Yes, after thinking about it, I'm going to archive this build so as to not mislead any more people.

Thanks for all the help,and see you guys later after I get the math fixed!

Haha, it's fine I was just kidding, well, I'm glad I could help somehow xD

Yep. Fixed the bug. Now I'm seeing the last whisper in all but the most lifesteal-heavy builds.

Well ****.

Thanks a ton, guys!

Edit: And sorry to ignore you, Wickedeye. Your comments did help, I just didn't respond to them.

+rep in apology. ;)

Edit Edit: Oh, I see what happened, Wickedeye! I didn't see your Edit to your post! Yes, that info about supports is really useful. When I try doing the "suggested build order" portion of my program, I will account for the possibility of the support rushing thornmail. Thanks for the heads-up!

Felt really ignored back there, but ok. You can apply to be a statistician or analyst for Riot, but I guess you would need experience and it also depends on where you live, because California seems to be the center of game analysts/staticians.

Aw, freak. After looking into it, you're completely right about the math.

It's an order of operations problem.

My math:

attackSpeedFromLevels = (ASHE_ATTACK_SPEED_PER_LEVEL*level-1)

What my math should have been:

attackSpeedFromLevels = (ASHE_ATTACK_SPEED_PER_LEVEL*(level-1))

Order of operations bit me in the butt. Hah! A grad student, still making lowly coding errors based on high-school algebra math mistakes. Do not follow my example, Emi-kun! It will kill your AP stats test!

So... all the math on my guide is now wrong. What do you think I should do? Archive, then Copy/Paste and try again come the 4.11 patch?

I'm not going as in depth as the others but not mentioning LW in your standard build is a massive mistake. You cannot take every factor into account as long as you don't combine the mathematical analysis with gameplay related knowledge.

Firstly, I don't know where you got the attack/second stat from but with PD + Zerker boots gives you about 1.3-1.5 attack speed (not sure about the exact amount). However, you've got to imagine that attack speed only gets used marginally due to loss through kiting, and that raw AD (and thus armor pen as well) also amplifies most of your spells.

I checked the math myself, assuming a base AD at level 18 of 101 and base AS of 1.05 (caitlyn base). Not taking runes/masteries into account for simplicity's sake.

Standard LCS build:
80+80+40 AD (bt, ie, lw) = +200 ad -> 301 AD
25+50% AS modifier = +75% as -> 1.8375 as (not sure if I did the AS calculation well)
35% armor pen
55% crit
So the dmg before armor reductions is 548 on average, attack speed of 1.8375 -> 1007 dmg.
Assuming 150 armor, 0.4 damage multiplier, after armor pen 0.5
so that's roughly 500 damage per second

Your build:
80+80 AD = +160 ad -> 261 AD
25+50+50 as modifier = +125 -> 2.3625 as
85% crit
So the dmg before armor reduction is 594 on average, attack speed of 2.3 -> 1365 dps
Assuming 150 armor = 545 dps

So that means without taking spells or kiting into account, the DPS are already very close. You also forget to take blade of the ruined king into account which is a very common first item but getting PD on top of that would mean hitting the AS cap and some of its effectiveness going to waste.

That's not to mention most bruisers in the game will have OVER 150 armor which brings it even closer, and that LW is cheaper overall.

Well considering the effort you've put into this guide, I'll give you my upvote :) Like Wicked-chan said though, you should turn on Comment to Vote. This will prevent any people from downvoting your guide for no reason whatsoever. It'll essentially force them to give feedback until the 20th vote, and by then your guide will have an accurate standing amount the others.

Just one small thing though: For the build at the top of the guide, I'd switch the Barrier to Heal and add Last Whisper instead of the double life steal option. Ashe is long ranged, so the short range active on BotRK isn't all that useful.

To be honest, I would go ahead and just use one mastery page. A lot of masteries seem like really good "in theory", like running 30/0/0 masteries, but in the actual game have a lot of flaws. Some individual choices might be debatable, such as choosing between Butcher or Double-Edged Sword , but for the most part the page is the same.

I feel like 200 armor is still kinda low, but maybe that's just me, haha. Like I said in my last post, it might be useful to look through some builds to reduce any bias one might have and have a better understanding of how tanky tanks really are. Also, you have to keep in mind that you're calculating a full build, level 18 ADC. This means that you also have to take into account a full build, level 18 Tank. I think you said you're using your wife as an example with her Udyr jungle, but 9 times out of 10, you won't get to complete 6 item build.

