Despite this post being quite heavy on the maths I’ve gone through it in easy steps and included an example along the way to hopefully make this more understandable.

A big thank you to Lolwiki for the help with this article.

How are we going to do this? Well let’s start from the beginning and work through an example so the later stuff is easy to understand. I’ve attempted to colour code the article a bit so it creates a logical path that can be easily followed. If you’ve got any questions feel free to contact me on twitter.

Glossary and Definition:

• AD – Attack Damage – Attack Damage stat of a champion
• AS – Attack Speed – How Frequently Auto Attacks occur
• DPS – Damager Per Second – The amount of damage a champion can do every second before mitigation is applied, ie the damage is reduced by armour/magic resistance and then taken off the health pool
• TDPS – True Damage Per Second – The amount of damage a champion can do every second after mitigation, ie it’s the amount of damage taken off a targets health pool
• EHp – Effective Health – How much of a given type of damage, physical or magic, a character can take before dying
• PEHp – Physical Effective Health – How much physical damage a character can take before dying
• Gold Efficiency – How much gold the item is worth relative to the number of stats it gives

Example, Ashe attacking Singed

Attack Damage (AD) is the amount of damage you do to a target with a basic attack when you right click on a champion. Some skills also scale with attack damage, or bonus attack damage, which will increase the damage done by the spell when you use it to hit a target. This scaling will be represented by an orange number in game.

Our Ashe has 200AD, so every hit against a target has the potential to do 200 damage. This damage potential will be lowered by a targets armour..

Attack Speed (AS) is the frequency with which you are able to trigger your basic attack. It is represented by a number between 0 and 2.5, which is the maximum attack speed you can have. The number represents the number of attacks you get per second.

Our Ashe has an attack speed of 1.7 so she gets 1.7 attacks every second or 1 attack every 0.59 seconds.

Calculating Damage Per Second

If we multiply AD by AS we get the amount of damage you do every second, which i’m calling (cleverly) Damage per Second (DPS).

Damage Per Second = AD x AS
DPS= 200 x 1.7
DPS = 340

Working out the effect Armour has is a more complex calculation because armour provides a % based decrease in damage taken that slows towards 100% damage reduction but never gets there. This means that armour alone can NEVER reduce damage done to zero no matter how much armour you have.

What this DOESN’T mean is that armour is less valuable the more you have because 1 armour provides the same increase in Effective Health (EHp) regardless of how much armour you have, i’ll discuss this later on.

Calculating Damage Reduction through Armour

(100 / (100 + Armour)) = Damage multiplier

Example of Singed with 100 Armour

100/(100+100) = Damage multiplier
100/200 = Damage multiplier
0.5 = Damage multiplier
50% damage reduction

 So for 100 armour, you get 50% damage reduction against physical damage.

Back to DPS.

To calculate the actual damage you take from a hit, you factor in the amount of AD you have and the targets damage reduction from Armour.

Damage Taken

Damage taken = AD x Damage Multiplier
Damage Taken = AD x (100 / (100+Armour))

In order to calculate the True Damage Per Second (TDPS) we have to include how frequently the attacks occur and this factors in Attack Speed.

Calculating TDPS

True Damage Per Second = Damage Per Second x Damage Multiplier
TDPS = (AD x AS) x (100 / (100+Armour))

Now this might sound complicated but all it states is that:

The Damage you do every Second is equal the amount of damage your attacks do, how frequently you can do them and how much the enemy character’s armour reduces this damage.

Let’s add in some numbers here from our example:

Ashe has 200 AD and 1.7 Attack Speed while Singed has 100 Armour.

Calculating Ashe’s Damage vs Singed

TDPS = DPS x Damage Multiplier
TDPS = (AD x AS) x (100 / (100+Armour))
TDPS = (200 x 1.7) x (100 / (100+100))
TDPS = (340) x (0.5)
TDPS = 170

So Ashe does 170 tue damage per second against Singed.

Now it might not seem much to start with but that’s because Armour is such a huge deterrent against physical damage! Only 100 Armour has cut our Ashe’s DPS in half.

Let’s make Singed even tankier and have him pop his ultimate, which gives him an additional 65 armour for 25 seconds:

Calculating Ashe’s Damage vs Singed with his ult up

TDPS = DPS x Damage Multiplier
TDPS = (AD x AS) x (100 / (100+Armour))
TDPS = (200 x 1.7) x (100 / (100+165))
TDPS = (340) x (0.38)
TDPS = 129.2

So by popping his ultimate Singed is able to reduce Ashe’s TDPS from 170 down to 129.2 which is a drop of 31%!

So by using his ultimate our Singed has increased the time it takes Ashe to kill him using auto attacks by almost one quarter and this is how Effective Health is different to Health, because it factors in the enemy’s resistance values. I’ll pick up on effective health in a bit but let’s stick with TDPS for now.

