"Balance can be perceived as a standard Normal Distribution Curve"

Except from what I've seen it isn't. Having recorded the data I've so far found that it doesn't seem to resemble a normal distribution curve at all. It lacks all of the defining features.

It matches a weighted normal distribution curve, which is slightly different from a standard normal distribution curve.

Except it doesn't seem to match that either.

Care to share your recorded data then?

I'm sorting it out and actually plan to upload it to my page of math but for now:
http://i48.tinypic.com/1zezi0z.png
Before anyone says "that's a weighted standard distribution", no it's not (you're evidently misunderstanding what that means). In any given normal distribution the function is symmetrical, the median (which this page uses), mode and mean are all the same and you can define where a percentage of the values will fall within x amount of standard deviations (a standard normal distribution has additional characteristics on top of these). I have an incomplete set of results with a larger damage range and 80% balance and the results are looking to reaffirm that the damage rolls don't present a normal distribution of any sort. The current writeup on the page implies (and as I quoted in the discussion title, it explicitly says standard normal distribution) that there will be a symmetrical function returned by damage values and that this function moves around with the balance % representing where the mode/median/mean falls.

Hold the phone, I've had something of a realisation and if what I'm now thinking is right, both the wiki and I are kinda right and wrong. Taking more data but results will be soon.

How are you factoring criticals?

Critical hits are Max * Crit modifier + Damage_Roll, so make sure that when you calculate the roll of a critical (R1) you are using Damage - (Max * 1.5) as the roll and not Damage / 2.5, otherwise you'll skew the data.

Believe me I know. In fact I verified it back in C1 and made a point of telling others.

The damage distribution looks like its approximately on a normal curve. Mabinogi's RNG is very poor though. That is to say it uses an old fashioned RNG method of cross weighting and is, itself, "poorly random".

Cross weighting can best be expressed as Flip a coin with a 50% chance of heads or tails. If you get heads 2 times in a row, the chance of the next heads is slightly decreased while tails will be slightly increased. This effect amplifies to prevent random chaining. On the extreme long term, the true probability (50%) will be preserved, but on the short and mid term, it leads to minor clustering and dis-regulation of the probability curve due to short term probabilities being modulated.

http://i47.tinypic.com/106xxyo.png

It really isn't.

What it looks like is that the client itself does use a normal distribution for the random part of damage but if this goes beyond max (or min likely) it caps it at max. This means that for high (or low) balance your mode will likely be max (min) damage and your mean damage shifts slightly towards the 50% mark.

Several reasons, not enough data, last i recall, there is still a debate on crit cap being 30% or 33.3% even with over a thousand tests. Or could have been changed recently, the magic thing that was debated was one of those things that changed over time.

Personally, the people who gave us the max damage needed to make R types better by averaging also tested the balance and magic formulas. I trust their results to this day.

Source or bust. And I've done max damage needed to make R type better than S. My results match up to theirs but there's strong evidence that the basis for it is wrong.

It's just mathematical logic, you need a large pool in order to determine how the average will be, i tested stuff too, and my results were skewed... Then three different mathematical people debunked my results because my pool was not large enough, as they said... Everything always averges out. It's like me running their method with 150% crit and not a single WM crit for 2 runs. No offense but 500 is waaaay too small a pool.

It's sufficient in the face of the shape it makes. At the very least it completely destroys the traditional idea that damage distribution is presented as a normal distribution because there's no way for this to smooth out into one. There's simply too many values lying outside the expected ranges.

It doesn't debunk the shape at all. The general shape IS a normal curve. It just shows signs of unevenness that could potentially normalize within another 1000 runs. You can't rule out the possibility of a normal curve just yet. Remember, you must calculate the statistic significance of the shape of the curve AND the ratio of outliers. That means you'd have to take multiple sets in order to confirm that the outliers have exact intervals.

A normal distribution curve does not have to have the same Z value as a standard normal distribution curve, don't forget that.

Edit: Why does your frequency add up to greater than 100%?

No it isn't, using the rule that a normal curve is symmetrical alone means that this isn't a normal curve. The frequency doesn't add up to greater than 100% either, that's just excel being unclear. The frequency adheres to the y axis on the left, the cumulative frequency adheres to the right and maxes at 100%.

It seems like as you're getting more samples its normalizing. Thats the only explicitly evident pattern.

Its still following a probability bell curve. Though at 75% balance its following as 20% to max, which is a statistic anomaly under a weighted bell curve.

Try testing the data against a shifted bell curve (Shift median value according to calculated mean, leave the curve normal, and apply new omega values accordingly. Max will occur more frequently, since Statistic Max = Balance*2*Max - Balance*Min, but truncates all values over Max to Max).

Yes it follows a bell curve pattern but that doesn't necessarily mean it's a normal distribution and the fact that it caps at max means that you can't say that balance is represented by a bell curve because it distorts the shape. Going by this the only time damage is truly represented by a normal distribution is if you have 50% balance.

