I was asked a question on the EUW General Discussion forums the other day to look at the DPS difference between building Last Whisper after purchasing Zeal or completing Phantom Dancer:

“Out of personal interest, I would like to see a comparison between building
BT->Zeal->LW |x| ->PD
and
BT->PD->Longsword |x| ->LW
at the |x| spot.”

It’s a great question and we can answer it easily, while also discussing Crit. I’ve added the colour coding again, like I did in the previous article, so it should be easier to follow.

Building Our Marksmen

I’m going to look at Ashe because she has no Attack Damage or Attack Speed steroids in her kit and I can ignore her passive.

I’m going to assume she’s got Attack Damage Marks and Quints (15AD) and that she builds the classic 21/9/0 mastery set that gives us 0.67 AD/level from deadliness, 8% Armour Pen from Weapon Expertise, 3 AD from Brute Force and 5 Flat Armour Pen from Sunder.

I’m going to ignore Havoc (2% increased damage) and Executioner (5% Increased damage below 50% hp) because I’m not interested in making this more complex than it needs to be and having the numbers jump around. Crit has to be set to average additional damage because this is an average damage test and it’s easier to calculate that than adding in crits at random times. Also, I take the 4% CDR mastery rather than the 4% Attack Speed so i’ll do the same with this build.

Finally we’ll just look at auto attacks because that applies to every Marksmen and you can then use common sense to decide on how the damage changes with whatever champion you are playing. For example, AD is more favourable on characters with AS steroids and abilities that scale from AD (like MF), while AS is more favourable on champions with on hit damage and AD steroids (like Vayne).

Base Build

The base build we’ll be using is one Doran’s Blade, Berserker’s Greaves and a fully stacked Bloodthirster.

Then we’ll modify it with a Zeal and a Last Whisper (Build 1) or a Phantom Dancer and a Longsword (Build 2).

Build 1

Build 2

So let’s look at the important stats for our attacking Ashe at level 11, a time when she could concievably grab the base build and then look to expanding the item set:

So the big differences we can see are that Build 1 has more AD and more Percentage Armour Pen, while Build 2 has more Crit and Attack Speed. If you go back to my old article on auto attacking, you can see many of the formulas I list here. They’re explained in a lot more detail there, but I’ll give a quick explanation as I’m going through.

Damage per Second:

Base Build:

DPS = AD x AS
DPS = 213 x 1.079
DPS = 229.8

Build 1:

DPS = AD x AS
DPS = 253 x 1.198
DPS = 303.1

Build 2:

DPS = AD x AS
DPS = 223 x 1.408
DPS = 314.0

So the first thing we can see from this is that Build 2 does more damage than Build 1 if we just factor in Attack Damage and Attack Speed. This is ignoring Crit, Armour, and Armour Pen entirely, but will be a relevant calculation when we move on to the next stage.

In order to work out True Damage Per Second things get complicated, but I’ll break the equation down into simple steps:

Step 1:

Calculate the Damage Per Second.

It’s easy:  multiply how hard you hit with how frequently you hit. We’ve done that above.

Step 2:

Calculate the contribution of Crit Chance and Crit Damage.

Well this is fairly simple again. Because we’re averaging it out, you just allow DPS to be multiplied by ((0.Crit Chance) x (1.Additional Crit Damage))). Now this may seem complicated, but all we’re saying is that on average every 0.Crit Chance (e.g. 0.2 for 20% Crit), this attack is going to do additional Crit Damage worth of damage. Everyone starts with +100% Crit Damage, for a 200% damage attack, and we’ve got an additional 10% from masteries. So the additional attack does an extra 110% damage.

Crit Calculation:

TDPS = (DPS) x ( 1 + (( CritChance / 100 ) x Crit Dam )
TDPS = (DPS) x ( 1 + (( CritChance/ 100 ) x 1.1)

Step 3:

Add the contribution from Armour.

Armour is factored in by calculating the Damage Multiplier and will be a number below 1 for all positive values for Armour  (read: not zero or below zero).

Damage Multiplier Calculation:

100 / (100 + Armour)

If we add this into our equation from above:

TDPS =  (DPS ) x (1 + (( CritChance / 100 ) x Crit Dam )  x ( 100 / (100 + Armour ))

Step 4:

Factor in Armour Pen.

Armour Pen comes in two forms:  Percentage and Flat. Percentage is applied first and then Flat after. Penetration can never drop Armour below zero.

This is done by modifying the Damage Multiplier as follows:

Factoring in Armour Pen:

100 / (100 + ( 1 – ( %Pen / 100 )) + Armour – FlatPen)

So let’s add this into the overall equation to get what we need to do the DPS calculations:

TDPS = (DPS) x (1+(CritChance/100) x (1+Crit Dam/100)) x (100/(100+(1-(%Pen/100))+Armour–FlatPen ))

It’s quite the equation, but hopefully it’s not too intimidating given how easy it is to break down into little segments.

The Results

If we plug the numbers from the different builds listed above, we get the following table:

First, against a zero Armour target, we can see that Build 2 is more powerful and this is because all of the Armour Pen is completely wasted in Build 1 as only Armour reduction can reduce Armor below zero. So if the target has zero Armour, you shouldn’t really be buying Last Whisper, but I hope that would be obvious.

At 50 Armour, Build 2 does more damage than Build 1, so if you’re able to get on to the squishies at the back line they will die quicker with PD than with LW. Around level 11 about half of the characters in the game have less than 50 base Armour, while the other half have more. This will be adjusted by masteries, runes, and any items purchased.