If we assume Singed has 3000 Health it will take our Ashe 17.64 seconds to kill him at 100 armour and 23.2 seconds to kill him with his ult up, the gain of 31% we mentioned before. So he’s gained 6 extra seconds of Ashe’s constant damage just by using his ult and this is because of his increased damage reduction from armour.

Now of course we’re assuming that Ashe is able to fully damage Singed everytime her auto attack is ready but we’re also assuming that she isn’t using her Volley, critting or has items like Blade of the Ruined King. But let’s just keep it simple for this article.

Effective Health

Effective Health is a great way for characters to investigate how tanky they are against a certain kind of damage. Effective Health takes into account both Health and Armour or Magic Resistance to give the actual amount of damage you can take of that type. So Physical Effective Health (PEHp) is the amount of damage it takes for physical damage to kill you.

PEHP calculation

Physical Effective Health = Max Health x (1 + (Armour/100)

Singed has 3000hp in the two differing armour cases above, so we can see his Physical Effective Healths are:

Physical Effective Health = Max Health x (1 + (Armour/100)
PEHp = 3000 x (1 + (100/100)
PEHp = 6000
Physical Effective Health with ult up = Max Health x (1 + (165/100)
PEHp with ult up = 7950

The limitations of Effective health is that it varies depending on the kind of damage being dealt to you so Physical EHp isn’t the same as Magical EHp and True Damage will bypass resistances and just adjust your raw Health. Against a character, or a team, who deal a varied kind of damage then effective health falls down a little because its hard to calculate your tankiness relative to the damage you might recieve. If you build a high Magical EHp and then only get focused by the carry you’re more or less wasted all of the MR you purchased.

Although this post has been about single target attack damage from basic attacks, the same maths can be applied to spells that deal physical damage like Talon’s Rake, Riven’s Broken Wings or even Ashe’s Volley. If you want to calculate it for spell damage then just replace Armour with Magic Resistance and you’ll be calculating Magical Effective Health rather than Physical Effective Health.

Additionally you can do the same for calculating the DPS you do, change AD to the amount of damage the spell does + the AP

Building Last Whisper

17.64 seconds is a long time to spend attacking a tank and will probably get your team killed before you have any hope of killing Singed, so let’s equip Ashe with a Last Whisper and see what happens. In order to do a comparison calculation we’re not going to increase Ashe’s AD above 200 because I think investigating the value of the armour pen is a more interesting discussion than an increase in attack damage. Though if you want to you can use the equations listed above.

Armour Penetration is quite a complex calculation but let’s look at reducing armour as a whole first.

Armour can be decreased in 4 ways and they are applied to the armour in a certain order.

1. Flat Armour Reduction
2. Percentage Armour Reduction
3. Percentage Armour Penetration
4. Flat Armour Penetration

So any Flat Armour Reduction you have is applied first, then Percentage Reduction, then Percentage Penetration and finally Flat Penetration. Flat Armour Reduction and Flat Penetration stacks additively so if your target has 25 armour and have 10 Reduction and 10 Penetration then he’ll lose 20 armour in total down to 5 armour. Percentage Reduction and Penetration stack multiplicatively so you lose a % based on the penetration amount you have.

For our example we’re looking at Percentage Armour Penetration from Last whisper and the Percentage Penetration from masteries.

TDPS calculations using Armour Pen

TDPS = DPS x ( 100 / ( 100 + ( 1–%Pen ) x Armour – FlatPen ))

It might look complicated but the second part of the formula is just the Damage Reduction through Armour equation adjusted to allow penetration to factor in. So its Damage Per Second multiplied by the Damage Multiplier but with Armour Penetration factored in.

Armour Reduction can also be factored in but because the only difference between Armour Reduction and Armour Penetration in the game is that reduction can drop armour below zero while penetration can’t. So for the purposes of this Percentage Penetration and Percentage Reduction are the same thing because we’re not reducing armour below zero.

Applying this to the example

Ashe’s previous TDPS was 170 against a singed with no ult and 124.9 against a Singed popping his ult. Let’s look how Last Whisper changes this. Lets look at Ashe running the 8% Penetration from Masteries: (She has no Flat pen at this point)

Ashe’s Damage vs Singed with 8% Pen mastery

TDPS = DPS x ( 100 / ( 100 + ( 1-%Pen ) x Armour – FlatPen))
TDPS = (AD x AS) x ( 100 / ( 100 + (1-0.08) x 100 – 0))
TDPS = (200 x 1.7) x ( 100 / ( 100 + (1-0.08) x 100))
TDPS = (340) x 0.52
TDPS = 176.8

Before Singed pop’s his ult the 8% penetration gives Ashe 6.8 more TDPS or a3.5% increase in damage done.

Let’s add Last Whisper’s 35% extra penetration into the mix: (Because pen stacks multiplicatively, we need to multiply the reduction % as an amount of what’s left. So for 35% and 8% we multiply 0.65 (1-0.35) and 0.92 (1-0.08) respectively to get 0.598. This means we have 40.2% ((1-0.598)*100) penetration rather than 43%.

