explain xkcd - User contributions [en]

1991: Research Areas by Size and Countedness

2025-10-22T02:08:46Z

2607:F140:400:6F:9832:655B:9E48:6AC2: /* Explanation */ Mentioned alternative interpretation of title text

{{comic
| number = 1991
| date = May 9, 2018
| title = Research Areas by Size and Countedness
| image = research_areas_by_size_and_countedness.png
| titletext = Mathematicians give a third answer on the vertical axis, "That question is poorly defined, but we have a sub-field devoted to every plausible version of it."
}}

==Explanation==

This comic is a [[:Category:Scatter plots|scatter plot]] that ranks different research fields according to the precision of the knowledge of the number of the studied object (vertical axis) vs. how large (the size of) the studied object is on the horizontal axis.

For instance, the facts pertaining to the number of United States presidents are well known (although the exact number can be disputed, in that Grover Cleveland's non-consecutive terms are usually counted seperately, so the official count exceeds the number of individuals who have become President; Donald Trump repeating this feat), so the study of their history is at the top of the Y-axis. This study is placed close to the Y-axis as the size of a president is about midway in size between the two extremes of the X-axis, elementary particles to the left (small) and the entire cosmos (cosmology) to the right (big).

On the X-axis, Presidents are close to the middle. Both presidents and other larger life forms (as a research area) including extinct animals (paleontology) and exobiology are all close to the same central position just right of the Y-axis, with smaller animals like birds and insects just to the left of the Y-axis. But where the number of presidents is well known (aside from the handling of split-terms), then the number of exoplanet life forms (exobiology) is completely unknown (and would likely be affected by other disputes, such as whether something the size of Pluto counts as a planet) and thus it will be found at the very bottom of the Y-axis, since we have no idea if there is life elsewhere and if so how many places will it be and how varied.

The 19 research areas are listed and explained in the [[#Tables of research areas|tables]] below.

In the title text, mathematicians may give a third answer that the concept of counting the things being studied is not reasonable, because the things are abstract or otherwise not discrete. There are many different types of math that blend into each other, and many have turned into separate sub-disciplines based on different interpretations of fundamental rules. As a specific example in geometry, different interpretations of how many lines you may draw parallel to another line through a given point has lead to {{w|hyperbolic geometry|hyperbolic}} (infinite parallel lines) and {{w|spherical geometry|spherical}} (0 parallel lines) geometric systems that are just as valid (and valuable, in some contexts) as the more commonly known {{w|Euclidean geometry|Euclidean}} (1 parallel line) geometry. As a specific example of the blending, {{w|number theory}}, {{w|set theory}}, and {{w|topology}} all interrelate and it is difficult to concretely say whether many theorems belong to one branch of math or another.

Another way to interpret this is that mathematicians do not know exactly how many mathematical objects there are. For example, under one interpretation of contingent, mathematicians do not know whether there are any infinities between {{w|beth zero|| to {{w|beth one}} (and analogously for {{w|Large cardinal axiom#Hierarchy of consistency strength|large cardinal axioms}}, so mathematicians do not know how many types of infinity there are, but based on set theorists investigating how the existence of certain types of infinity implies the existence of certain other types of infinity, one can say that set theory is a field devoted to studying every plausible version.

For a table with the coordinates given in percentage for each research field, see the table in the [[#Trivia|trivia]] section

===Upper left quadrant===
This is the section with the small items with count known.