(Again, I might be a little biased here) Runaan's Hurricane isn't all that great of an item due to the awkward component items and it giving nothing but pure, unadulterated attack speed. Mathematically, it might seem really good, but in the actual game its lack of other stats make it a poor choice to buy.

And it's no problem ^_^ Ashe was my first champion and to this day is one of my favorite ADCs. It's wonderful to know she's getting some love and appreciation in this guide!

@Emi-chan

These are some good points. Thanks for the help!

I am not factoring in Runes/masteries. This is mostly because there are slightly more than 600,000 different combinations of Physical Attack quints and marks, so if I test each build with each combination of runes, my program will run about 600,000 times slower. And right now, it takes about five minutes to run (and checks literally millions of different item combinations!) I could perhaps find clever ways to "prune" it and short circuit combinations that are completely non-viable (Like Crit Chance runes on any build with 100% crit chance...)

So, here's my plan for next update (Which I want to coincide with the 4.11 patch, because I'll have to rerun my numbers anyway...)

• Assume a "base" of 200 armor instead of 150, because that seems to better represent "average" armor on the enemy team.
• Instead of looking at Damage per second, I will consider "Damage per ten seconds." I will assume you get off five seconds worth of auto-attacks, the first of which is an auto-crit. You will also fire two volleys. (or three if the build contains enough CDR). Youmuu's Ghostblade will only be active for three of the five seconds of auto attacks. (Making it less powerful than the "always on" I assume now.)
• I realized recently that the statikk shiv builds more quickly if you have a Ruunan's Hurricane. I didn't account for that. If you have both, I'll assume the shiv activates once every four shots instead of once every ten.
• I'm working on a way to try and calculate optimal build order to maximize Damage integrated over time. That would be awesome to have.
• If I can't find some way to optimize runes, I'll probably just take the runes/masteries of JhoiJhoi's Ashes to Ashes guide and assume you have those on. It shouldn't affect the builds too much, but for the sake of completeness, I think I should include them anyway.

Thanks for all your feedback, Emi-Chan!

Also, if Counter-Logic Gaming has a statistician, I'd love to meet him and ask him how he got his job. I'd love to crunch numbers for games all day. I'm in the process of getting a Masters in Computer Science (In fact, I should probably be working on my thesis right now... -_- ) and I think that job would be really sweet.

Heh, well I'm not quite as experienced in this game as most other people here on the site is (Silver II and been playing just over a year actually ^_^) so some of my information may or may not be completely accurate. I'll try and answer your questions to the best of my ability though :)

I'm just a high school student, so I'm really don't know what to tell you what you could add to make your statistics more accurate (I am taking AP Probability and Statistics next year though! And I'm only 16 :D). I only know from experience that in teamfights, 90% of the time I won't be able to get all of my AAs off like your guide suggests. One thing that I did indeed notice you left out was stats from runes and masteries. They can give you around 20 extra AD, which can give that edge to damage dealt. In addition, masteries like Frenzy , Executioner , and Havoc can allow you to do more as well. Was that factored in already?

I think you mentioned somewhere in the guide that you left out champion abilities as well. In this case, since it's Ashe, you would be leaving out her W, which you can weave in between autoattacks to deal additional damage. This damage benefits from pure armor pen, which is part of the reason why items like Last Whisper outshine a second Phantom Dancer.

Regarding your points as to why LW isn't matching with the calculations:

• Again I'm not all that experienced, but from what I've seen in my games, at level 18 most tanks have about 250-300 armor. I mean armor without any enhancements from abilities, when they're just walking around creating map pressure. Added on with additional things like you mentioned (Soraka's Astral Blessing), it could easily surpass it. If you want a more accurate look at the average armor for a tank, I'd recommend looking around on some pro builds and calculate how much they have in their example full builds. It should give you a less biased look on how tanks are played :)
• I would honestly say it's something in between "poking" and "shredding". From what I understand, if the tank is constantly in your attack range, then you constantly attack him and attempt to "shred" his/her health. If he/she is not in your range, then you try and get in range to deal damage while maintaining a safe distance yourself, hence it is more like poking.
• That is indeed part of why people choose to build their champions the way they do. Since LoL only has a limited amount of item slots, you want to increase the efficiency of the 6 you are provided (7, if you include the trinkets). Since you want to make your item slots are useful as possible, you want your items themselves to be worth the money you are spending. If you can get an item that's cheaper, and deals almost as much (if not more) damage than another item, why not get it?
• Like Wicked-chan (yes you may add the -chan ending ^_^) said and reasoned, the pros are usually the basis for what we build. Not only do they have billion dollar companies sponsoring them, they often have the most experience in the game itself (a lot of them have been playing since Season 1 or 2, so for several years now). They know the game mechanics and how everything runs in teamfights/individual skirmishes way better than the average person like you or me do, and their judgement is usually based on said experience.

