MabiMythBusters - Balance!

• Mystickskye wrote on 2013-02-11 11:39
  Investigation is still ongoing, I don't assume that everything written is 100% correct. Assumptions and guesses have been made.
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  For a long time it's been assumed that damage distribution in mabi follows a normal distribution where your balance indicates your median damage.
  

  [Image: http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/400px-Standard_deviation_diagram.svg.png]

  
  This is your standard normal distribution
  
  The "peak" represents:
  -median; if you took all the numbers and put them in order this value would be in the middle of that data set.
  -mean; if you took all the numbers, added them up and divided them by how many numbers there were (averaging in other words!) it would be this value.
  -mode: the most commonly occurring value.
  It was a nice theory and made damage calculations nice and simple (eg, based on this you can say that at 80% balance each point of maximum damage is worth 0.8 average damage and each point of minimum damage is worth 0.2 average damage). However it was never shown that this really was the case, it was an assumption made and accepted without any real proof.
  
  So! I took it upon myself to record damage values and see what really happened. I used a giant with a warhammer and attacked Gray Town Rats and recorded what came out. I chose Gray Town Rats because their defensive stats are known (independently verified but that's a topic for another time), they're weak and numerous. My giant had:
  -135-298 damage
  -75% balance
  -lots of crit
  -R1 crit
  I note crit because you can't just leave critical hit values out, it skews the data somewhat and decreases the accuracy, increasing the number of values you need for the data to be representative. So then, with a data range like that and 75% balance we'd expect the median/mean/mode to be:
  298 - 135 = 163
  163 * 0.75 = 122.25
  135 + 122.25 = 257.25
  257.25! This should be our peak/median/mean/mode. Regarding crits, each critical hit adds your maximum damage adjusted by your crit modifier (150% at rank 1) so:
  298 * 1.5 = 447
  We can remove this much from each crit hit to find out what the actual damage roll was. On top of this, add one to every single value to account for the Gray Town Rats' defense.
  
  So with all this knowledge in hand we can go forth and begin rat genocide! For the purposes of this test I went and recorded 500 normal attacks against Gray Town Rats. It was extremely tedious and boring. With data in hand though it can be plugged into a program like excel and analysed; getting figures like the mean, median,mode, standard deviation and you can even make a histogram which should give us that nice bell shaped curve shown above.
  
  Before we go any further, some might argue that 500 samples isn't enough. To that I say pish posh. Anyone that's actually taken statistics will know that in this case it's enough to get a general idea of the trends in the data and that taking further samples will generally only smooth the curve out some and make it nicer looking (in other words, if there are certain clear characteristics taking more samples won't change them!).
  
  Anyway on with the show!
  

  [Image: http://oi47.tinypic.com/106xxyo.jpg]

  
  The money shot
  
  Well, that doesn't look quite right. Ignoring the uneven curve there's the issue that the very maximum value of 298 all by itself manages to spike up above the median area (looking at the cumulative percentage we can in fact see that the max damage value accounts for almost 20% of all the recorded values, that is faaaaar too much). Already we can tell that the mode isn't the median and looking at the descriptive stats we can see that the calculated mean is different from the calculated median too. What gives?
  
  Well firstly, it's not a complete wash. The median is close to where we thought it might be and from the median the values falling under it seem to be within three standard deviations too (251 - 40.7713 * 3 = 128.69). In fact looking at it by percentages the values seem to fall under a rough 50/50 split around the median too.
  
  So what's happening? Based on the data my guess is that the game actually does use a full symmetrical bell shaped distribution that stretches out beyond the maximum (in our case anyway). It's just that because we do have our maximum damage, any values in the distribution that go beyond max are rounded to max and this causes the massive spike solely in the max damage roll. More simply the "cut off" section of probability is added to the probability we'd roll max damage.
  
  So, we roll max damage a lot more than common thought would have us assume. Good right? Not really. Because it still looks to be approximately equal going each way from the median, the maximum damage rolls aren't able to make up for the "further away" lower damage rolls that occur. This means that your average damage would actually be lower than your median/balance indicates. This is most evident in magic with 100% balance. To maintain an average at 100% balance you have to roll nothing but max rolls and that's not the case. Rolling anything below max would lower the average and this is what happens in practice.
  
  tldr
  -Balance isn't actually represented by a normal distribution unless you have 50% balance (the key defining features of a normal distribution standard or otherwise are that the graph is symmetrical and median=mean=mode)
  -Balance doesn't indicate average damage. If you have more than 50% balance your average is less than your balance, if you have less than 50% balance your average is more than your balance.
  
