Your typical dunce (at least in fiction) has the intellect of a snail. But attaching numbers to everything is fun, so why not stick him or her with a lower IQ score to prove his or her stupidity?
Unlike the opposite end of the spectrum, it's much more rare to see one of these scores reach ridiculous levels that don't match up with real world IQ testing. Of course, when it does get ridiculous (to the point of giving negative scores), it's much more jarring. On the other hand, most really ridiculously low IQs - single digits, or negative numbers - are found in comedies, where the Rule of Funny can be blamed, or in-character hyperbole.
See Idiot Hero, The Ditz, and The Fool for character concepts that may invoke this. If the character isn't managed properly, this may be an Informed Flaw and result in fans calling Obfuscating Stupidity.
Contrast Improbably High IQ. For more information about IQ testing, go here.
In Forrest Gump, Forrest has a somewhat more probable IQ of about 75, which is considered the lowest for "normal" intelligence, or mildly retarded. While low, someone with an IQ of 75 could still go to school, get a job, get married, etc., just as Forrest did. It's notable that in the old days with harsher (or one might say different) terminology, one wasn't considered a "moron" until an IQ of 70, psychologically speaking.
Alien³ also averts this. Aaron, Superintendent Andrews' assistant, is commonly known by the nickname "85", which is his IQ. He demonstrates some noticeably stupid behavior, but manages to avoid lapsing into complete idiot territory.
Idiocracy demonstrates this. The two intelligent people who never bred had a believable IQ of about 130, while the football player and the trailer park people, had scores in seventies and eighties, which are believably low-average. However, five hundred years later, the average score was five. Granted these people were so stupid that their society was collapsing around them and most of them probably would qualify as retarded, but given how IQ is defined, it's impossible for the average IQ to be anything other than 100, since the norms are adjusted to ensure that average. Justified, as they may be too stupid to realize that.
In one of the spoof trailers for Tropic Thunder, the main character of Simple Jack is stated to have an IQ of five.
What do you get from a glut of TV? A pain in the neck and an IQ of 3
In the Discworld novel Thud!, the rather drug addled troll named Brick is described at one point as having a possibly negative IQ for comedic effect. Sergeant Detritus is at least once jokingly commented to have a room temperature IQ, but in fact he is no fool and an excellent policeman, if not especially bright. Also subverted in that the comment can be taken literally, as Troll's brains in Discworld work faster in low temperatures, so his intelligence really is likely to be affected by the room temperature.
However, with trolls, there's an inverse relationship between temperature and intelligence. Detritus is brilliant when he's about to freeze to death in the Pork Futures Warehouse, and Mr. Shine is considerably sharper than the run-of-the-mill Discworldian even in "normal" temperatures, being composed of stuff that naturally refracts light instead of absorbing it.
It has been theorized that if a troll would actually sit down in a chair and take an IQ test, it would score lower than the chair it was sitting in.
The Tsumi wrestlers in "Interesting Times" are apparently chosen in part for their savage lack of intelligence; One Big River (a guard who is dim but functional) supposedly failed the application test when he didn't try to eat the table.
In The Westing Game, in Sandy's notes of the other heirs to Westing's will, a note on Otis Joseph Amber is that his I.Q is 50 (Mild to moderate mental retardation, depending on the scale you're looking at). A person with this I.Q would have difficulties on their own, but it's stated that Amber is a delivery boy and even marries Berthe Erica Crowe after the contest ends. When he's around, he acts more nutty yet competent than mentally impaired
Less so if it's interpreted as a shorthand way of saying "the average (contemporary for the speaker) person would, on a (contemporary for the protagonist) Stanford-Binet IQ test, score somewhere around the point at which the results become too low to be reliable, say 45." This still seems awfully low, though.
The narrator of Stuck in Neutral has no conscious control of his muscles, and is typically assumed to be mindless, so his IQ was once measured at 3. (From his narration, he seems to be of above-average intelligence—he's just completely uneducated.)
Live Action Television
Hazel Wassername of 30 Rock has an IQ of 70 - which she seems to think is pretty high.
Red Dwarf: Holly the computer, once said to have an IQ of 6000, has succumbed to computer senility. While trying to prove he's not completely gone, he he fabricates a back-up computer program called Queeg, who "takes over" Red Dwarf in the episode "Queeg". In the episode, Queeg claimed that Holly's IQ was actually 6. ("It's got a six in it, but it's not six thousand...")
Garfield once gave us this list of beings, in order of intelligence (from 0 to 1000): Amoeba - slug - dog - stone - ape - human - kitchen machine - cat.
Even further, a special anniversary edition printing of Garfield told us that Odie's IQ is so low, "you can't test it, you have to dig for it." Despite these claims, Odie actually HAS beaten Garfield on rare occasions.
In the Dutch comic Dirkjan an extra has this to say about Marini: "You gotta keep an eye on that one. He has an IQ in negative numbers"
Averted in Tropico. Having certain hindrances puts El Presidente's IQ at "just below room temperature", which, if one measures temperature on the Fahrenheit scale, puts him near the cutoff for "profound retardation".
The animation "Tramps Just Wanna Have Fun" reveals "an IQ of 1".
Suzaku from Code MENT scored negative on his IQ test and is surprised when Lelouch tells him that´s a bad thing.
Homer Simpson of The Simpsons has an IQ of 55 (taking a crayon out of his brain raised it 50 points to 105), but lives in a normal (not really, but within the show, yes) suburban home without any assistance.
Interestingly, his 105 IQ is portrayed as ungodly high, to the point where he devises a mathematical equation that flawlessly disproves the existence of God. Of course, 105 is actually only barely above average. Fridge Brilliance sets in when you realize that compared to everyone else around him, 105 probably would look downright brilliant.
Peter Griffin of Family Guy has an IQ of 40. (The episode "Petarded" puts him on a graph below both normal people and the line for retarded, but above creationists.) The episode in which this is revealed blatantly said Peter was retarded, and he can't live without assistance, temporarily losing custody of his kids after Lois was injured. However, he still does a little more than he should be able to. While his job performance is very poor, he apparently is good enough to keep it. Also, he's legally allowed to drive and consent to major surgery, even though his "informed consent" to the latter is complete guesswork.