.33333
  • M.Red. = 100 / (100 + 140) = 0.41666
Old: 100 Armor, 100 Magic Resist, and no other autoattack modifiers yield the following.
  • AA.Red. = (1 - 0) * (1 - 0) = 1
  • P.Red. = 100 / (100 + 100) = 0.5
  • M.Red. = 100 / (100 + 100) = 0.5
        (1500 * 0.5 * 1) + (1000 * 0.5) + (1500 * 0.5) + 500                
(1500 * 0.333 * 0.85) + (1000 * 0.333) + (1500 * 0.416666) + 500
       


= 1.328
Therefore, you are now 32.8% tankier with 
Frozen HeartNegatron Cloak for the above conditions
(or take the reciprocal of that which is 0.753 so you take 24.7% less damage relative to before)

Concept:
Given the damage type you can reduce the amount with armor, magic resist, and attack speed slow when applicable.
Then you can compare the difference of the actual damage you receive.





© 2015 League of Legends Math       InvertedComposer