  What does this change?
  In all honesty, almost nothing. It just means on average people with high balance aren't doing as much damage over time as commonly thought. It also means that a lot of calculations based on average damage aren't 100% correct but the difference is probably very minor.
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  To be edited, I want to add some more diagrams for explaining and such. Please report any mistakes/errors! I am not infallible!
• Aubog007 wrote on 2013-02-11 12:24
  Well to me, as i pointed out a while ago, it's like the bell curve smashed into a wall
• Sukima wrote on 2013-02-11 12:28

  Quote from Mystickskye;1030603:

  tldr
  -Balance isn't actually represented by a normal distribution unless you have 50% balance (the key defining features of a normal distribution standard or otherwise are that the graph is symmetrical and median=mean=mode)
  -Balance doesn't indicate average damage. If you have more than 50% balance your average is less than your balance, if you have less than 50% balance your average is more than your balance.

  
  
  I thought this was common knowledge.
  
  

  What does this change?
  In all honesty, almost nothing. It just means on average people with high balance aren't doing as much damage over time as commonly thought. It also means that a lot of calculations based on average damage aren't 100% correct but the difference is probably very minor.
  ---------------------
  To be edited, I want to add some more diagrams for explaining and such. Please report any mistakes/errors! I am not infallible!

  
  
  I don't see how. It's skewed to the right, and i'm not reading that tl;dr of hell to find out how you've come to that conclusion. Balance is important, nuff said.
• FallenDanny wrote on 2013-02-11 13:57
  I actually found this helpful, + rep for the information.
• JoeyDee9 wrote on 2013-02-11 14:06
  As a helpful tidbit of knowledge, provo barrier spikes on basic are very good for statistical analysis. They don't move and don't normally die.
  
  And your findings do make sense, while proffing some knuckles for a friend the other day I noticed my absolute max damage hit more often that one would think. The frequency it came up supports the idea that the bell curves is centered at the point of balance and extends past your max damage.
• tacundavid wrote on 2013-02-11 14:07
  Critical damage is not (damage)*(critical modifier), but (damage)+[(damage)*(critical modifier)].
  Thus (damage)*[1+(critical modifier as %)].
  
  Edit:
  After passively thinking about it while roaming other threads, this mistake in your calculations will correct some of the skewed data where the absolute max damage continues to be "rolled". Taking into account that critical hits effectively multiplies your damage by 2.5 will even out your histogram.
• Sukima wrote on 2013-02-11 14:10

  Quote from JoeyDee9;1030670:

  As a helpful tidbit of knowledge, provo barrier spikes on basic are very good for statistical analysis. They don't move and don't normally die.
  
  And your findings do make sense, while proffing some knuckles for a friend the other day I noticed my absolute max damage hit more often that one would think. The frequency it came up supports the idea that the bell curves is centered at the point of balance and extends past your max damage.

  
  
  Right, this explains a lot. I think the OP worded their conclusion wrong.
• JoeyDee9 wrote on 2013-02-11 14:24
  Note: Topic for another discussion about critical modifiers. My ego cylinder seems to break the current notion of critical modifiers. With r1 crit neither Water cannon or flame burst follows the damage*(1+critical modifier) formula, as in a 200 1 charge crit hits like 390 damage and not 450.
• Sukima wrote on 2013-02-11 14:27

  Quote from JoeyDee9;1030684:

  Note: Topic for another discussion about critical modifiers. My ego cylinder seems to break the current notion of critical modifiers. With r1 crit neither Water cannon or flame burst follows the damage*(1+critical modifier) formula, as in a 200 1 charge crit hits like 390 damage and not 450.

  
  
  Perhaps that's an attribute of alchemy or ego cylinders. More testing needs to be done.
• JoeyDee9 wrote on 2013-02-11 14:48

  Quote from Sukima;1030688:

  Perhaps that's an attribute of alchemy or ego cylinders. More testing needs to be done.

  
  
  My thoughts right now are that it the ego percent it done after crit, but that's another topic for another thread.
  
  As for critical hits in this statistics shouldn't they be divided by 2.5 for the most accurate numerical value?
• Nobelium wrote on 2013-02-11 16:43
  My brain is kind of fried right now, so I probably can't digest this even if I tried.
  