{| class = "wikitable sortable"
! Research field
! Size of the thing
! Knowledge of #
! Explanation
|-
|{{w|Elementary particle physics}}
| The smallest subjects that we have actually detected are the {{w|elementary particles}}. In the {{w|Standard Model}} of particle physics, they are considered point masses (i.e. to have zero width). They may be made of smaller {{w|String theory|strings}} but if so these have still not been detected.
| We think we have a fairly good estimate of how many elementary particles that are known. There could be some uncertainty though, so it is not at the very top.
|Elementary particle physics is concerned with the study of subatomic particles (the smallest things that we know of), of which there are 17, not including antimatter. Most notably, until recently it was uncertain whether the {{w|Higgs boson}} was one of the elementary particles, but scientists have a "pretty good estimate" because the mathematical models don't predict the existence of many other particles.
|-
|{{w|Dentistry}}
|Several mm to several centimeters
|Most teeth are visible to the naked eye, and dentists have x-ray technology to see what's not visible, so counting them is pretty straightforward.
|Dentistry is the study of teeth (pretty small, both in size as well as in quantity). Humans adults grow 32 teeth, which is a "pretty good estimate" since it is very rare for {{w|Hyperdontia|more than 32 teeth to grow}} and it is rather common for {{w|wisdom teeth}} to be surgically extracted or in some cases never to develop. Children may only have 20 teeth before they start falling out, but each tooth that falls out is because another tooth is growing underneath, so a child might have as many as 52 teeth, counting the child teeth that haven't fallen out yet plus the adult teeth that are starting to form. So while a dentist will usually have a good idea how many teeth will be in a patient's mouth, they won't know for sure until they look or consult dental records.
|-
|{{w|Shakespeare}} studies
|Most are the size of typical book. In printed form, they would be in the range of tens of centimeters in height and width and ~1 centimeter in depth. Although, if stored in digital form, they could be much smaller than a tooth, so it seems to refer to print or handwritten originals.
|Generally, 36 plays are attributed to him, but between 1 and 3 additional plays are considered "lost" (i.e. at some point between being first published or performed and scholars seriously studying Shakespeare, all known copies, references, and fragments were destroyed, making it impossible to determine whether Shakespeare actually wrote them or whether they actually existed as separate plays), and {{w|Shakespeare apocrypha|some 20 more}} are believed to have been written by him, but not signed. To make matters worse, some plays that ''were'' published or performed under Shakespeare's name are believed to have been written as collaborations (not fully by him) or mis-attributed (we don't know who wrote them but most people say it was him).
There are also {{w|Shakespeare authorship question|various fringe theories}} that say very few, or even none, of the 'Shakespeare' works are actually by that man from Stratford. None of the alternative origins are widely accepted, however, and "Shakespeare studies" could still be considered the best umbrella term for the same group creative works that all the different Anti-Stratfordians are wanting to reattribute to numerous other people.
|Shakespeare studies is concerned with the works ascribed to William Shakespeare. These works are studied fairly commonly, so the position in about the middle makes sense. We also have a "pretty good" estimate of how many works are 'by Shakespeare'. Although the exact number is unknown, relative to other items on this list, we have a good estimate.
|-
|{{w|Ornithology}}
|Birds tend to be small, with most species able to be held comfortably in hand; even the largest known flying bird, the {{w|Condor}}, stands smaller than the average human, with a handful of non-flying avians such as the {{w|ostrich}} being larger, but still weighing less than 2-3 humans.
|The number of known bird species is [https://en.wikipedia.org/wiki/Bird#Diversification_of_modern_birds estimated at about 10,000], though [https://www.amnh.org/about-the-museum/press-center/new-study-doubles-the-estimate-of-bird-species-in-the-world a 2016 research result] suggested a near-doubling of this figure. As for the number of individual birds, a paper aptly titled [https://link.springer.com/article/10.1023/A:1018341530497 "How many birds are there?"] examines a number of ways of counting them; the results are "surprisingly consistent", with counts of approximately 200-400 billion individual birds.
|We do have a "pretty good estimate", to within perhaps a factor of two.
|-
|Ancient {{w|literature}}
|As above, with Shakespeare plays, original or print reproductions would be the size of a book, typically. Although ancient {{w|scrolls}} may have different dimensions with similar total volume.
|Because of the high number of {{w|lost works}}, it is hard to have a solid estimate of the number, although rough lists have been made (e.g. {{w|Ancient literature#List of ancient texts}}).
|While it is fairly straightforward to look up how many books [http://www.proquest.com/products-services/Books-in-Print.html are currently in print], or how many books [https://mashable.com/2010/08/05/number-of-books-in-the-world/ all currently printed information would fit into if bound into equal-length volumes], and then limiting those estimates to those that date before a specific year, counting how many books from the period of interest haven't survived to the present day (books that were "{{w|lost work|lost}}" either by deliberate discontinuation, or accidental destruction such as in the {{w|Destruction of the Library of Alexandria|Library of Alexandria}}) is a bit more difficult. However, because we know the work existed (it is mentioned by name in some other text), we have "pretty good estimate" that the number of lost works is "only" in the tens of thousands, as is the number of surviving works.
|}

===Upper right quadrant===
This is the section with the big items with count known.

