I'm pretty sure we're wasting EVs left, right and center here (Defence: Uncovered)

Pikachu of Doom

Section 1: Block Damage

While I was messing around trying to write a formula to work out the lowest attack value needed to OHKO a pokemon I noticed that in manual testing it often took 3-5 stat increatments to do one extra damage, obviously because of flooring the variables. The number of stats per damage gain goes up with Defense while going down with Attack and power, this means it is very important for pokemon like Salamence with fairly good defenses who's opponents rely on low power moves with 4 times effectiveness which means Salamence might need 20-30 Def IVs for each 4 times 'block' of damage, Ev's that could be invested in HP for a lower percentage damage.

Obviously this doesn't affect much apart from really specialised ev spreads at the moment and it would be difficult to find a universal value for each pokemon in its current state. However, since there is a regularity in base stats as well as base powers it might just be possible to work out much better ev spreads for certain pokemon (this can include attacking stats as well).
Click to expand...
This was a PM thats been sitting in my sent messages for a while (to Dragontamer if you're interested) unfortunately it was never replied to.

Unlike what's implied in the PM, each block of damage can be much more significant than 4 damage for every stat in defence... or much less. Take this for example:

Code: 
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Max damage from Hidden Power Ice 70 from 250 Sp.Attack on a Salamence with 196 to 206 Defense.

As you can see putting 12 EVs into Defense can save you upto 8 HP while the same amount of HP EVs save you a one off damage of 3 HP which is automatically inferior to the 8 damage you save for each time you were attacked by HP Ice. Obviously this varies with attack and move power but if you want to survive those Ice type attacks or any other X4 Effective hit, you're best investing in your Defenses.

Hold your horses a second before you start whamming in those heavy Defense EVs into poor Bronzong, there's a catch!

Code: 
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Max damage from Close Combat from 369 Attack on a Bronzong with 268 to 278 Defense.
In this example you only get 1 less damage for a whole 4-8 EVs which can always be spent better in HP. This is because Bronzong's Defense is so high and the opposing pokemon's attack (value as I like to think, simply Attack times Move Power) to be rounded down much more often.

As a summary a simple rule of the thumb is:
- As the opposing Attack and Move Power goes up, the less Defense EVs you need to invest to reduce the damage by 1.
- As your Defense goes up the EVs required to reduce the damage by 1 point go up.
- Never invest Defense IVs in order to take NVE hits and if you have any Defense above around 250 you shouldn't bother at all unless you are expecting to take SE hits.

Section 2: Block Damage

Ceil(Ceil(HP/0.85 - 2) * Def / 0.84)

This was the formula I devised to calculate what Attack Value (Attack times Base Power) you need in order to KO a certain Pokemon. Since we are talking about being defensive I'll use this one:

Ceil((HP - 2) * Def / 0.84)

As any mathematician knows the highest product you can make with two numbers which add up to a certain number are the two numbers closest together so if we want to maximise the Defense Value we have to keep increasing Defense until it is as close as possible to two less than HP. Since the same Base Stat for HP as Defense is always 105 more than the latter, there is very little point putting a single drop in HP to maximise one defense unless that Defense has more than 20 base more than HP (not 51 and a half for that 105 stat difference since you get 31 and a half back if the Def is maximised meaning you may want 1 or two HP EVs).

You'll notice I mentioned I've been talking about maximising one defense, however, since HP counts for both Defenses 4 EVs are effectively giving you 2 extra stat points. Ideally your stat total would look like X+2 HP /X Def/X Sp.Def but it doesn't always work like that. For this reason you might want to invest some more EVs into HP than advised.

Closing Comments

I really wanted to write more on this but I've run out of time, might write some later. I'm a bit worried that it's a bit unreadable. Anyway, some closing points:

- The first part is very difficult to impliment fully into competative spreads. Does anyone have any imput on this?
- My Attack and Defense Values. Do you like the idea?
 
I'm surprised that nobody's replied to this, because to me, this is new data that I didn't know about, and it may change many EV spreads that we use now.

I assumed that your "resistance" to physical attacks was based purely on your Defense stat, and then the HP was just a bottle that "life", or HP, was drawn from after the Defense stat determined how much damage you would take. That is true, right?

I had to read through your post a couple of times to get a good idea of what your were saying, so I may still be a bit confused.
 
The second part of it confused me a bit.

Since the same Base Stat for HP as Defense is always 105 more than the latter, there is very little point putting a single drop in HP to maximise one defense
Click to expand...

