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In "Deja Q," the now-human Q talks to Guinan in Ten Forward, when he states that he is "just one of the boys with an IQ of 2005."

Where did Q get this information? Did Beverly give him an IQ test? Was he informed by the Continuum before being sent away that he'd retain his intellect? If so, that means Q's IQ while fully Q would be 2005. It seems strange that the Q would use intelligence quotient, so that seems unlikely.

Ham Sandwich
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The current convention is that IQ test results are scaled to fit a normal distribution with mean 100 and standard deviation 15. This means that an IQ of 2005 corresponds to a score which is 127 standard deviations above the mean (since (2005-100)/15=127), which would mean that roughly 1 in 10^3505 people are as intelligent as Q is.

The population of the galaxy cannot possibly be anywhere near as large as 10^3505, and contains more than one member of the Q continuum.

So either:

  1. Q is full of it (this sounds the most likely to me)
  2. Q is massively more intelligent even than other Qs (ha ha ha)
  3. IQ has been redefined in the 24th century to be an absolute measure of intelligence rather than a measurement which is relative to the general population (in which case we don't have enough information to answer this question).
Micah
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    We know from various episodes that the galaxy is merely one of an infinite number in a multiverse. Have you taken that into account? – Valorum Feb 18 '17 at 18:17
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    I think trying to count populations across multiverses is fraught with difficulty when many of the people in question are effectively duplicates of each other. If you like, I suppose you could imagine that Q knows some way of doing it, is aware of enough different universes to make that determination, and is telling the truth. Personally I'd rather stick with a general rule of "anyone claiming an IQ over 300, in any work of fiction, is either lying or using a very different definition of IQ than ours." – Micah Feb 18 '17 at 18:45
  • Do we know Q is from our galaxy? – 1252748 Feb 18 '17 at 18:51
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    @1252748 - We do not. They also don't seem to be bound to it since Q(uinn) takes the Voyager outside the entire universe in Voy: Death Wish – Valorum Feb 18 '17 at 18:52
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    @1252748: This answer would not change significantly if you replaced "the galaxy" with "the observable universe", or even "a number of copies of the observable universe which is equal to the number of atoms in the observable universe". – Micah Feb 18 '17 at 19:01
  • Definitely needn't change since the first suggestion is absolutely the correct answer :-) – 1252748 Feb 18 '17 at 19:25
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    Fourth possibility: IQ is still normally scaled, but only to humans (or close), and not to all possible sentients. E.g. if snails had a SpeedQ metric, normalized to the standard top speeds of snails and slugs, a cheetah would be many thousands of standard deviations off the top of it. How exactly that extrapolation would work without good calibration is a little iffy, though. -- That said, I agree Q being full of it (picking an arbitrary large number to point out he's terribly smart) is the most likely answer. – R.M. Feb 18 '17 at 20:51
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    @R.M. That's actually what I was trying to get at with possibility #3. The problem is, IQ scoring is entirely ordinal — that is, it purports to tell you who's more intelligent than who, but has no notion of how much more intelligent they are except by counting the number of people there are of intermediate intelligence. It's not (at the moment) like speed, which we can measure independently and then perhaps notice that it falls into a normal distribution; the normal distribution is part of the notational convention. – Micah Feb 18 '17 at 21:02
  • I appreciate all of the effort in this thread. Very good thoughts. – Ham Sandwich Feb 18 '17 at 23:08
  • "An intelligence quotient (IQ) is a total score derived from one of several standardized tests designed to assess human intelligence." It's not unrestricted to any theoretically sentient beings; so this answer's analysis is invalid. (https://en.wikipedia.org/wiki/Intelligence_quotient) – Daniel R. Collins Feb 18 '17 at 23:18
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    Just because it's designed to assess human intelligence doesn't stop you from administering it to a non-human — and if the non-human is a sapient Star Trek alien, you'll likely get a reasonable result. If you want to argue that the reference population should consist solely of humans, that seems plausible, but it only strengthens my point — there are likely at most a few trillion humans in the Trek-verse, meaning that there's no good basis to assign anyone an IQ score of more than 205 or so. – Micah Feb 18 '17 at 23:26
  • Fifth possibility: Q took a quick break through time and took a human IQ test, at some point relevant to 24th century human IQ scores. – Jason C Feb 19 '17 at 00:40
  • @Micah - A normally scaled measure (as opposed to, say, a straight-up percentile) wouldn't actually be useful without some basis for believing there is an underlying real quantity that is normally distributed. – Random832 Feb 19 '17 at 05:29
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    @Random832: Converting between normally scaled measures and percentiles is a pretty trivial computation or table lookup. If one is useful, so is the other. (The normally scaled measure wouldn't be the optimal way to report scores unless there was an underlying real quantity that was normally distributed, but people do all kinds of sub-optimal things in the name of convention, so that's not very strong evidence of anything.) – Micah Feb 19 '17 at 05:56
  • Plus one for the math. – Reinstate Monica -- notmaynard Feb 19 '17 at 15:51
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    Why is the galaxy median IQ 100 ? Also, we have seen Q shrinking an entire spaceship to subatomic levels. So why can't Q make 10^252525252 more Q ? – nurettin Dec 09 '19 at 13:13