{| class = "wikitable"
! Research field
! Size
! Knowledge of #
! Explanation
|-
|{{w|Marine mammal|Marine}} {{w|Mammalogy|Mammology}}''[sic]''
|They range in size from the {{w|Marine Otter}} (about 1m) to the {{w|Blue Whale}} (up to 30m).
|About 125 non-extinct species.
|Marine mammals are the largest extant animals. The US Government [http://www.nmfs.noaa.gov/pr/species/mammals/ recognizes] 119 marine mammals. However, what constitutes each species is [https://www.marinemammalscience.org/species-information/list-marine-mammal-species-subspecies/ constantly being revised], with new studies indicating either that what used to be considered a subspecies is actually a separate species, or that what used to be considered a separate species is actually a subspecies. As the depths of the ocean are further explored, species that were outright unknown are spotted and need to be classified. However, since marine mammals breathe air and thus must surface, it's likely that all species have been observed by scientists.
|-
|{{w|List_of_Presidents_of_the_United_States|Presidential History}}
|All presidents are {{w|Heights of presidents and presidential candidates of the United States|human-sized}}, with the tallest being {{w|Abraham Lincoln}} at 6 ft 4 in and the shortest being {{w|James Madison}} at 5 ft 4 in.
|As of 2021, 46 people (only 45 are unique; Grover Cleveland is counted twice because his terms were not consecutive) have served or are serving as President of the United States.
|Presidents are generally considered "big" men in history. Therefore, each one is fairly well known and documented. There is, however, some discussion on how many presidents there have been in the history of the United States, since prior to the {{w|Twenty-fifth Amendment to the United States Constitution|25th amendment}}, it was unspecified whether vice presidents counted as presidents during the President's absence. Most notably, this ambiguity is the reason {{w|David Rice Atchison}}'s tombstone is inscribed with the words "President of the United States for one day" (he was not eligible and did not accept the duties even if he was).
|-
|{{w|Railway engineering}}
|Railways can span across countries, and therefore are fairly large
|As railroads are built by humans, we know pretty well how many there are. However small systems (parks, mines) may make this number uncertain.
|A railway can span anywhere from a few hundred feet, to thousands of miles, so they're pretty big. The type of a railway is generally given by its {{w|track gauge}}, which is defined as "standard" (the usual gauge for a region or country), "narrow" (rails closer together than that standard) and "broad" (rails farther apart than that standard). Since what is standard varies from country to country, and indeed from line to line, how many kinds of "narrow" gauge and "broad" gauge exist depend on who you ask. However, whereas every region has ''a'' standard gauge, "{{w|standard-gauge railway}}" has a specific meaning used by rail technicians and enthusiasts worldwide, of a track with rails 1435 mm (4 ft 8.5 in) apart. Anything narrower than that is often described as a narrow-gauge line, even if it is the standard gauge for a particular rail network.
|-
|{{w|Geology}}
|The {{w|Earth}} is larger, by far, than everything else on the chart except the universe (Cosmology), black holes, and God (at least under some conceptions, see "Theology" below).
|There is only one Earth (at least if you set aside the possibility of multiverses, see below in Cosmology).
|Geology is generally considered the study of rocks (small rocks being considered fragments of mountain layers, so what counts as a "rock" for a geologist can be pretty big). There is no universally agreed upon number to how many {{w|List of rock types|types of rock}} there are, but all geologists agree they can be grouped into igneous, metamorphic, and sedimentary rock. Alternatively, geology can be construed as the study of the planet Earth's composition ( *geo*- meaning "Earth" ), and geologists are confident that the planet Earth is big and there is only one of it.
|-
|{{w|Cosmology}}
| As this encompasses (at least) all of the visible parts of the {{w|universe}} we live in, there can be no other "items" to study that would be larger.
| There is only one visible universe, but there could be multiverses/parallel universes, and also an infinite universe beyond the borders of our own part of this universe's event horizon. So it depends on who you ask if they say there is one of and infinite number of universes to study, thus it is placed close to the middle of the two extremes.
|Cosmology is the study of the universe. There is an asterisk with the note "Depends on who you ask", relating to the estimate of how many universes there are. While it might seem obvious that there is only one universe, some branches of physics believe that our universe is part of a {{w|multiverse}}, and this remains an open and contested subject in the field.
|}

===Lower left quadrant===
This is the section with the small items with count unknown.

