In 1907, a Massachusetts physician named Duncan MacDougall performed several experiments on dying dogs and people. MacDougall hoped to weigh his subjects’ souls as they died, measuring their wasting bodies on precisely calibrated scales. In the finish, one of the dogs lost weight, and v of the people lost and gained weight in unpredictable ways. But one subject lost 21 grams at the moment of death. After millennia of development from the
of ancient Chinese humoral medicine,
this figure of 21 grams was the first attempt at quantifying the vital forcefulness of a human being. MacDougall was never able to put any more than dying people on his scales, and his experiments were fundamentally flawed, of form, but his dismal nonsense represents an attempt to bridge the gap not merely between the quantitative and the qualitative merely betwixt the profane and the sacred, between the known and the felt. It’s difficult to blame people for being tempted by Promethean power: they learn the error of their ways, one way or another.
I practice blame Akira Toriyama, though, for introducing this morality play in the pages of
Dragon Ball Z
in 1988. Just as information technology backfired on MacDougall, it backfired on Toriyama. And now, after mining
Dragon Ball Z
power-level numbers, it backfires on me. I wanted to understand merely how stiff Goku and his swain fighters were based on years of fan-driven mathematical speculation. I wanted to come to a greater decision. Just I severely underestimated the imagination of forum-dwellers.
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The soul of Toriyama’s was once mysterious. In the original
was a mystical free energy developed through meditation and training with the world’s masters. Its depiction resonates with historical
techniques, kung fu movies, and
Journey to the Westward,
distant source material.
Loftier jumps had the potential to evolve into flight. Focused punches could get focused energy blasts.
Dragon Brawl’southward characters could larn to sense each other’s
— by and large its forcefulness, valence, and personal flavor — but in
Dragon Ball Z,
Toriyama introduced a technological complement to the technique in the form of power levels and scouters. Members of the villain Frieza’southward army, a paramilitary for a galactic existent estate company, were equipped with augmented reality lenses that would display the power level of the person in their sights. They could decide whether a planet’s inhabitants were weak plenty to bulldoze or stiff enough to crave reinforcements. Early ability levels in the series include a farmer with a gun (5) and our hero Goku without his weighted clothing (416). Goku’s former rival Piccolo charges upward from 408 to 1480 when unleashing his Special Beam Cannon, fatefully and fatally surprising the villain Raditz.
Toriyama invented power levels equally a short-hand for the audience to sympathise the stakes of a Dragon Brawl run across, only the stats too meant Goku and the other heroes could learn to fool them by suppressing their power levels. They overcame their numerical limits in an array of lightning, floating rocks, colorful auras, and grunting. Ability levels ceased to exist predictive. Appropriately, around when Frieza was vanquished — just over 1 third of the way through the 291-episode
Dragon Brawl Z
— ability levels were never once more mentioned in the anime
or official secondary sources. The highest official power level, and i of the concluding to exist canonized, is found in the reference work
Information technology gives Goku’s full Super Saiyan power when contesting Frieza: 150,000,000. Afterward that, power levels were at all-time extratextual doodads, occasionally popping up aslope manga chapters every bit a fashion to promote a new movie.
Though Toriyama gave up on ability levels, their quantitative precision took hold in many a child’southward mind, including mine.
Dragon Brawl Z
blew upward in popularity in the Us in 1999, when Cartoon Network began airing a newly dubbed Frieza Saga. These episodes would exist the final ones boasting ability levels. In 2000, at that place was a whole world of many, many, many fights — clear victories, nearly losses, and everything in between — and no power levels attached to them. The loyal viewer was told the new cyborg villains were powerful ... but how powerful were they? Luckily, a viewer had all the data that they would need to extrapolate and estimate new power levels: the boilerplate gulf in skill betwixt heroes and villains at the kickoff of an arc, the furnishings of various power-up and training techniques, and how much more powerful Saiyans become after beingness browbeaten to the brink of death.
A fan could create a power level logic that could explicate the whole
Dragon Ball Z
universe all on your ain. Then they could proudly put their findings up on some random cyberspace forum as a monument to their fan dedication and intellectual prowess.
Though I never took part in coming upwardly with new power levels for characters, at around this time I was regularly trawling anime forums to see what other people thought. How much more powerful was the nefarious Cell with each Android he captivated? And how many times more than was that than Frieza’s final grade? How much ability did going Super Saiyan two or 3 give a fighter? What
the more than powerful fusion technique? The fact that I only found speculation rather than definitive answers only made me want to look harder for concrete proof.