  Can someone please tell me in simple words how balance relates to damage dealt now? More balance= better?
• Zekkii wrote on 2013-02-11 19:09

  Quote from Nobelium;1030728:

  My brain is kind of fried right now, so I probably can't digest this even if I tried.
  
  Can someone please tell me in simple words how balance relates to damage dealt now? More balance= better?

  
  
  Same as always, just get 80% balance and live a happy life. They're just testing balance distribution, you'd have to obsessive compulsive to actually tell the difference.
• Mystickskye wrote on 2013-02-11 21:17

  Quote from Sukima;1030614:

  I thought this was common knowledge.

  Given that the wiki used to describe it otherwise, that people continued to cite a normal distribution, that people continue to make calculations of average damage based off this assumption, apparently not.

  Quote from Sukima;1030614:

  I don't see how. It's skewed to the right, and i'm not reading that tl;dr of hell to find out how you've come to that conclusion. Balance is important, nuff said.

  Of course you're not seeing how because you're not reading it. Yes balance is important, I haven't said otherwise. What I'm saying is that it doesn't accurately indicate your average damage as many people believe. As an aisde it's actually skewed to the left, not the right. Direction of skew is described by the "tail end".

  Quote from JoeyDee9;1030670:

  As a helpful tidbit of knowledge, provo barrier spikes on basic are very good for statistical analysis. They don't move and don't normally die.
  
  And your findings do make sense, while proffing some knuckles for a friend the other day I noticed my absolute max damage hit more often that one would think. The frequency it came up supports the idea that the bell curves is centered at the point of balance and extends past your max damage.

  Man, I can't believe that the provo barrier spikes didn't even occur to me @@ I'll keep that in mind for future tests! And yeah, even as I was recording the data I was thinking it was odd that my specific max damage was being rolled so much. I still didn't realise just how much it was though till making the histogram!

  Quote from Sukima;1030675:

  Right, this explains a lot. I think the OP worded their conclusion wrong.

  I'd already mentioned this occurrence within the body (as he read and you didn't) and this actually reinforces my conclusion. Not much changes overall but your balance doesn't indicate your average damage.

  Quote from tacundavid;1030671:

  Critical damage is not (damage)*(critical modifier), but (damage)+[(damage)*(critical modifier)].
  Thus (damage)*[1+(critical modifier as %)].
  
  Edit:
  After passively thinking about it while roaming other threads, this mistake in your calculations will correct some of the skewed data where the absolute max damage continues to be "rolled". Taking into account that critical hits effectively multiplies your damage by 2.5 will even out your histogram.

  You haven't read correctly. Try again :|

  Quote from Nobelium;1030728:

  My brain is kind of fried right now, so I probably can't digest this even if I tried.
  
  Can someone please tell me in simple words how balance relates to damage dealt now? More balance= better?

  As I said after the tldr marker, it changes very little. More balance still equals more damage over time, it's just that balance doesn't accurately indicate your average damage as thought (unless you have 50% balance).
  
  To describe what this does, instead of just accepting that things fall down we now have a clearer picture of how and why things fall down.
• Nobelium wrote on 2013-02-11 22:31

  Quote from Mystickskye;1030884:

  As I said after the tldr marker, it changes very little. More balance still equals more damage over time, it's just that balance doesn't accurately indicate your average damage as thought (unless you have 50% balance).
  
  To describe what this does, instead of just accepting that things fall down we now have a clearer picture of how and why things fall down.

  
  
  Thanks. I just found this sentence particularly confusing, and I didn't want to take too close a look while I am studying for other things: -Balance doesn't indicate average damage. If you have more than 50% balance your average is less than your balance, if you have less than 50% balance your average is more than your balance.
  
  I understood that as something like less balance= actually better for average damage or something. Maybe I'll come back to read this in more detail one day.
• JoeyDee9 wrote on 2013-02-11 22:34

  Quote from Nobelium;1030943:

  Thanks. I just found this sentence particularly confusing, and I didn't want to take too close a look while I am studying for other things: -Balance doesn't indicate average damage. If you have more than 50% balance your average is less than your balance, if you have less than 50% balance your average is more than your balance.
  
  I understood that as something like less balance= actually better for average damage or something. Maybe I'll come back to read this in more detail one day.

  
  
  Basically 80% of your maximum damage isn't your average damage with 80% balance. The reason so is because you have a max damage limiting the bell curve as this data set shows.