{| class = "wikitable"
! Research field
! Size
! Knowledge of #
! Explanation
|-
|{{w|Mycology}}
|microscopic to a few miles
|Estimated at 2.2 million to 3.8 million species.(Though of these only about 120,000 have been described.)
|Mycology is the study of fungi (since fungi tend to grow flat -- excepting for mushrooms, which are their sexual organs, and do not exceed a foot in height (see [http://www.isciencetimes.com/articles/5740/20130729/giant-fungus-china-mushroom-world-s-largest-size.htm World's Largest mushrooms] -- mushrooms are generally considered small). Many fungi are microscopic, but some get to be a few miles in diameter.[http://www.nationalgeographic.com.au/nature/the-worlds-largest-living-organism.aspx The World's largest living organism.] It is a lot harder to discern which species a fungus is, and therefore classify it, so we "have no idea" how many kinds of fungi there are. Studies [https://www.ncbi.nlm.nih.gov/pubmed/21613136 vary wildly] between about 70,000 to over 5,000,000. There is a comic named after this study: [[1664: Mycology]].
|-
|[[1012: Wrong Superhero|Entymology]]
| For insects, from a fraction of a mm to several 100.
| Estimated from 1,000,000 to 3,000,000
|It is unclear whether [[Randall]] means {{w|entomology}} or {{w|etymology}} (probably neither; it's likely that this wasn't a mistake and it is possibly a direct reference to [[1012: Wrong Superhero]]). He may be referring to both fields as insects and words overlap in size. In either case, [https://www.ncbi.nlm.nih.gov/pubmed/28938083 estimates for insects] (entomology) vary from less than 1,000,000 to 30,000,000; and [https://www.quora.com/How-many-root-words-are-there-in-the-English-language estimates for root words] (etymology) reaching hundreds of thousands. Entomology was mentioned in the title text of [[1610: Fire Ants]].
|-
|-
|{{w|Microbiology}}
|The {{w|Smallest organisms|smallest viruses}} are around 30nm long. The largest bacterium may reach almost 1mm.[https://curiosity.com/topics/the-worlds-largest-bacterium-is-visible-to-the-naked-eye-curiosity/].
|120,000 to 10,000,000+.
|Microbiology studies microscopic (too small to see) organisms, of which some 1,400 are known and "estimates for the total number of microbial species vary wildly, from as low as 120,000 to tens of millions and higher", according to [https://www.quora.com/How-many-root-words-are-there-in-the-English-language Nature magazine].
|-
|{{w|Pharmacology}}
|{{w|Drugs}}, including {{w|medications}} and {{w|recreational drug use|illegal and recreational drugs}} are molecules which are sub-microscopic (in the range of nanometers).
|Although it is possible to tally all the known drugs, this is at the extreme low end of the pile because the number of possible organic compounds is nearly infinite and the fraction of those are bioactive is completely unknown.
|The number of drugs (pharmaceuticals) discovered and synthesized is not tallied, according to [https://www.raps.org/regulatory-focus%E2%84%A2/news-articles/2014/10/how-many-drugs-has-fda-approved-in-its-entire-history-new-paper-explains recent studies], but an estimate can be obtained by seeing how many have passed through the {{w|Food and Drug Administration|U.S. FDA}} (1,453). Many home remedies, which might technically qualify as drugs, have not been approved because {{w|Novelty (patent)|"everybody knows that"}}, as well as many solely recreational drugs since regulation might result in outlawing. Because of this, "we have no idea" how many drugs truly exist. Since drugs are extremely powerful molecules that are only administered in choice amounts, they are generally perceived as small.
|}

===Lower right quadrant===
This is the section with the big items with count unknown.