In that location was never a consensus on which unofficial power levels were accurate, and no one was interested in building one. Toriyama may take dropped power levels considering they were unnecessary — quite explicitly telling rather than showing — but they distilled a fundamental component of
anime: the definitive power to control and defeat someone, to conquer, to be better. It is an iron police of an emotional universe, i that children on the verge of preteenhood can learn to wield.
I thought I could return to this world over 20 years after and have a more than sophisticated insight into the logic backside fans’ ability level ratings on a computational level. Given how slight differences in methods of power scaling from battle to battle snowball into exponential differences between raters, I knew any attempt to observe value in Dragon Ball Z power level information would exist quixotic. Information science is always messy. Only, so over again, it wouldn’t really exist quixotic if, past the time I gathered my sources, cleaned my data, and did my analyses, I didn’t really believe there would definitive results. And I did detect
My method for assessing Dragon Ball power levels: establish search terms and scope, extract the data from the sources, make some comparisons, and use Gaussian descriptive statistics on hopelessly non-Gaussian data. To do so, I scoured every relevant remnant of Web 1.0, from rickety Angelfire and Tripod pages to living-dead forums (who nonetheless hosts these things?). Bless the work of various fandom wikis as the new host for this kind of content, even if I may at present take that special kind of computer virus unstuck from time.
To demonstrate how power-scaling in the DBZ fandom works, and why it’due south only made definitive results more challenging to pin downwardly, hither’southward an example from how forum-affiche “SkullMac” considered the Dragon Ball characters on the defunct fansite Planet Namek (now copied to an Angelfire site). SkullMac had some strident thoughts about what sources and methods should be used to understand and create power levels:
Opinions tin’t be wrong as long equally its based on fact anyway, or did your English teacher never tell you that? That’s why all of my guesses are based STRONGLY around the stated levels.
Also, some people have east-mailed me about a “Dragonball Guide Book” which lists levels for the entire series. Let me fix the record straight on this one: those levels mean zero. Come up on people! Fifty-fifty those of you who spotter the English version of DBZ know that Freeza has over 1,000,000 in his second grade. The guide book lists him at virtually half that. Now, which do you believe? The manga written past Toriyama himself? Or an unofficial guide volume written years later past a 3rd political party. I rest my case.
SkullMac’south list starts with official Daizenshuu power levels, but immediately hits some snags. Similar and so many results, character names are translated with verve; “Ozouru” is one of the infinite spellings of “Oozaru,” the “big monkey” that Saiyans turn into later on seeing the full moon. There are mistranslations of stats: Near of the numbers in
are not written in decimals but in a relatively common combination of decimals and Japanese identify value demarcations for 10,000 and 100,000,000. The Daizenshuu power level for Super Saiyan Goku should be 150,000,000, every bit previously noted. SkullMac writes 15,000,000. Frieza’due south peak power level is similarly decimated. By miscalculating Super Saiyan Goku, the error throws off the power scaling estimates for the rest of the series. And yet, SkullMac’s interrogation of the canon text is off-white: Would Goku really have a base power level bound from 90,000 to 3 million in a matter of days, as the Daizenshuu says? Mistake or not, I would take guessed something like SkullMac’s power levels back in the solar day.
These quirks and a g others are littered throughout the ability level lists. The hope is that these errors will fall abroad every bit sources get aggregated. Some volition autumn abroad but not all of them.
Constraints helped me focus my questions and data cleaning. To map out the full spectrum of power levels, I only took data from lists that included power levels beyond the Frieza Saga, focusing on the leaning towers of hypotheticals rather than the basics. I likewise included information from the
movies and from the noncanonical
Dragon Brawl GT, just for some flavour. I outright ignored anything from
Dragon Brawl Super, a series that started 20 years afterwards
Dragon Ball Z
ended and is and then boring that watching information technology makes me believe I am already dead. (I also ignored the OVA considering, even though information technology has some plot connection to
GT, information technology is very disruptive.) In curt, for the purpose of this analysis, if a 2000s-era me couldn’t download it equally an AMV off Kazaa, older me did not care almost information technology. In the cease, the
data was so thin that I couldn’t do much with it other than reminisce about Super Saiyan 4 and Limp Bizkit.
