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 end, one of the dogs lost weight, and five of the people lost and gained weight in unpredictable ways. But one subject lost 21 grams at the moment of death. After millennia of evolution from the pneuma in The Iliad and qi of ancient Chinese humoral medicine, this figure of 21 grams was the first attempt at quantifying the vital force of a human being. MacDougall was never able to put any more dying people on his scales, and his experiments were fundamentally flawed, of course, but his dismal nonsense represents an attempt to bridge the gap not only between the quantitative and the qualitative but between the profane and the sacred, between the known and the felt. It’s hard to blame people for being tempted by Promethean power: they learn the error of their ways, one way or another.
I do blame Akira Toriyama, though, for introducing this morality play in the pages of Dragon Ball Z in 1988. Just as it 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 just how strong Goku and his fellow fighters were based on years of fan-driven mathematical speculation. I wanted to come to a greater conclusion. But I severely underestimated the imagination of forum-dwellers.
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The soul of Toriyama’s was once mysterious. In the original Dragon Ball, ki was a mystical energy developed through meditation and training with the world’s masters. Its depiction resonates with historical qigong techniques, kung fu movies, and Journey to the West, Dragon Ball’s distant source material. High jumps had the potential to evolve into flight. Focused punches could become focused energy blasts.
Dragon Ball’s characters could learn to sense each other’s ki — mostly its strength, 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’s army, a paramilitary for a galactic real estate company, were equipped with augmented reality lenses that would display the power level of the person in their sights. They could determine whether a planet’s inhabitants were weak enough to bulldoze or strong enough to require reinforcements. Early power 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 up from 408 to 1480 when unleashing his Special Beam Cannon, fatefully and fatally surprising the villain Raditz.
Toriyama invented power levels as a short-hand for the audience to understand the stakes of a Dragon Ball encounter, but the stats also 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. Power levels ceased to be predictive. Accordingly, around when Frieza was vanquished — just over one third of the way through the 291-episode Dragon Ball Z — power levels were never again mentioned in the anime or official secondary sources. The highest official power level, and one of the last to be canonized, is found in the reference work Daizenshuu 7. It gives Goku’s full Super Saiyan power when battling Frieza: 150,000,000. After that, power levels were at best extratextual doodads, occasionally popping up alongside manga chapters as a way to promote a new movie.
Though Toriyama gave up on power levels, their quantitative precision took hold in many a child’s mind, including mine. Dragon Ball Z blew up in popularity in the United States in 1999, when Cartoon Network began airing a newly dubbed Frieza Saga. These episodes would be the last ones boasting power levels. In 2000, there was a whole world of many, many, many fights — clear victories, near 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 information that they would need to extrapolate and estimate new power levels: the average gulf in skill between heroes and villains at the beginning of an arc, the effects of various power-up and training techniques, and how much more powerful Saiyans become after being beaten to the brink of death.
A fan could create a power level logic that could explain the whole Dragon Ball Z universe all on your own. Then they could proudly put their findings up on some random internet forum as a monument to their fan dedication and intellectual prowess.
Though I never took part in coming up 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 absorbed? And how many times more was that than Frieza’s final form? How much power did going Super Saiyan 2 or 3 give a fighter? What was the more powerful fusion technique? The fact that I only found speculation rather than definitive answers only made me want to look harder for concrete proof.
There was never a consensus on which unofficial power levels were accurate, and no one was interested in building one. Toriyama may have dropped power levels because they were unnecessary — quite explicitly telling rather than showing — but they distilled a fundamental component of shonen anime: the definitive power to control and defeat someone, to conquer, to be better. It is an iron law of an emotional universe, one that children on the verge of preteenhood can learn to wield.
I thought I could return to this world over 20 years later and have a more sophisticated insight into the logic behind fans’ power 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 find value in Dragon Ball Z power level data would be quixotic. Data science is always messy. But, then again, it wouldn’t really be quixotic if, by the time I gathered my sources, cleaned my data, and did my analyses, I didn’t actually believe there would definitive results. And I did find something.
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 still hosts these things?). Bless the work of various fandom wikis as the new host for this kind of content, even if I may now have that special kind of computer virus unstuck from time.
