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The use of IQ as a qualification or 'proof' of superiority. Thanks in part to the common belief that IQ is a flawless, accurate representation of every aspect of human intelligence and partly because it's an easy way to show an intelligent character's arrogance, a character might use their supposed IQ as an argument for their taking control of a matter, or otherwise being better than others.
The characters who do this will usually be on the border of Improbably High I.Q., if not over it. If it's aliens doing this, it can be part of a Puny Earthlings routine.
In Real Life, by the way, IQ is a relative measure; the average is always 100. While we're at it, "genius" is kind of a vague term as an official assessment of intelligence; it's generally accepted that the line is at either 130 or 140, but there's no actual agreed-upon documentation making it official. For more information, see the Useful Notes page for IQ Testing.
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In the opening narration of Scorpion: "My name is Walter O'Brien. I have the 4th highest IQ ever recorded - 197. Einstein's was only 160".
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On The Simpsons, Springfield is left under the control of local Mensa members after the mayor skips town in "They Saved Lisa's Brain". They disagree on how the town should be run and eventually start arguing by stating their IQ at each other. Frink insists his 199 IQ qualifies him to be in charge — but is soon "outranked" when Stephen Hawking arrives on the scene!
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In The Paper Chase, when James Hart first meets him, Franklin Ford III boasts of having an I.Q. of 190 as a sign of his superiority.
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In The Big Bang Theory, Sheldon often uses his IQ as a 'proof' that he's correct.
There's also the first episode when Sheldon and Leonard list their combined IQ of 360 as evidence that they can figure out how to get into a locked apartment building. Two girl scouts then come up, ring every bell, and are immediately buzzed in.
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 The Big Bang Theory
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Used in the Red Dwarf episode "Holoship". The holocrew members of the holoship state their IQ along with their name and rank, and there seems to be a positive correlation between IQ and rank. Rimmer introduces himself with "IQ unknown."
Also used in "White Hole", where Rimmer points to Holly's IQ to explain why he thinks she, as opposed to Lister, should be in charge of their attempts to destroy the white hole.
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Bakuman。: invoked Ashirogi's Money and Intelligence manga is meant to be a Deconstruction of how overemphasized both tend to be, especially the idea that the more you have of either, the better off you're supposed to be. Nanamine reads the series and concludes that money and intelligence really are all that matter.
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In the Commander Keen games, Big Bad Mortimer McMire brags that his IQ is 315, one point higher than our hero (which is 314 in tribute to Pi).
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 Commander Keen (Video Game)
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A variation of this happens in Baka and Test: Summon the Beasts. Students are assigned to a corresponding classroom depending entirely on the score they received during the entrance exams, ranging from the prestigious (Class A) to the insignificant (Class F). The smarter students get better amenities and classroom equipment.
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 Baka and Test: Summon the Beasts
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In C.I.A.: Agents the five principal agents get their "Place Rank" by this.
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One episode of Malcolm in the Middle has the teacher of the gifted class Malcolm is in introduce a system where they are ranked by intelligence. Malcolm obviously see this is B.S., but his classmates take the system to heart, and even forget their names after they memorize their rank number. Eventually Malcolm demonstrates how dumb this system is by being the highest ranked student, and everyone else having a complete mental breakdown over how "stupid" they are.
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Tokimeki Memorial Girl's Side: In the archenemy scene, Takafumi Wakaouji is mentioned to have an IQ of 200.
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Blake's 7:
In "Hostage", a villain holding Jenna and Cally at gunpoint warns them not to try anything as he has a high intelligence quotient. They soon demonstrate he's Too Dumb to Live.
The Federation has a class system apparently ranked on this. Vila claims he engaged in Obfuscating Stupidity to avoid a high ranking and the required military service. Avon of course says that no faking on Vila's part was required.
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One World uses this to kickstart the Fandom-Specific Plot of several characters being deemed to be incredibly powerful wizards.
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GURPS uses "IQ" as the short-form name of the Intelligence stat — a measure of all forms of intelligence, independent of culture or species — although it has barely any relation to actual measures of IQ. Apparently it wasn't meant to.
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Rifts and other Palladium Games does the same, with a note that the stat x10 is roughly equal to the character's IQ. Since stats for a human are on a 3-18 scale, this completely fails to work, and falls into both Improbably High I.Q. and Improbably Low I.Q..
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In Ender's Shadow, the Battle School commander creates resentment toward young Bean from his classmates by revealing that Bean scored highest among them not just on IQ, but on every aptitude measure but one — that of physical ability, since Bean is much younger and smaller.
It is later revealed that he also scored extremely low in what is arguably one of the most important traits, ambition. Or specifically, the need to dominate and destroy others.
It's also revealed that Bean was the sole surviving subject of an illegal genetic engineering experiment (all the other babies, Bean's siblings, were killed to cover up the experiment) aimed at creating extremely intelligent people with the side effect of a short lifespan caused by the Square-Cube Law. Bean's brain doesn't have the same "off" switch as everyone else's and continues to grow and form new neural pathways like an infant's. As a result, Bean is fated to die at a relatively young age.
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Frasier: Niles and Frasier are reminded that they took an IQ test as children, and all their mother would tell them is that they were two points apart, and when they discover the results in an old box of childhood memorabilia, Niles reads them and announces that he has the higher score. When a suspicious Frasier insists on seeing the results for himself, he finds out that not only is Niles' score higher, but it's by considerably more than two points. Frasier spends the episode angsting about being dumber than his brother.
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The following is a list of statements referring to the current page from other pages.

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Intelligence Tropes
 Lupin III: Part 5 / int_d42367cf
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