You didn't read what I said. A shifted distribution curve looks like like a normal distribution curve with the omega points shifted with the median value, and a physically non-existant statistical max. This has the effect, assuming that non-existant upper bound "rolls" are included, will cause the max to be rolled excessively. i.e, if shifting the median upward would cause values greater than the max at a 50% (normal) to be rolled 10% of the time, then the max's adjusted rate would be its calculated rate + 10%.

I read what you said (and what you've said now is basically what I said earlier), I'm just pointing out that we can't just call it a normal distribution as people keep insisting. i.e. the current writeup on the page doesn't explain how balance works well at all.

If its a shifted distribution curve, its still normal so what would we call it?

Actually it's technically not a normal distribution even if you say shifted (following a normal distribution pattern doesn't necessarily mean that the data is normally distributed, certain characteristics must be present to truly call it normal distribution curve) but regardless the current write up on the page doesn't accurate depict what happens. People assume that an actual normal distribution is formed when that's not the case (along with the implications that go along with that).
http://mabiumby.files.wordpress.com/2010/05/80balance.png

However if you apply this sort of thinking to more extreme cases like 0% balance (or at least, extremely close to it) or 100% balance (magic), it's clear that the median isn't the mean. Following what the data collection suggests, at 100% balance you'd be hitting maximum damage for half the time and the other half of the time you'd be hitting less down towards the minimum though the probability lessens as you move away. This sort of scenario whilst requiring confirmation is already alluded to by common observations of magic, they don't always hit for 100% damage with max balance. This is even mentioned by the page and thus contradicts itself by mentioning the average.

Basically, the segment needs a rewrite to accurately depict how balance affects damage.
Even saying that balance is represented by a truncated normal distribution would be better if still not entirely accurate.

That's why I was saying its a shifted distribution curves. It has all the properties of a normal shifted distribution curve (Mean > Median @ %50+, Mean < Median @ 50%-, Upper bound occurrence approx. equal to calculated value + overflow curve sum, moving median upwards decreases occurrence of lower bound relative to the rest of the curve, curve remains normal at 50% median).

I'm not seeing why you're saying its not a shifted curve?

Edit: Can't find any good visuals. Basically, you shift the median to the left but its still 50% probability on either side, the probability on the right side would normally cause lesser variance due to only having 20% of the values (assume 80% balance weight. for the upper 49.99% of probabilities, however in a shifted curve this will not occur. The curve remains mostly normal on either side and the ceiling values accumulates all the "lost" probability.

I'm saying it's not a normal distribution even if you call it shifted or not. You can't call something normal if it's not approximately symmetrical and doesn't have mode=median=mean. These are defining characteristics of a normal distribution. Calling it just a shifted distribution curve is okay.

I see what you're saying.

The problem I've run into is that I can't accurately determine whether its a simple shifted curve (as I described above), or a more complex weighted distribution with shifting. They can be very similar depending on the formula used. I only know that the curve is being shifted (most notable example is hitting max 50% of the time @ 100% magic balance).

Yeah, it's not something I'm sure on myself. At any rate, we have enough to attempt a simple rewrite at least.

I regarded it like what tellos said, i basically envision the curve slamming into a wall because of not enough "space" to finish the bell, or just replacing the numbers with "higher than 100%" and slamming it down to 100% in my head. But the way you put it, i guess i understand what you are saying, to me though it is still a bell curve, just graphed differently, i will try to plot out a graph later to visualize it.

Or i can try to explain, when it slams into a wall, the part that is "slammed" gets inverted and attached to the highest possible damage. Basically at 80% balance, it curves up to 80, dips down to 100% and at 100% it shoots back up, that is how i can best explain it.

But to me, and considering it has been almost 10 years since i have had any math education whatsoever, that this is still a bell curve, just with a "limit", so forgive my rustiness on my math, may be a bit "dumbed" down.

Guys I've already described how anything over a limit is just made to be the limit way back, we can stop reiterating it now :B

was on phone. This discussion threading nonsense makes my phone freeze.

If I may ask, what the hell are "Normal Distribution" and "Bell Distribution" curves?

Wikipedia:Normal distribution

Is there a TL;DR version? Walls of text are hurting my brain QQ

You've never seen this before? http://upload.wikimedia.org/wikipedia/commons/0/06/De_moivre-laplace.gif

Dumbing it down as much as I can, it's a graph with a bell-shaped curve somewhere on it. There are other requirements, but given how my previous attempt to explain the idea of probability to you went, I won't bother mentioning them.

Best way to learn it is to take a data management course, or go to college/university. The latter will make you love bell curves or hate them.

Heck, high school statistics class.

Yeah... I don't know if I'm the only one but...... I think someone needs to describe balance on the wiki in layman's terms. I have no idea what 50% of this means. Darkessmager (talk) 13:47, 20 March 2013 (PDT)

Maybe later. For now, we don't even have all the information yet.So far, we don't even have enough data to be certain that it's a bell curve, maybe we will when Mystickskye publishes the rest of his data.

Half of what's in the balance subsection should be moved either to the damage calculations area or a separate damage randomization subsection.