Assuming I understood you correctly, I'm not altogether sure that's true. If, as you say your goal is to maximise one defense, then after you've maxed out EVs in that defense, wouldn't the logical step be to then max out HP so that it also adds to that defense. Unless you want that pokemon to be able to tank hits on both sides equally well, then those EVs are still better placed into HP than any other stat.
 
How well I followed got worse and worse the longer I went. I would have assumed that it's just me being a newb, but it seems others are having trouble too :p

Correct me if I'm wrong but Section 1 essentially said that if you're expecting to be eating earthquakes, close combats and the like you'll want to invest in HP EV's. If youre expecting 4x ice fangs or something like that you'll be better off investing in Def/SDef EV's.
 
I'm dense today, I think.

Can someone explain it simply to me? I think it's a neat topic, by the way.

Edit: Dirtybird did a decent job I guess...but wouldn't that mean all this is based on prediction?
 
This is partial excerpt a post of mine on a different forum board on EV distribution:

The Numeric Effect:

In most instances, you always want to boost your highest stat with a nature and reduce either your lowest or an irrelevant attacking stat. Without EV’s, Steelix has 436 Defense and Hariyama has 276 Attack. By giving Steelix a +Defense Nature, its defense rises to 479. The numeric effect here is 43 points. That is the equivalent of172 EVs. By giving Hariyama a +Attack nature, its attack rises to 303. The numeric effect here is 27. That is the equivalent of 108 EVs. This is the most efficient use of natures as far as overall stats are concerned. However, the most efficient use is not always the most effective.

The Percentage Effect:

How you use your EVs depends entirely on your pokemon’s purpose. In Steelix’s case, its defense is so ridiculous that even if you poured in all your EVs, the percentage effect would be negligible. Steelix’s max defense with 252 EV’s adnd nature is 548. 548/436 is is 1.26. In other words, pouring all those EVs and nature only gave a 26% increase. 10% of that came from the Nature alone.

Steelix’s initial HP is 261. If you use all your EVs on it, it raises to 354. This particular HP number is awesome for another reason which will be explained later. 354/261 is 1.36. Using 252 EV’s on HP nets an increase of 36%. That means Steelix will be able to take 36% more abuse from both physical and special attacks than it would have before. Putting the EV’s in defense would have netted a 14% increase only in physical attacks.

Steelix’s initial special defense is 136. If you use all your EVs on it, it increases to 229. 229/136 is 1.68. That means with max special defense, Steelix can take 68% more abuse from special attacks than it would have without.

The standard Steelix EV spread is 252 HP/120 Atk/136 SD with a Defense boosting nature. It leaves it with 354 HP, 479 Defense, and 200 Special Defense. This confers upon Steelix: 36% boost in overall durability, a 17% boost in attack power, a 10% boost in defensive power, and a 47% boost specifically in taking special attacks.

Hariyama’s initial HP is 429. Pouring all 252 EVs into it nets 492. 492/429 is 1.14. Pouring all those EVs into HP only gave you a 14% increase in overall durability.

Hariyama’s initial Defense and SD are 156. By pouring 252 EVs into either of them nets 219. 219/156 is 1.40. Or a 40% increase in a particular defense. Even if you split the 256 EVs into each defense for 128 each, you would get a better overall durability with the same EVs (19%).

Hariyama is interesting because its set is a better determinant of how much you should put into defense.
 
Basically this means that the damage you get is not as simple as: you use your (sp.)def to resist the attack, and the you have some amount of hp.

The idea that we can conclude is the following (you have to use basic differential calculus):

Given a certain poke with it's base stats and a certain attack (let's assume it's physical based, but the idea is the same with special based ones), once you have decided how many ev's you can invest on hp and def, the spread that will make you take the lowest % damage (notice that what matters is the damage in terms of % and not in amount of hp) is this one that makes both def and hp stats be as close as possible.

Let's see an example: if you have a blissey and you have 252 ev's to spread between hp and def to take an earthquake, giving 252 ev's in hp and 0 ev's in def is a bad idea, but if you swap distributions you will be taking much less damage.

This is the reason why blissey needs 252 ev's in def. You all know blissey has outstanding hp and horrible def, so people's first approach is saying: it's supposed to take special hits, let's invest ev's in hp and sp.def. But that's wrong. If you give it max def, you'll resist a lot more, and you will still take special hits like nothing.