{| class = "wikitable"
! Research field
! Size
! Knowledge of #
! Explanation
|-
|{{w|Botany}}
|Plants tend to range from few centimeters to hundreds of meters. Therefore, on average plants are about the same size as human beings.
|Plants estimated from 295,000 to 305,000 in total.
|Botany studies plants, which can reach {{w|List of superlative trees|hundreds of feet by any measure}}. Some {{w|Pando (tree)|clonal colonies of trees}} spread for miles. However, plant tend to clump together in forests and jungles, which makes it hard to get to them and document them. Every year, thousands of new plants are discovered, with the best estimate being that there are [https://news.mongabay.com/2016/05/many-plants-world-scientists-may-now-answer/ nearly 400,000 vascular plants] and an additional [https://www.britannica.com/topic-browse/Plants/Nonvascular-Plants 12,000 non-vascular plants]. However, the rate of discovery doesn't appear to be slowing down significantly, so we truly "have no idea."
|-
|{{w|Paleontology}}
|Paleontologists study fossils, which range in size from very small to very large. When most people think of paleontologists though, they tend to think of them as studying large animals such as dinosaurs.
|Estimated at around 5 billion species.
|Paleontology studies fossils, particularly those of extinct animals, which can reach {{w|Largest prehistoric animals|huge sizes}}. However, since fossils form under very special circumstances, if the animal did not die under those special circumstances, there will be no record of their existence. Therefore, the number of extinct animals can never truly be known, but we've found [http://scienceblogs.com/authority/2010/01/12/how-do-we-know-that-most-of-th/ around 250,000]
|-
|{{w|Black Hole}} {{w|Astronomy}}
|Compared to most astronomical objects, black holes are fairly small. However, most of them (that we are able to detect) are still larger than the Earth, so they would still fall on the "big" end of this chart. Alternatively, Randall may be referring to their mass, which is on the scale of stars.
|It has been estimated that the number of black holes in the {{w|Milky Way}} is around 100 million ([http://hubblesite.org/explore_astronomy/black_holes/encyc_mod3_q7.html]), although there is uncertainty in that estimate and the total number in the universe depends on the size of the universe (see "cosmology", above).
|"Most stellar black holes [...] are impossible to detect. Judging from the number of stars large enough to produce such black holes, however, scientists estimate that there are as many as ten million to a billion such black holes in the Milky Way alone." ([https://science.nasa.gov/astrophysics/focus-areas/black-holes NASA Black Hole information page])
|-
|{{w|Exobiology}}
|The comic puts this in the size range of paleontology, which can include many sizes (see above), and also marine mammalogy, which tends to have individuals that are in the range of tens of centimeters to several tens of meters. However, {{w|life|life as we know it}} is dominated in numbers by {{w|microbes}}, and {{w|Evolutionary history of life|life on Earth}} began {{w|Abiogenesis|microscopic}}, leading most {{w|Astrobiology|Astrobiologists}} to hypothesize that life on other planets would necessarily include microbes and [https://en.wikipedia.org/wiki/Fermi_paradox#No_other_intelligent_species_have_arisen only possibly include macroscopic life].
|The estimate of {{w|List of potentially habitable exoplanets|how many planets with life there are}} varies from 16 to 40,000,000,000; additionally, [https://en.wikipedia.org/wiki/Habitability_of_natural_satellites#In_the_Solar_System multiple moons] are believed to be potentially habitable for some forms of life in our own solar system. However, the number of bodies apart from Earth confirmed to have life is currently zero. Even more uncertain than the number of potentially habitable exoplanets is the {{w|Rare Earth Hypothesis|huge uncertainty}} in the likelihood of life arising on a habitable planet.
|Exobiology refers to the study of life outside Earth, which requires {{w|SETI|scanning the entire universe for life}}. Currently, exobiology seeks to find a planet or similar body with life (and, {{w|definition of planet|to qualify as a planet}}, bodies capable of sustaining life are big). The uncertainty about how many planets have life in the Milky Way relates to the {{w|Fermi Paradox}}. For life, of the type we know, to exist outside of the Solar system there need to be planets around other stars. Such planets are called Exoplanets, and they have been a [[:Category:Exoplanets|recurrent subject]] on xkcd.
|-
|{{w|Theology}}
|Presumably, any god transcends the bounds of spacetime, making this the largest.
|Depends on who you ask.
|Theology is not a strict science, but as presented here it is the field concerned with the study of one or more {{w|deity|deities}} which is a sacred supernatural being. In particular, theologians study the question of whether {{w|theism|one or more gods exist}} {{w|atheism|or not}}, and, in the former case, whether there are {{w|polytheism|multiple gods}} or {{w|monotheism|just one}} or indeed whether there is literally only {{w|pantheism|one god}}. Although the existence of any supernatural being(s) is unfalsifiable by any known means, the entire human race has very strong opinions on the subject, so this field probably deserves the “depends on who you ask” disclaimer as well. Quantitative uncertainty is also mentioned in [[900: Religions]].
|}

==Transcript==
:[An X-Y scatter plot of research areas, written in gray font, where both axes have arrows in both ends. At the end of each arrow is a label. Above the left part of the X-axis there is a line which goes to a text about the meaning of the X-axis. Similarly there is a line to from the top of the Y-axis to a questions “asked” to those that study the given subject, their answers being somewhere between the two labels on the Y axis.]

:[The X-axis from left to right, text first and then labels:]
:Size of the thing you study
:Small
:Big

:[The Y-axis from top to bottom, question first and then labels:]
:"That thing you study - how many of them are there?"
:"We have a pretty good estimate."
:"We have no idea"

:[The research areas names are listed here below by sorting them into the four quadrants from top left to bottom right. In each quadrant the areas are listed after most left first, and then top to bottom for those at the same x position.]

:[Upper left quadrant (Small & count known):]
:Elementary particle physics
:Dentistry
:Shakespeare studies
:Ornithology
:Ancient Literature

:[Upper right quadrant (Big & count known):]
:Presidential History
:Marine Mammology
:Railway Engineering
:Geology
:Cosmology*
:<small>(*Depends who you ask)</small>

:[Lower left quadrant (Small & count unknown):]
:Pharmacology
:Microbiology
:Entymology
:Mycology

:[Upper right quadrant (Big & count unknown):]
:Botany
:Paleontology
:Exobiology
:Black Hole Astronomy
:Theology