So, here's an attempt to summarize everything we once thought/assumed and everything we now know(everything Mystickskye just proved in this discussion) with explanations in layman's terms: Basically, what we know right now(because of the very long discussion up there) is that our old theory(of how Mabinogi's damage system works and how balance works) is wrong, and that the actual system is simpler than what we previously assumed, but our calculations (of the relationship between balance, and the effect of min dmg relative to max dmg, on average dmg output) may get more complicated.

The old theory, which is explained with pictures | in the article Mystickskye linked to earlier, was that the damage RNG's(Random Number Generator) probability-density-function(basically a graph representing how often each outcome will happen) was much like a normal distribution curve(the famous bell shaped curve) except that it was skewed(asymmetrical) and fitted to the damage range so that the significant(worth calculating and/or not ridiculously close to 0% probability) part of the curve started at min dmg and ended at max dmg. (Technically, a normal distribution curve extends from negative infinity to positive infinity on the x(horizontal) axis, but for most of that it's infinitely close to 0 on the y axis, so only the "bell shaped" portion near the average is meaningful and can be calculated reasonably.) In the old theory, the RNG was thought to adjust the weight or skew so that the mean(all numbers added together, then the result divided by the number of numbers; the most popular definition of "average" in math) damage would be (max-min)*balance+min(the number that is balance percent of the way up the dmg range). The second graph in the obsolete guide from the earlier link shows how the skewed/weighted probability density curve resembles a normal curve if the scale of the axis on the smaller side of the average is stretched.

The new theory, based on Mystickskye's data, is that the RNG's probability density curve is perfectly symmetrical(not skewed or weighted at all) and is not limited to the min-max range. When the RNG produces a number within the min-max range, the dmg is that number; when the rng produces a number higher than the max, dmg becomes max; when the rng result is lower than min, dmg becomes min. In other words, the probability density curve for actual dmg will be identical to that of the randomizer for the range lower than max and higher than min, but for max, the probability density of actual dmg will be the sum(total) of the randomizer's probability density for max and higher, and the probability of min will be the total probability of min or lower. In the new theory, the probability density curve is still presumed to be bell-shaped, though we don't have enough data to be certain yet. Balance can be thought of as shifting the position of the curve over the damage range such that average, before cappinging to max and min, would be (max-min)*balance+min(same formula as before).

In the old theory, the probability density curve was thought to be warped but not shifted by balance; and limited to the min~max range.

In the new theory, the probability density curve is believed to be shifted but not warped by balance; and capped by max and min but calculated beyond them.

In the obsolete model, probability density decreased toward 0 near/at max and min.

In the current model, probability density decreases when approaching max or min, but jumps at the end.

To roughly illustrate the new model for the the Mabinogi damage system; here is a frequency histogram of both the for a plausible outcome of 57 hits with 5 min, 15 max, and 65% balance; with both the initial bell curve randomization and the capped probability density, overlapping on the same graph:

^ = hit on the final probability density curve that was added to the min or max.

> or < = hit on the initial randomised "bell curve" that will need to be puched to max or min.

* = both curves overlap; a hit that did not need to be pushed over to max or min.

										*	*											9	Frequency
										*	*											8
									*	*	*	*		^								7
									*	*	*	*		^								6
								*	*	*	*	*	*	^								5
								*	*	*	*	*	*	^								4
							*	*	*	*	*	*	*	*								3
				^		*	*	*	*	*	*	*	*	*	<							2
			>	*	*	*	*	*	*	*	*	*	*	*	<	<	<	<				1
01	02	03	04	05	06	07	08	09	10	11	12	13	14	15	16	17	18	19	20	21	22	dmg
			1	1	1	2	3	5	7	9	9	7	5	3	2	1	1	1				frequency (in RNG)
				2	1	2	3	5	7	9	9	7	5	7								frequency (in dmg output)

note: this is is just to illustrate how the dmg values are capped and is not based on actual data, and the width and shape of the bell curve is probably off. The data we have on this system so far is not enough to determine the standard deviation, how far past the min and max the curve is calculated for, or even that it's indeed a bell shaped curve.

Did any of that help? Is there anything i left out?

i seriously was under the assumption that's how balance was always perceived in mabinogi, just a really crappy way of wording it.

Actually it's pretty clear that it's a bell shaped distribution in general. The chance that it's not is pretty slim.

And no, chances are average damage is not shown by (max-min)*balance+min. This is clear with 100% balance.

However, this may be mostly moot. Current test notes say that balance is calculated differently into damage in test and these changes may make it through to public. So who knows, maybe things will go back to the way we originally envisioned. Maybe it'll go some other way. At this point I don't want to put too much more into things given that a schematically significant change might make it in. The definition was outright wrong for some years, I'm not going to lose sleep over it. If anyone thinks they can word it better though they're welcome to try.

In what I've read so far about the experimental balance changes, the part that changed was how DEX adds Balance, not how Balance affects dmg output.

Like i said before, i thought it was like the bell curve smashed into a wall, and that's how everyone i asked understood this concept.

Your data only proves my point though.

"bell curve smashed into a wall" is ambiguous in itself >_>

I have odd logic. i meant like the "excess" that is runoff from 100% is folded back into itself.