To demonstrate how power-scaling in the DBZ fandom works, and why it’s only made definitive results more challenging to pin down, here’s an example from how forum-poster “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 can’t be wrong as long as 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 e-mailed me about a “Dragonball Guide Book” which lists levels for the entire series. Let me set the record straight on this one: those levels mean nothing. Come on people! Even those of you who watch the English version of DBZ know that Freeza has over 1,000,000 in his second form. The guide book lists him at about half that. Now, which do you believe? The manga written by Toriyama himself? Or an unofficial guide book written years later by a third party. I rest my case.
SkullMac’s list starts with official Daizenshuu power levels, but immediately hits some snags. Like 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 after seeing the full moon. There are mistranslations of stats: Most of the numbers in Daizenshuu 7 are not written in decimals but in a relatively common combination of decimals and Japanese place value demarcations for 10,000 and 100,000,000. The Daizenshuu power level for Super Saiyan Goku should be 150,000,000, as previously noted. SkullMac writes 15,000,000. Frieza’s 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 fair: Would Goku really have a base power level jump from 90,000 to 3 million in a matter of days, as the Daizenshuu says? Mistake or not, I would have guessed something like SkullMac’s power levels back in the day.
These quirks and a thousand others are littered throughout the power level lists. The hope is that these errors will fall away as sources get aggregated. Some will fall away 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 also included data from the DBZ movies and from the noncanonical Dragon Ball GT, just for some flavor. I outright ignored anything from Dragon Ball Super, a series that started 20 years after Dragon Ball Z ended and is so boring that watching it makes me believe I am already dead. (I also ignored the OVA because, even though it has some plot connection to GT, it is very confusing.) In short, for the purpose of this analysis, if a 2000s-era me couldn’t download it as an AMV off Kazaa, older me did not care about it. In the end, the GT data was so sparse that I couldn’t do much with it other than reminisce about Super Saiyan 4 and Limp Bizkit.
While cleaning the data, I found a couple sources that would have derailed the results, numbers so large as to be incomprehensible. Unlike the average power level list, which would more gently move up the orders of magnitude, these extreme datasets exploded far beyond those mere trillions. These lists felt more like the work of a sixth-grader who was just introduced to scientific notation and the orderly way English denotes numbers that will never be said aloud. The source itself is incoherent. The power levels already scrape the ceiling in the sextillions in the Cell Saga but then explode to “octoquinquagintillion” (10^179) to kill Cell and then onto alleged numbers like “novensexagintillion” (10^210). From the perspective of entropy, the amount of information in the universe in detectable particles is around 10^80 bits. If each particle in the universe was its own universe, there still wouldn’t be enough information to represent this number. It is a special thing that humans can conceive 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. Contained 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 data science competition site Kaggle. At the same time, it doesn’t seem big enough compared to what I found when I was a child. I didn’t remember any particular list, but I hope that was just a fault 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 most of those childhood lists were lost, coming from an age of the internet that was never meant to be permanent. I’m sure there are more lists out there, but I never may find them. A moment of silence for them.
Quantitative nihilism led me to freedom as I found myself rewriting character names, and changing how modifiers like “anger” and Super Saiyan stages were written, over and over again. Sure the data is logarithmically distributed over time, but it is a gentler increase within sagas, so I let it pass as “normally distributed.” Some trends in the data look roughly exponential: Why not assume it’s signal rather than noise? Why should I deign to consider using medians when they are boring? I let myself have a kind of surly fun, smashing action figures together in my mind and seeing what happened.
Let us now revisit questions that my childhood self wanted answered. First, how much more powerful does someone become as a Super Saiyan?
Below is a histogram of just that. The x axis is the Super Saiyan multiplier, and the y axis is the number of times people used something near that multiplier. I made comparisons within sources because of the wide differences in orders of magnitude between raters and some math reasons that make that hard to deal with otherwise.
The peak here indicates that most fans think going Super Saiyan makes you 50 times more powerful. This is also 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 Cell alongside his dad’s ghost, is at a 100-fold increase in power level, only twice the power of Super Saiyan. Super Saiyan 3 is a 400-fold increase, according to most 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 GT is an ambivalent memory outside of the canon these days.