==Trivia==
Sortable table with the coordinates in percent:
{|class="wikitable sortable"
! Research area
! Size (%)
! Estimate (%)
|-
|Elementary Particle Physics ||7 ||72
|-
|Pharmacology ||12 ||6
|-
|Microbiology ||15 ||13
|-
|Dentistry ||21 ||84
|-
|Entymology ||24 ||25
|-
|Mycology ||29 ||38
|-
|Ornithology ||34 ||62
|-
|Shakespeare Studies ||37 ||88
|-
|Ancient Literature ||38 ||53
|-
|Botany ||60 ||40
|-
|Presidential History ||62 ||89
|-
|Marine Mammology ||66 ||68
|-
|Paleontology ||68 ||31
|-
|Exobiology ||68 ||5
|-
|Railway Engineering ||79 ||81
|-
|Geology ||90 ||90
|-
|Theology ||91 ||5
|-
|Black Hole Astronomy ||92 ||26
|-
|Cosmology ||94 ||62
|}

{{comic discussion}}

[[Category:Scatter plots]]
[[Category:Rankings]]
[[Category:Science]]
[[Category:Physics]]
[[Category:Astronomy]]
[[Category:Math]] 
[[Category:Fiction]] 
[[Category:Religion]] 
[[Category:Animals]] 
[[Category:Exoplanets]] 
[[Category:Politics]] 
[[Category:Geology]]
[[Category:Scientific research]]
[[Category:Mycology]]

2435: Geothmetic Meandian

2025-10-22T01:52:00Z

2607:F140:400:6F:9832:655B:9E48:6AC2: /* Explanation */ Added italics for minima

{{comic
| number = 2435
| date = March 10, 2021
| title = Geothmetic Meandian
| image = geothmetic_meandian.png
| titletext = Pythagorean means are nice and all, but throwing the median in the pot is really what turns this into random forest statistics: applying every function you can think of, and then gradually dropping the ones that make the result worse.
}}
==Explanation==
This is another one of [[Randall|Randall's]] [[:Category:Tips|Tips]], this time a stats tip. This came as the first tip comic after the statistics tip in [[2400: Statistics]].

There are a number of different ways to identify the "{{w|average}}" value of a series of values, the most common unweighted methods being the {{w|median}} (take the central value from the ordered list of values if there are an odd number - or the value half-way between the two that straddle the divide between two halves if there are an even number) and the {{w|arithmetic mean}} (add all the numbers up, divide by the number of numbers). The {{w|geometric mean}} is less well-known but works similarly to the arithmetic mean. The geometric mean of ''n'' positive numbers is the ''n''th root of the product of those numbers. If all of the numbers in a sequence are identical, then its arithmetic mean, geometric mean and median will be identical, since they would all be equal to the common value of the terms of the sequence. However, if the sequence is not constant, then {{w|Inequality_of_arithmetic_and_geometric_means#Geometric_interpretation|the arithmetic mean will be greater than the geometric mean}}, and the median may be different than either of those means.

The geometric mean, arithmetic mean, and the {{w|harmonic mean}} (not shown) are collectively known as the {{w|Pythagorean means}}, as specific modes of a greater and more generalized mean formula that extends arbitrarily to various other possible nuances of mean-value rationisations (cubic, etc.).

{{w|Outlier}}s and internal biases within the original sample can make boiling down a set of values into a single 'average' sometimes overly biased by flaws in the data, with your choice of which method to use perhaps resulting in a value that is misleading, exaggerating or suppressing the significance of any blips.

In this depiction, the three named methods of averaging are embedded within a single function that produces a sequence of three values - one output for each of the methods. Being a series of values, Randall suggests that this is ideally suited to being ''itself'' subjected to the comparative 'averaging' method. Not just once, but as many times as it takes to narrow down to a sequence of three values that are very close to one another.

It can be shown that the xkcd value of 2.089 for GMDN(1,1,2,3,5) is validated:

{|-border =1 width=100% cellpadding=5 class="wikitable"
!
! Arithmetic mean
! Geometric mean
! Median
|-
! F1
| 2.4 || ''1.974350486'' || 2
|-
! F2
| '''2.124783495''' || 2.116192461 || ''2''
|-
! F3
| 2.080325319 || ''2.079536819'' || '''2.116192461'''
|-
! F4
| '''2.0920182''' || 2.091948605 || ''2.080325319''
|-
! F5
| 2.088097374 || ''2.088090133'' || '''2.091948605'''
|-
! F6
| '''2.089378704''' || 2.089377914 || ''2.088097374''
|-
! F7
| 2.088951331 || ''2.088951244'' || '''2.089377914'''
|-
! F8
| '''2.089093496''' || 2.089093487 || ''2.088951331''
|-
! F9
| 2.089046105 || ''2.089046103'' || '''2.089093487'''
|-
! F10
| '''2.089061898''' || 2.089061898 || ''2.089046105''
|}

Each row in this table shows the set Fn(..) composed of the average, geomean and median computed on the previous row, with the sequence {1,1,2,3,5} as the initial F0. While GMDN is not differentiable, due to the median, this can be interpreted as somewhat similar to a heat equation which approaches equilibrium through averaging. Interestingly, the maximum value alternates between the average and the median (bolded in the table), while the minimum value alternates between the geometric mean and the median (italiticzed in the table). This observation holds for many inputs.