Another example: never give skarmory a single ev in def until you have 252 ev's in hp (this is, if you have 180 ev's to be spent for defending purposes in skarm, all of them should go on hp. If you already have 252 ev's in hp and some extra ev's left, then you should start giving it def ev's to make it even more resistant). Notice that skarmory hp is lower than it's def, so we want to make the difference as low as possible, and those hp ev's are helping to resist better special attacks.

But don't make this mistake:

some poke: hp=352 def=256
some poke: hp=368 def=256

So second poke has a bigger difference between it's two stats than first poke. Does it take bigger damage than first poke? No. I think it's obvious, but i 'llexplain it just in case. The bigger your stats, the lowest the damage, but given an amount of ev's, there's a way to spread them to be more efficient.

Also, if you're using a (sp)def boosting nature, pick that one in which you have the highest base stat.

Back to the topic. Trouble comes when you have to spread the ev's between hp, def and sp.def. like, say, swampert. It is a bit more complex now since reducing the stat difference between hp and a certain defense, may make the distribution not being the best one for the other defense.

But there's someone in this forum that invented a program that searches for the best ev spread for any poke, related to defenses.
 
He is x-act, look for that program, it's very useful. But when i started using it i realised that the ev spread i gave every poke without a previous math analyisis was very close to the best spread, so you may already do it well by instinct, and don't know it.
 
What this basically is is that Pokemon is not a game of addition and subtraction, but multiplication and division. The difference in taking hits between 100 Defense and 200 Defense is actually much larger than it is between 300 Defense and 400 Defense. The first change means you'll end up taking about 1/2 as much physical damage, but the second means you'll be taking about 2/3 as much. Even though the actual numbers are both 100 apart, it's their % difference that's important. The same is true for HP.

The relation of these two is simply HP * Def. This gives a measure of a Pokemon's ability to take physical hits. A Pokemon with 400 HP and 200 Defense takes physical hits just about as well as a Pokemon with 200 HP and 400 Defense.

Some people may find this to be highly unlikely, because the Pokemon with more Defense is going to be taking a lot less damage in terms of nominal HP. However, this will be a larger % of its max HP, and that, as I said, is the real stat that matters.

For example, let's look at Blissey and Celebi, two tanks that don't like to die. If you do 300 damage to Blissey, it's just going to use Softboiled, as that's not even a 2HKO. To Celebi, however, you've just left it with, at most, a quarter of its max HP. Celebi is closer to death.

So, how do you maximize your ability to take hits from one side of the spectrum?

Well, we have a simply formula here.

Power = HP * Defense

HP and Defense are both regular numbers. You can find the average of any two numbers by just adding them and dividing by 2. So this means that we can rewrite the above formula as

Power = ((HP + Defense)/2 + x) * ((HP + Defense)/2 - x)

Here, x is just however far above or below the actual stat the average is. For instance, if HP were 10, and Defense were 8, the average of these numbers would be 9. 10 is 1 more than 9, and 8 is 1 less, so x would be 1. Because we found the average of the numbers, HP will always be as much below the average as Defense is above it, or HP will always be above as much as Defense is below (this is just how averages work). Those of you familiar with binomial multiplication will recognize the above as giving a product of a difference of squares. That is to say, we can rewrite the above as:

Power = ((HP + Defense)/2) ^ 2 - x^2

When you square a number, it's always positive. This means that it doesn't matter if HP or Defense is higher, the sign is essentially removed when you square x, so it works for all values of HP and Defense. So what we have is some number, ((HP + Defense)/2)^2, and we are subtracting another number from that. We want Power to be as big as possible, and this happens when ((HP + Defense)/2)^2 is as big as possible, and when x^2 is as small as possible (because we are subtracted x^2. x^2 is obviously smallest when x is equal to 0.

If you'll remember what we did above, we defined x to be how far apart HP and Defense were from the average of those two stats. If x is equal to 0, then both numbers must be equal to the average of HP and Defense, or, in other words, they must be equal to each other.

This means that to maximize your ability to take physical hits, you want to have HP and Defense be as close together as possible. This is why maxing Skarmory's HP is so important: it actually means that Skarmory takes less damage from physical attacks than if you maxed Defense (and you get a bonus in your ability to take special hits to boot!). This is also why you want to give Blissey 252 Defense EVs before touching that HP.

But what if you want to take hits from physical and special hits? Then things get a bit more complicated. There are a few formulas that people use. The most commonly seen one is

HP * (Def + SpDef)

However, this gives a rather large weighting to the HP stat. This leads others to prefer

HP / (1/Def + 1/SpDef)

as this doesn't weigh HP as heavily.