At first, I was confused that the Super Saiyan 2 and Super Saiyan 3 power level gains were so meager after the 50-fold increase of the first Super Saiyan stage. But, thinking about how the show told its story, the reasoning began to make a bit more sense. It almost goes without saying that the outcomes of fights were based on the ratio of the winner’s power level over the losers. Frieza kept increasing his power multiple times with each form change, finally reaching an impossible peak above Goku’s Kaioken multipliers. Only leaping to legendary heights could Goku really gain an advantage over Frieza. In subsequent arcs, it was always the case that the villain — the androids or Cell or Majin Buu — was just out of reach 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 two or three, could turn the tide in the end.
How much more powerful do characters get when they fuse? That’s less clear. Compared to Goku and Vegeta’s powers combined at the time, the Vegito fusion is a 10- to nearly 300,000-fold increase. The median is around 300. The Gogeta range isn’t quite as wide but still cracks 100,000 times on the high end. This median is around 250. The Gotenks multiplier is more stable, perhaps because 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 after 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 Big Charts, noting the mean power levels of main characters throughout the series and the movies alongside the main villain of each saga and movie. (It is convenient that present and future Trunks don’t overlap in these.) The means are above. The medians are below. Each measure of central tendency has its faults. On the one hand, means are more sensitive to outliers, especially higher ones when the data is relatively log-distributed, as you see with the power levels. On the other hand, the median doesn’t get close to describing the wide range in power levels; you can see that with the standard deviation.
The fandom’s calculations of the mean suggest a massive power bump in the Cell arcs that increases steadily over the various 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 series is over. (Please ignore the single Gohan measurement in the Kid Buu Saga.) The mean and median power levels in the movies show three phases: pre-Super Saiyan, post-Super Saiyan, and the spitballing that occurred in the last 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 but in the steady increase in power saga-by-saga. Sure, as you see below, characters can become 10-20 (by the median) or 100-10000 (by the mean) times stronger within sagas or even single battles.
Heroes might discover some hidden power inside of themselves a couple of times, but those sudden paradigm shifts are supercharged by the way that the baseline power shifts over time. It’s apparent that Goku is doing push-ups in 100-times Earth’s gravity or shooting energy balls in the Hyperbolic Time Chamber or whatever, but the effect of that training is what matters more over time. There isn’t enough data to get good 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 least an order of magnitude more powerful just by training for ten days before the Cell Games, we only remember the lightning crackling around him as he turns Super Saiyan 2 and only doubles his power. (For a comparison, Goku becomes at most 30 times stronger over the entire course of the original Dragon Ball series.)
You even see it in Krillin’s power level in the Big Charts. By the Android Saga, he’s a pugilistic nonentity: he trains half-heartedly, but as the most powerful human he sidelines himself first as a kind of errand boy and then as a family man. Nevertheless people think he’s getting ten times stronger with each passing saga, keeping the same multiplicative relationship between him and the “real” fighters as if to track the stability of the emotional relationship. Conversely, Piccolo stops being a competitive fighter after fusing with Kami and losing to Cell, so his power level drops after that.
All of the fan data and drawn conclusions lead us to where the average DBZ fan starts: changes in power are changes in narrative. The multiplicative logic of power levels doesn’t need to make sense; it just needs to feel right in the moment. Vegeta threatened the Earth with his Galick Gun attack at a power level of 18,000. Frieza does destroy a planet, relatively easily, at a power level of around 100,000,000. By the Cell Saga, fights should destroy planets by accident, but they don’t. The outside world never matters: in Dragon Ball Z, the fate of the universe always hinges on to two sweaty men gasping on the knife’s edge between death and self-transcendence.
The deciding factor in any fight is someone becoming powerful “enough.” There is always an “enough” in simple stories. It is a metaphysical safety net. “Enough” is clearly “enough.” You always know when something is not “enough.” “Enough” comes with pivotal moments, pivotal decisions. You clear the bar or you don’t. If you 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 one — high-stakes testing, hiring decisions, weddings — but we worry about the ones that don’t: What if there is a judgment that suddenly falls on my head about being “enough” as a son? As a partner? As the author of an overly analytical article about Dragon Ball Z?
The power-level mindset is comforting in comparison. It offers a person a deterministic toy universe and crowns you the little despot. And yet things start to dissolve if one looks at it for too long. The edges of your certainty blur, and you are free in the tearing wind.
For the curious, here is a link to the data spreadsheet: If you find a source that I did not, send me the link. I won’t update this article, but I will tell you if the analyses change at all.