While cleaning the information, I found a couple sources that would have batty the results, numbers and so large as to be incomprehensible. Unlike the average power level list, which would more gently move upward the orders of magnitude, these extreme datasets exploded far beyond those mere trillions. These lists felt more like the piece of work of a sixth-grader who was just introduced to scientific annotation and the orderly fashion English denotes numbers that will never exist said aloud. The source itself is incoherent. The power levels already scrape the ceiling in the sextillions in the Prison cell Saga but so explode to “octoquinquagintillion” (ten^179) to impale Cell and and so onto alleged numbers like “novensexagintillion” (x^210). From the perspective of entropy, the amount of information in the universe in detectable particles is around 10^lxxx bits. If each particle in the universe was its own universe, there still wouldn’t be enough data to stand for this number. It is a special thing that humans can excogitate of something that points at this number, but it is not useful.
In the end, I wound up with 34 sources of data, and about 12,000 power-level ratings cleaned. “Over 9000,” one might say. Independent in this spreadsheet is the largest
Dragon Ball Z
power level dataset I know of — it is 10 times larger than the one on crowd-sourced information science competition site Kaggle. At the aforementioned time, it doesn’t seem big enough compared to what I found when I was a child. I didn’t remember whatsoever item listing, but I hope that was just a mistake of my memory and that I was visiting some of the pages that I did years ago, ships passing in the very long night. Many if not virtually of those childhood lists were lost, coming from an age of the internet that was never meant to exist permanent. I’m sure there are more than lists out there, simply I never may discover them. A moment of silence for them.
Quantitative nihilism led me to freedom every bit I institute myself rewriting graphic symbol names, and changing how modifiers similar “anger” and Super Saiyan stages were written, over and again. Sure the information is logarithmically distributed over fourth dimension, only information technology is a gentler increase within sagas, so I permit it pass every bit “unremarkably distributed.” Some trends in the data await roughly exponential: Why not assume information technology’s indicate rather than noise? Why should I deign to consider using medians when they are irksome? I let myself take a kind of surly fun, great activeness figures together in my mind and seeing what happened.
Let united states of america now revisit questions that my childhood self wanted answered. First, how much more powerful does someone go as a Super Saiyan?
Below is a histogram of just that. The x centrality is the Super Saiyan multiplier, and the y axis is the number of times people used something about that multiplier. I made comparisons inside sources because of the wide differences in orders of magnitude between raters and some math reasons that brand that hard to deal with otherwise.
The peak here indicates that most fans think going Super Saiyan makes you lot 50 times more powerful. This is as well the factor increase implied by the Daizenshuu power levels; raters either pulled from the source or from other lists that pulled from the source.
The consensus for Super Saiyan 2, which helped Gohan defeat Jail cell aslope his dad’southward ghost, is at a 100-fold increase in power level, but twice the power of Super Saiyan. Super Saiyan iii is a 400-fold increase, according to almost of our sources.
Super Saiyan 4 has many fewer ratings, leading to a huge spread in proposed multipliers, from five to 4000, without a clear consensus. Perhaps it doesn’t matter as much because
is an clashing memory exterior of the catechism these days.
At starting time, I was confused that the Super Saiyan 2 and Super Saiyan 3 power level gains were so meager subsequently the l-fold increase of the showtime Super Saiyan stage. But, thinking about how the show told its story, the reasoning began to make a bit more sense. Information technology near goes without saying that the outcomes of fights were based on the ratio of the winner’s ability level over the losers. Frieza kept increasing his power multiple times with each class change, finally reaching an impossible peak above Goku’s Kaioken multipliers. Merely leaping to legendary heights could Goku really proceeds an advantage over Frieza. In subsequent arcs, it was ever the case that the villain — the androids or Cell or Majin Buu — was just out of attain of the main characters’ skill throughout their arc. Heroes could hold their own, if only for a few moments. A smaller power bump, a factor of ii or three, could turn the tide in the cease.
How much more powerful do characters get when they fuse? That’south less clear. Compared to Goku and Vegeta’s powers combined at the time, the Vegito fusion is a 10- to near 300,000-fold increase. The median is effectually 300. The Gogeta range isn’t quite equally wide but however cracks 100,000 times on the loftier cease. This median is effectually 250. The Gotenks multiplier is more stable, perhaps considering Gotenks is a more prominent fusion that engages in more battles: the mean multiplier is dragged up to 1500 by outliers.
The median for Gotenks is 100. I’d lean more towards the Gotenks multiplier as the canonical one for fusions achieved via the fusion dance, given its greater frequency in the dataset. I had always imagined the fact that the fusion dance timed out later on 30 minutes, leaving its constituents scrambling to save their skins, would imply that it was more powerful, but I guess the permanence of the Potara Earrings outweighs the humiliation of the fusion dance.