At 100 Armour, which is a value most tanky characters could easily attain at this level, is where it starts to get interesting. Build 2 only does a small amount more damage and the DPS numbers equal out at around 125 Armour. So by 100 Armour you could probably still build Phantom Dancer first, but above 125, Last Whisper is stronger.

At the higher Armour levels it starts to swing much further into Last Whisper’s favour.

So, returning to the question above, the answer seems to be:

For auto attacks alone, a champion will do more damage buying LW when the target has 125 Armour or more.

I should add an addendum here. PD provides Attack Speed and Crit Chance, both of which do nothing for abilities unless they can crit, while LW will increase the damage done to anything so long as it does physical damage, namely most of a Marksman’s moves. So if your carry has a large number of physical damage moves, LW will be better below 125 Armour.

Additionally, if your champion has an Attack Speed steroid then this makes LW even stronger because it provides AD and an increase in damage through Armour Pen. As a result, if the champion you are playing has an AS steroid, LW will be better than PD lower than 125 Armour. To use a quick example, an Attack Speed steroid of 20% will decrease the point at which Build 1 beats Build 2 down to 110 Armour rather than 125.

It is more efficient, when increasing your DPS, to buy the stat you have the least of because it is a multiplicative sum.

To use a simple example:

If you have the sum 100 x 1 and it costs the same amount to increase either side of the equation by 1 then 100 x 2 will yield a significantly higher value than 101 x 1.

This math can also be applied to tanks. If you’re playing a tanky initiator champion and you see that the enemy ADC or top laner has rushed an early LW then grabbing more HP rather than Armour would be stronger. I find at my ELO that ADCs tend not to build Last Whisper until quite late so stacking up on the Armour early can make you very difficult to kill, provided the rest of your team can do the damage for you.

What if you buy IE instead of BT?

This  is where we start to get into the realm of interesting possibilities. What if you build IE first instead of BT?

Well the build is 600g more expensive at this point across all of the builds, so the DPS numbers will be higher.

Stats are shown below:

Now we can see that the Crit Chance is higher on all the builds but the natural AD is lower. IE’s passive (+50% Crit Damage) will favour builds with higher Crit Chance (i.e. Build 2).

In terms of DPS, which again is ignoring Crit and Armour Pen, all 3 builds are lower than the Bloodthirster build, but that’s because IE’s passive is a larger part of its gold cost and has less AD.

Looking at the TDPS numbers:

TDPS is higher by purchasing IE before BT, with Build 1 gaining 80 TDPS, from 337 up to 417, and Build 2 gaining almost 100 TDPS against targets with near zero Armour levels. This makes sense because more of Infinity Edge’s gold budget is taken up by offensive stats rather than Life Steal in addition to it costing more.

The base build doesn’t gain such a huge increase in damage over BT. As soon as you add Zeal, and then PD later on, you get a huge spike in damage because of the scaling of Attack Damage with both Attack Speed and Crit, on top of IE’s +Crit Damage passive.

Back to the comparison, we can see that the point at which the new Build 1 beats Build 2 is between 100 and 200 Armour, much like if we built BT first. The interesting thing is that it actually brings the point at which Build 1 beats Build 2 down from 125 Armour to around 110, despite IE’s passive complimenting PD.

But why is this?

Well, Armour Pen is such a huge damage multiplier and is one of the main reasons characters like Zed or Kha’zix get so much damage from building it. So although it would seem like the +Crit Damage passive would inflate Build 2’s numbers, the extra damage gained actually benefits the LW build more.

So for champions who have under 110 Armour building PD first will kill them quickest, while above 110 Armour Last Whisper after Zeal will net you the earliest kill.

Again, this number will decrease as your DPS increases and whether your character has additional steroids and/or non-Crit Damage scaling (from abilities or items).

Bonus Analysis: LW Straight After BT

Build 5 costs 2300g more than the base build while Build 6 will cost you an additional 2050g, so there is a little discrepancy in the gold cost. However, this analysis is merely to illustrate how damage will change depending on the item built and whether LW can replace Zeal as the second item purchased.

TDPS table:

From this we can see that Build 6 is stronger than rushing LW as a second item at low Armour levels because of how little Attack Speed we have in Build 5 relative to the amount of AD we have. If you expect team fights soon, LW will be stronger than Zeal; however, if you’re just expecting to skirmish with a squishy target then Zeal is more powerful.

Last Whisper is a strong second item but will be out damaged by Zeal + Pickaxe at low Armour levels.

If we exchange Ashe for Graves or MF, they actually do more damage to targets with 50 Armour using Build 5 during their attack speed steroid. So if you can keep this up often or have Nunu support for example, then purchasing LW as a second item is probably one of the strongest items you can buy for damage.

Conclusion

LW is an effective item for increasing Marksmen damage even at low Armour and damage levels. Against targets with 100+ Armour, LW is one or the most effective items for increasing your mid game damage, even more so if you’re champion has an Attack Speed steroid. As such I would recommend LW as a second item on champions like Graves, Tristana, or MF even if you have purchased IE first over BT.

LW is probably still the best second item on most other Marksmen as they tend to have some kind of AD scaling on their abilities and will benefit from the Armour Pen as long as the targets you are fighting have around 75 Armour. If the laning phase is going on for a long time and you’re likely to only be attacking other squishy targets then Zeal second will give you a larger return for your gold.

If there are any external Attack Speed sources that can be reliably used on your champion, such as Nunu’s Blood Boil, Jarvan’s flag, or Warwick’s Hunter’s Call, then LW will give you more damage as a second item over Zeal/PD/SS.

The value of Last Whisper will increase as DPS increases so should always be part of a Marksmen build, and in most cases, it is better if you buy it as a 2nd or 3rd item rather than a 5th or 6th.