I really like this guide because although I'm not really good with math, I like seeing the math behind common things like the games we play only for fun. Also, I want to suggest you to turn up the Comment2Vote option because some people might come here, don't like what they see and just downvote without a reason.

Also, you said something aimed for Emi-chan (sorry if that's too intimate, but your name just asks for a chan at the end y.y), but I will take the liberty and say what I think. Doublelift plays for Counter Logic Gaming, these e-sports teams are from famous companies that invest their precious money onto these teams, so they have analysts, coaches and everything to maximize their teams performance during a game. So my logic, and the logic I think other people follow is that the pros play the way analysts think is the better way at the current patch/meta, thus, they try to follow the same thing.

Of course I might be wrong, but that's just the way I see it.

EDIT: I play most at the support role because I really like it, so I have to protect myself from the enemy ADC, when he's getting fed, the first thing I'll build is Thornmail and if he's still a problem I'll go for Frozen Heart or Randuin's Omen, and I absolutely love seeing they be like: OMG, I can't deal any damage to this support, simply because they don't have armor penetration. So I think Last Whisper is a must, specially if you're becoming the main carry, since people will stock up armor and you'll have nothing to counter that. Thornmail can totally ruin an ADC without armor pen, was my favorite item on low elos where ADCs constantly would forget to build Last Whisper and go 3x Bloodthirsters xD.

@Emikadon: I'm grateful you see my guide for what it is; an attempt to mathematically determine the best builds for Ashe that is obviously myopic on some respects. You seem like a thoughtful, articulate, and educated person with a lot of reputation points, so perhaps I could ask you a few questions?

First, is there anything you think would be more useful than DPS to run my numbers on? I admit that DPS seems to only apply when people are "all in" duelling you, like when you accidentally run into the enemy jungler while passing through your jungle. When you kite, attack speed helps, but not as much as the numbers show. In a teamfight situation, DPS is more useful than when kiting, but still not as good as the multiplier would seem. How would you recommend I adjust my core assumptions to more accurately model an in-game situation?

Second, I like your comment in tiny text about the last whisper. You'll notice that I stop just short of recommending the second phantom dancer. This is because I have seen many professional players and high elo players swear by the last whisper. I am familiar with the correlation between elo and LW purchases. Thing is, mathematically, I can't quite get it to stack up.

I can see perhaps four reasons for this:

• The first is that perhaps my assumption of enemy armor is wrong. Mathematically, in a four item build, the last whisper becomes a better pick when your opponent has 259 armor. My wife's "Tanky Udyr" build reaches 270 armor, making last whisper just barely superior. But Udyr has no natural armor steroid. If I were fighting a Rammus, or Udyr had an Astral Blessing or something, the armor would be much higher and a Last whisper would be the obvious pick. Am I just underestimating the armor of tanks because of my over-experience with Udyr?
  
• Second, you have a point that attack speed is only as good as my formula indicates if you're constantly attacking. I tend to do this against enemy tanks if I'm not in danger of getting bursted by their APC. I steal enough life from the tank that I can pretty much ignore whatever damage they do. But perhaps in higher elo play, ADCs "poke" tanks instead of "shredding" tanks. Is my "constant fire assumption" incorrect?
  
• Third, last whisper is cheaper than a phantom dancer by about 500 gold. If you assume your opponents have about 259 armor, making phantom dancer and last whisper equal, then you might as well buy the cheaper item so you can get to full build faster. Is that why people prefer it?
  
• Finally, there is a possibility that double-lift's... er... emphatic endorsement on the Last Whisper makes everybody afraid to question it's supposed supremacy. I've noticed that the League community often tends to parrot what high elo players do, who in turn parrot what the pros do. But I wonder if Doublelift's opinion is based on math or on anecdotal experience? Does Counter-Logic Gaming have a statistician on staff to assist in item builds? It would be nice to have a realistic conversation about the pros and cons of the last whisper without it turning into a witch-hunt. (I'm allergic to fire; it gives me large red rashes.)

Thanks again for your comment. I hope to make this guide more useful in the future based on reader suggestion. These numbers should reflect your games as much as they can.