The problem with both of these is that they assume you want your defenses to be balanced. The second stat heavily punishes Pokemon that split their defenses, and the first simply cannot take into account how much you'd want to take on type of hit compared to the other.

X-Act uses a slightly better formula. In his case, Overall Harm = (k(D + S) + 4DS) / HDS, as explained in that thread linked above. He also has a Java applet that will calculate this for you.

However, X-Act's applet rightfully forces you to make a choice. You have to decide where the balance lies. Skarmory, for instance, needs quite a bit of EVs in Defense to stand up to the physical assaults in ADV and in DP. Does Blissey need a similar Special Defense investment to take hits, or does that base 135 SpDef cover it enough?

There is one thing the OP mentioned that I feel I should address, and that's the issue of rounding. Yes, due to rounding, it's possible to waste EVs in a defense stat, or to fall 4 EVs short of reducing damage by another 2 points. However, this depends on the move's power and the Pokemon's stats, which are things outside of your control. All you can do is address what happens on the average, or in the worst-case.
 
Oh, I understand now. I was just confused by what the OP was saying, and I've known what he was explaining for quite a while, although this will help many newcomers. It's for the same reason that we make Blissey Bold over Calm and max out its Def over SpDef. Thanks everybody for clearing that up.
 
Oh i get it now... So you use bold on blissey to take physical hits 10% better versus 10% better for calm. But, 10% of a very hp damaging physical attack is much more than say for a relatively weak special atk.
 
Because of the amount of EVs you're giving Blissey and the +Def nature, its defense stat is more than twice of what it would be otherwise (with a +SpDef nature and no Def EVs), effectively halving the amount of damage you take from physical attacks, while still taking only 5-10% more from special attacks than you would with a Calm 252 SpDef Bliss. The tradeoff is worth it - taking 50% less from a Earthquake, while only taking 5-10% more from an Earth Power.
 
Hmm, I missed this PM >_> I deleted everything in my inbox a while ago, so sorry :-(

I should note that what you're doing is essentially where attack/defense tiers started. There were a few problems when you use raw HP * Defenses: people are far better at addition/subtraction than at multiplication, and second, the numbers are huge.

Thus, I decided to take the logarithm of both sides and just transform all the calculations into the logrithmic scale. That solved both problems.

Indeed, the entire formula is as follows:

Attack tier: log(attack * base power) - magic number.

The various magic numbers are (straight out of my attack tier thread:)

Magic Number #1: 3.53 (guarenteed)
Magic Number #2: 1.83 (chance)
Magic Number #3: 2.65 (average)
Click to expand...


I used base 1.1 for the logarithms, so if I reverse these transformations you get...

log1.1(attack * base_power) - 3.53 == OHKOable Defense Tier
1.1^(log1.1(attack * base_power) - 3.53) == 1.1^ OHKOable Defense Tier
1.1^(log1.1(attack * base_power) + log1.1(.84)) == 1.1^ OHKOable Defense Tier
1.1^(log1.1(attack * base_power * .84)) == 1.1^ OHKOable Defense Tier
attack * base_power / .84 == 1.1 ^ OHKOable Defense Tier
attack * base_power == 1.1 ^ OHKOable Defense Tier * .84
attack * base_power == (hp * defense) * .84

The other "magic numbers" correspond to average damage and the minimum damage from an attack tier.

I took a slight estimation in my calculations, and if my calculations are right, then your calculations are wrong... can you post your step-by-step on getting the formula Ceil((HP - 2) * Def / 0.84) ?? It is similar enough... but different enough... that makes me believe that we made an error somewhere. (if that made any sense >_>)
 
I assumed that your "resistance" to physical attacks was based purely on your Defense stat, and then the HP was just a bottle that "life", or HP, was drawn from after the Defense stat determined how much damage you would take. That is true, right?
Click to expand...

Correct which means that Attacks that 2HKO are better survived with more Defences than HP.

Assuming I understood you correctly, I'm not altogether sure that's true. If, as you say your goal is to maximise one defense, then after you've maxed out EVs in that defense, wouldn't the logical step be to then max out HP so that it also adds to that defense. Unless you want that pokemon to be able to tank hits on both sides equally well, then those EVs are still better placed into HP than any other stat.
Click to expand...

I meant with a limited number of EVs but logically with pokemon like Blissey you max out Def and HP.