Now here are the Large Charts, noting the mean power levels of main characters throughout the serial and the movies aslope the main villain of each saga and motion picture. (It is convenient that present and time to come Trunks don’t overlap in these.) The means are to a higher place. The medians are below. Each mensurate of fundamental trend has its faults. On the one hand, means are more than sensitive to outliers, specially college ones when the data is relatively log-distributed, every bit you lot encounter with the power levels. On the other hand, the median doesn’t get close to describing the wide range in power levels; y'all tin can see that with the standard divergence.
The fandom’s calculations of the mean suggest a massive power bump in the Cell arcs that increases steadily over the diverse fights with Buu. This tracks, considering the training that various characters went through to fight these villains. Meanwhile, the median shows a bump between the Frieza and Android Sagas and a steady, boring increase in log-space until the serial is over. (Delight ignore the single Gohan measurement in the Kid Buu Saga.) The mean and median power levels in the movies prove three phases: pre-Super Saiyan, mail-Super Saiyan, and the spitballing that occurred in the terminal couple movies. (Bio-Broly is an outlier across the board.)
What these plots tell me is that the secret of power-scaling in
Dragon Ball Z
lies not in new powers simply in the steady increment in power saga-by-saga. Sure, as you lot see below, characters can go x-20 (by the median) or 100-10000 (by the mean) times stronger within sagas or fifty-fifty unmarried battles.
Heroes might detect some subconscious ability inside of themselves a couple of times, but those sudden image shifts are supercharged past the way that the baseline power shifts over time. Information technology’s apparent that Goku is doing push button-ups in 100-times Earth’south gravity or shooting free energy assurance in the Hyperbolic Fourth dimension Chamber or whatever, simply the consequence of that training is what matters more than over time. In that location isn’t plenty data to go practiced multipliers between most sagas, but the official power scaling is an increase of 10-100 times in base power level between sagas. And yet, by the Android Saga, these training sequences are more for flavor than anything else. Even though Gohan became at to the lowest degree an club of magnitude more powerful simply past training for 10 days before the Cell Games, we only remember the lightning crackling around him as he turns Super Saiyan ii and only doubles his power. (For a comparison, Goku becomes at most thirty times stronger over the unabridged form of the original
Y'all even see it in Krillin’southward ability level in the Big Charts. Past the Android Saga, he’s a pugilistic nonentity: he trains one-half-heartedly, but every bit the most powerful homo he sidelines himself offset as a kind of errand boy so every bit a family unit man. Even so people remember he’s getting ten times stronger with each passing saga, keeping the same multiplicative relationship between him and the “real” fighters every bit if to rails the stability of the emotional relationship. Conversely, Piccolo stops being a competitive fighter after fusing with Kami and losing to Cell, then his power level drops after that.
All of the fan information and fatigued conclusions pb u.s.a. to where the boilerplate
fan starts: changes in power are changes in narrative. The multiplicative logic of ability levels doesn’t demand to make sense; it simply needs to experience right in the moment. Vegeta threatened the Earth with his Galick Gun attack at a power level of 18,000. Frieza
destroy a planet, relatively hands, at a power level of effectually 100,000,000. Past the Prison cell Saga, fights should destroy planets by accident, but they don’t. The exterior world never matters: in
Dragon Ball Z, the fate of the universe ever hinges on to two sweaty men gasping on the pocketknife’southward edge between expiry and cocky-transcendence.
The deciding factor in any fight is someone becoming powerful “enough.” There is always an “enough” in unproblematic stories. It is a metaphysical safety net. “Enough” is clearly “plenty.” You always know when something is not “enough.” “Enough” comes with pivotal moments, pivotal decisions. Yous clear the bar or you don’t. If yous are a hero, you will clear it in some way, eventually. Power levels help define the bar.
Few events in life have an “enough.” We focus on the moments that seem to have 1 — loftier-stakes testing, hiring decisions, weddings — but we worry about the ones that don’t: What if at that place is a judgment that suddenly falls on my head about beingness “enough” every bit a son? As a partner? As the author of an overly analytical article about
Dragon Ball Z?
The ability-level mindset is comforting in comparison. It offers a person a deterministic toy universe and crowns you lot the little despot. And nonetheless things get-go to dissolve if one looks at it for too long. The edges of your certainty mistiness, and you are costless in the violent current of air.
For the curious, here is
a link to the data spreadsheet
: If you find a source that I did non, send me the link. I won’t update this article, but I
tell y'all if the analyses modify at all.