To not distract from the comedic effect, the definition of the GMDN in the comic is left as a simplified sketch. To make the definition mathematically rigorous the implied infinite limit in the second line can be made precise, for example, as the result of a {{w|fixed-point iteration}} via <code>G = lim_{k -> infinity} m_k </code> where <code>m_0 = F(x_1, x_2, ..., x_n)</code> and <code>m_{k+1} = F(m_k) for k > 0</code>. This definition is well-defined only if we can proof convergence to a fixpoint of F for a set of inputs. Indeed, convergence holds if all numbers are non-negative (see discussions for proof and more cases). Note that the above definition yields a three-dimensional fixpoint G. Because all fixpoints of F are of the form <code>G=(g, g, g)</code>, with elements that are all equal, we can define <code>GMDN(x_1, x_2, ..., x_n) = G_1</code>, as the first element of G. This formal definition avoids the inconsistency present in the comic's definition sketch where the function GMDN as defined in the second line has the same three-dimensional output as F, but GMDN in the last line is shown to produce a single real number rather than a vector and is thus missing a final operation of returning a single component. Note also that the comic's definition of the median as the (n+1)/2-th {{w|order statistic}}, i.e. the (n+1)/2-th smallest value, coincides with the more regularly used sample median only on lists of odd length. For lists of even length the sample median is usually defined as the (arithmetic) mean of the two middle values <code>X_{n/2}</code> and <code>X_{(n+2)/2}</code> instead. Indeed, for lists of even lengths <code>X_{(n+1)/2}</code> is not well-defined without adding a flooring operation as (n+1)/2 is not an integer.

The comment in the title text about suggests that this will save you the trouble of committing to the 'wrong' analysis as it gradually shaves down any 'outlier average' that is unduly affected by anomalies in the original inputs. It is a method without any danger of divergence of values, since all three averaging methods stay within the interval covering the input values (and two of them will stay strictly within that interval).

The title text may also be a sly reference to an actual mathematical theorem, namely that if one performs this procedure only using the arithmetic mean and the harmonic mean, the result will converge to the geometric mean. Randall suggests that the (non-Pythagorean) median, which does not have such good mathematical properties with relation to convergence, is, in fact, the secret sauce in his definition.

The question of being unsure of which mean to use is especially relevant for the arithmetic and harmonic means in following example.
* Cueball has some US Dollars and wishes to buy Euros. Suppose the bank will exchange US Dollars to Euros at a rate of €5 for $6 (about 0.83333€/$ or 1.20000$/€).
* Megan has some Euros and wishes to buy US Dollars. Suppose the bank will exchange Euros to US Dollars at a rate of $7 for €6 (about 0.85714€/$ or 1.16667$/€).
[[Cueball]] and [[Megan]] decide to complete the exchange between themselves in order to save the {{w|Bid-ask spread}} of the {{w|Exchange rate}} which is the cost the bank imposes on Cueball and Megan for its service as a {{w|Market maker}}.
* Cueball offers to split the difference by averaging the rates €5:$6 and €6:$7 yielding a rate of €71:$84 (about 0.84524€/$ or 1.18310$/€).
* Megan offers to split the difference by averaging the rates $6:€5 and $7:€6 yielding a rate of €60:$71 (about 0.84507€/$ or 1.18333$/€).
In one direction (€/$), Cueball is using the arithmetic mean but Megan is using the harmonic mean while in the other direction ($/€), Megan is using the arithmetic mean but Cueball is using the harmonic mean. This creates two new exchange rates which are closer than the orginal rates, but the new rates are still different for each other. Megan and Cueball can then iterate this process and the rates will converge to the geometric mean of the original rates, namely:
* sqrt((5/6)*(6/7)) = sqrt(5/7) = 0.84515€/$ or
* sqrt((6/5)*(7/6)) = sqrt(7/5) = 1.18322$/€.