Correct me if I'm wrong but Section 1 essentially said that if you're expecting to be eating earthquakes, close combats and the like you'll want to invest in HP EV's. If youre expecting 4x ice fangs or something like that you'll be better off investing in Def/SDef EV's.
Click to expand...

Yeah although with Neutral hits you'd be better maxing out HP slightly more as it benefits both defenses. It's still a good idea to keep HP and Def as close together as possible.



I don't have time to answer anything else at the moment, I've got school.

If you use the first section, half the difference in each stage since HP benefits both stats therefore counts for 8 EVs and write a formula, you've got a nice calculator there.
 
Okay, since this thread somehow mentions me several times, it's appropriate that I post something, lol.

I'd define (HP x Def x SpDef) / (Def + SpDef) to be the best formula to know the overall "tankiness" of a Pokemon. My formula (k(D + S) + 4DS) / HDS written in the maximised defenses guide is basically the reciprocal of HDS/(D+S), with the "+2" of the damage formula factored in, which is only useful if you really want to be 100% exact.

The reasoning for HDS/(D+S) as being the best formula is the following. HD is the physical tankiness, and HS is the special tankiness. Combining them for the overall tankiness, we get 1/HD + 1/HS = 1/T, from which T = HDS/(D+S) follows.

The reason why Pokemon is more about the percentage damage dealt, rather than the actual amount of damage dealt, is the reason that Obi mentioned: that Pokemon is more a game of multiplication and division rather than one of addition and subtraction. And this is also the reason why Dragontamer uses logarithms in his attack and defense tiers, since logarithms convert multiplication and division to addition and subtraction respectively. (A sidenote to Dragontamer's attack and defense tiers: I think people should show more respect to these tiers as I find the idea behind them very cool and original. I had actually PMed Dragontamer with a suggestion of how to improve them a bit - by basically using whole numbers instead of decimals - but he decided that it would confuse the people who have already started using them.)

Dragontamer: his formula and yours are exactly the same, except that he took the "+2" in the formula into account, whereas you did not. The "+2" in the formula will not change things much, but taking it into account will make things more accurate. Things are usually very accurate even without it, though.
 
I like the Attack/Defense tiers as they're pretty easy to use although I think my Attack and Defense values are probably more applicable because YOu can actually work out percentage Damage.

The whole formula for Defense Values is this:

Ceil(Ceil(HP/M)/0.85) * D / 0.84

HP being HP, M being multipliers and D being Defense.

As you can see, there is no simple operation to change Effectiveness.

Of course if we wanted to be really clever we could Log the Defense Value, for example 170978.6 which is Def Value of 404/303 to base 1.1 is 126.421896... I've always wanted to try to base 2, crazy as it sound.

Log2(170978.6) = 17.38345624
Log2(170414.3) = 17.37868688 (404/302)
Log2(169285.7) = 17.36910058 (404/300)
Log2(84285.7) = 16.36300043 (202/300 or SE hit on 404/300)

Not worth it IMO.

My other choice was 1.31:
Log1.31(1.1) = 0.35296519 (Nature)
Log1.31(1.3) = 0.9716218421 (LO, ~2)
Log1.31(1.5) = 1.501571702 (CB, STAB etc.)
Log1.31(2) = 2.56695378 (SE hit)

Fairly memorable.

Edit: @Obi: I noticed that. It's more for certain pokes although since many Base powers are fairly regular and most EV spreads are similar it might just about be possible.

@X-Act: I can't get your App to work.
 
X-Act said:
I'd define (HP x Def x SpDef) / (Def + SpDef) to be the best formula to know the overall "tankiness" of a Pokemon.
Click to expand...

Ah yes. This is equal to my favored formula
HP / (1 / Def + 1 / SpDef)
except that you were smart enough to not use a complex fraction. :toast:
 
If Ive understood this right, this all trails back to a simple mathematic fact. If you want a high product, the factors should be as similar to each other as possible.

For example, we have 300 units to use to make 2 factors. If we make it 100*200=20000, we are not being optimal.

150*150=22500, see how the product is larger here?

This translates to Pokemon as trying to achieve (sp)defence and HP values to be as similar to each other as possible.

Thats what I can make of it anyway...
 
Soooo... say i ahve gastrodon. should i max out at 250 special defense in order to be tough as possible? If i have stockpile to boost it, would it help more to be in HP or defense... 252/252 arent that good for walls then...
 
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