Ashe’s damage vs Singed with Last Whisper + 8% Pen mastery

TDPS = DPS x ( 100 / ( 100 + ( 1-%Pen ) x Armour – FlatPen ))
TDPS = (AD x AS) x ( 100 / ( 100 + (1–0.402) x 100 – 0))
TDPS = (200 x 1.7) x ( 100 / ( 100 + (1-0.402) x 100))
TDPS = (340) x 0.63
TDPS = 212.8

Adding a Last Whisper increases the TDPS up to 212.8 giving us an increase in damage of 25%!

So by adding the penetration from masteries and Last Whisper Singed will die a whole 25% quicker. Now I know what you’re thinking, If Singed popped his ult before and it gave us a 24% drop in DPS buying Last Whisper just cancels out his ult right?

Well no, due to Last Whisper being a % based drop it actually works better at higher armour values. 35% of 100 is 35 but 35% of 300 is 105! So by popping his ult Singed is actually making your Last Whisper more valuable to you or, as we should probably put it, our Last Whisper is devaluing the stats from his ult because he gets 35% less armour. Let’s run the numbers:

Ashe vs Singed with his ult up

TDPS = DPS x ( 100 / ( 100 + ( 1-%Pen ) x Armour – FlatPen ))
TDPS = (AD x AS) x ( 100 / ( 100 + (1-0.402) x 165 – 0))
TDPS = (200 x 1.7) x ( 100 / ( 100 + (1-0.402) x 165))
TDPS = (340) x 0.503
TDPS = 171.1

So we do 171.1 DPS, an increase from 129.2, which is an increase of 32.4% TDPS compared to 25% when he had 100 armour.

If we compare this to the pre ult increase, which was 25%, we can see that our armour pen has given us a 7% damage increase against this tankier target for free. So LW will scale as the enemy gets tankier and this is the same with items like BotRK and Liandry’s Torment, who’s damage increases as they gain more HP.

When should we buy Last Whisper?

Well you see a benefit from Last Whisper as long as the enemy has Armour to penetrate but as a first item it probably isn’t the best. If we ignore the additional AD given by Last Whisper and just look at the passive then we can see that Last Whisper gives a percentage increase in your DPS depending on how much Armour the opponent has.

Let’s assume you’re playing a level 18 Ashe with Doran’s Blade, Bloodthirster, Beserker’s Greaves and Phantom Dancer at level 18: (Your AD would be ~236 and your AS ~1.5)

Table 1, How Ashe’s DPS changes with different armour values and with the addition of LW/8%pen mastery

DPS represents the raw damage done premitigation, so is independant of the armour value. TDPS represents the damage done by the build with 8% armour pen from masteries. LW+8% Pen TDPS column represents adding the LW and mastery armour penetration into the build.

The % increase is listed because it shows that the increase is a set value depending on the armour value of the target. So against a target with 200 armour you’ll always do 30% more damage with LW it just depends on the initial value of DPS as to whether you should get penetration items or not.

If the % increase in damage against a target would account for more damage than other items, then you should buy Last Whisper.

Let us look at a quick example:

Another Bloodthirster vs Last Whisper

This is a simple example because Bloodthirster provides a flat increase in damage because you get more AD and no damage multipliers like Crit, % health damage or Armour Pen. Let us assume that the Bloodthirster is fully stacked so it gives you 100AD. Let us then remove 40AD, the amount provided by Last Whisper, and this then allows a direct comparison between the damage provided by BT and the damage provide by LW.

To make the gold analysis a little fairer we’ll add a Vamp Sceptre to the Last Whisper Build, or take another 10AD off the Bloodthirster DPS numbers.

So if we suddenly give our character about 50AD more, representing buying a BT rather than getting a LW, their DPS will increase to 405 and give us the following table:

Table 2, How Ashe’s TDPS changes with the addition of 50AD.

What we’re comparing this to is the LW+8% Pen DPS column from table 1. If the Post Pen DPS gives us a higher value than the DPS value from this table then it is better to buy LW over Bloodthirster.

The values for TDPS are identical are higher at 100 armour, so around 100 armour you want LW, below you want the extra BT

The armour value at which LW will beat BT will decrease as your DPS increases because we’re dealing with a % increase in damage rather than a flat increase from AD. If you don’t have Last Whisper as part of your full item build on an AD carry then I think you may need to reassess your build.

LW also allows any physical damage you do to scale, so if you’ve built BotRK or have abilities that deal physical damage then the armour value at which LW does more damage than BT will be even lower.

Additionally in terms of non-AD carry damage dealers at end game Last Whisper may be one of your biggest damage increases.

IMPORTANTLY This is NOT a strict gold efficiency analysis and would further push the DPS numbers towards LW is we added a pickaxe rather than a Vamp sceptre.

Thanks for reading. In a future article i’ll add in discussions related to crit and look at how you might want to adjust how you build AD Carries.