There does exist an {{w|arithmetic-geometric mean}}, which is defined identically to this except with the arithmetic and geometric means, and sees some use in calculus. In some ways it's also philosophically similar to the {{w|truncated mean}} (extremities of the value range, e.g. the highest and lowest 10%s, are ignored as not acceptable and not counted) or {{w|Winsorized mean}} (instead of ignored, the values are readjusted to be the chosen floor/ceiling values that they lie beyond, to still effectively be counted as "edge" conditions), only with a strange dilution-and-compromise method rather than one where quantities can be culled or neutered just for being unexpectedly different from most of the other data.

The input sequence of numbers (1, 1, 2, 3, 5) chosen by Randall is also the opening of the {{w|Fibonacci sequence}}. This may have been selected because the Fibonacci sequence also has a convergent property: the ratio of two adjacent numbers in the sequence approaches the {{w|Golden ratio#Relationship to Fibonacci sequence|golden ratio}} as the length of the sequence approaches infinity.

Here is a table of averages classified by the various methods referenced:

{|border =1 width=100% cellpadding=5 class="wikitable"
|+ averages using various methods
! Method
! Value
! Formula
|-
! Arithmetic
| 2.4
| Add all numbers, then divide the sum by n, where n is the number of terms.
|-
! Geometric
| 1.9743504858348
| Multiply all numbers, then take the product's nth root, where n is the number of terms.
|-
! Median
| 2
| Find the term or terms which separate the upper half of the set from the lower set. If the set has an even number of terms, find the arithmetic mean of the middle two terms.
|-
! GMDN
| 2.089
| (see above)
|}

==Transcript==

F(x1,x2,...xn)=({x1+x2+...+xn/n [bracket: arithmetic mean]},{nx,x2...xn, [bracket: geometric mean]} {x n+1/2 [bracket: median]})

Gmdn(x1,x2,...xn)={F(F(F(...F(x1,x2,...xn)...)))[bracket: geothmetic meandian]}

Gmdn(1,1,2,3,5) [equals about sign] 2.089

Caption: Stats tip: If you aren't sure whether to use the mean, median, or geometric mean, just calculate all three, then repeat until it converges

==Trivia==
Geothm means "counting earths" (From Ancient Greek γεω- (geō-), combining form of γῆ (gê, “earth”) and ἀριθμός arithmos, 'counting'). Geothmetic means "art of Geothming" based on the etymology of Arithmetic (from Ancient Greek ἀριθμητική (τέχνη) (arithmētikḗ (tékhnē), “(art of) counting”). This is an exciting new terminology that is eminently suitable for modern cosmology & high energy physics - particularly when doing math on the multiverse. However, it is unlikely this etymology is related to the term "geothmetic meandian" as coined by Randall, as it can be more simply explained as a portmanteau of the three averages in its construction: '''geo'''metric mean, ari'''thmetic mean''', and me'''dian'''.

The following Python code (inefficiently) implements the above algorithm for a list of non-negative numbers:

<pre>
from functools import reduce

def f(*args):
args = sorted(args)
mean = sum(args) / len(args)
gmean = reduce(lambda x, y: x * y, args) ** (1 / len(args))
if len(args) % 2:
median = args[len(args) // 2]
else:
median = (args[len(args) // 2] + args[len(args) // 2 - 1]) / 2
return mean, gmean, median

max_iterations = 10
l = [1, 1, 2, 3, 5]
for iterations in range(max_iterations):
fst, *rest = l
if all((abs(r - fst) < 0.00000001 for r in rest)):
break
l = f(*l)
print(l[0], iterations)
</pre>
Here is a slightly more efficient version of the Python code:

<pre>
from scipy.stats.mstats import gmean
import numpy as np

def get_centers(a, tol=0.00001, print_rows = True):
a = np.array(a)
l_of_a = len(a)
if l_of_a == 1:
return a[0]
elif l_of_a > 2:
result = all(
(
np.abs(a[0] / a[1]) <= tol,
np.abs(a[0] / a[2]) <= tol,
np.abs(a[1] / a[2]) <= tol,
)
)
if result:
return a[0]
res = [np.mean(a), np.median(a), gmean(a)]

if print_rows:
print(res)
return get_centers(res, tol)

</pre>

And here is an implementation of the Gmdn function in R:

Gmdn <- function (..., threshold = 1E-6) {
# Function F(x) as defined in comic
f <- function (x) {
n <- length(x)
return(c(mean(x), prod(x)^(1/n), median(x)))
}
# Extract input vector from ... argument
x <- c(...)
# Iterate until the standard deviation of f(x) reaches a threshold
while (sd(x) > threshold) x <- f(x)
# Return the mean of the final triplet
return(mean(x))
}

{{comic discussion}}


[[Category:Math]]
[[Category:Statistics]]
[[Category:Portmanteau]]
[[Category:Tips]]