I vaguely remember a short story, I believe written decades ago, where the electoral system was so refined that, rather then polling millions of citizens, it allowed to individuate the single person so average that his opinion reflected perfectly the average opinion of the whole population. I am not sure whether he was just asked his preferences or even made President or some other eminent position.

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    "Franchise", as Mike Scott says, is about the society where voting was reduced to one person. But he wasn't "perfectly average"; that's a different story, where a man was discovered who by a statistical fluke exactly matched all the "average" figures (height, weight, number of kids etc) and he became so popular he was elected president. Can't find the name of that one at the moment. – Daniel Roseman Sep 2 '15 at 11:34
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    @DanielRoseman, please tell me he didn't have 2.2 kids. Eww. – gowenfawr Sep 2 '15 at 12:22
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    @gowenfawr: Reminds me of Norton Juster's The Phantom Tollbooth, in which there is a minor character who is 0.58 of a boy. He explains that he comes from a perfectly average family. – Nate Eldredge Sep 2 '15 at 15:10
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    I could swear I've read a story like this but I haven't read either of the stories in the answers so far. That makes me wonder if there are any more stories with similar themes - particularly by Philip K. Dick, of whom I've read maybe everything and this sounds like an idea he might play around with. It has shades of We Can Remember It For You Wholesale. – Todd Wilcox Sep 2 '15 at 19:13
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    @ToddWilcox Also "The Mold of Yancy" about (spoilers!) a politician who is only ever seen on TV because he's actually a computer-generated composite of all the things the average person likes in a leader. – dodgethesteamroller Sep 3 '15 at 0:03

I found the story I was thinking of, where an "average man" was discovered who by a fluke exactly matched all the averages. It's "Null-P", by William Tenn.

The world had been devastated by a nuclear war, and the population was desperate for a bit of normality, so jumped on this paragon of averageness, electing him president and starting a trend (similar to the theme of stories like "Harrison Bergeron", "The Marching Morons", etc.) where "averageness" is celebrated.

In the end,

the descendants of the "average" people are out-competed by evolutionarily advanced dogs who keep humans around simply to throw sticks.

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    Sounds like a better version of idiocracy – slebetman Sep 2 '15 at 15:41
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    If I understand correctly, this is not an answer to the original question, but to a different story-ID you asked yourself in a comment. Wouldn't it be better to re-ask that question as an actual question and then self-answer it? – SQB Sep 2 '15 at 16:06
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    Thanks for both answers, Daniel and @MikeScott! Actually, Tenn's one rings a stronger bell, for the name of the main character, Abnego, his averageness and the fact that he “goes on to become a 6-term US president and then a World leader” (found here). But apparently I was in my mind conflating the two stories. – DaG Sep 2 '15 at 16:12

That is "Franchise" by Isaac Asimov. To quote from the linked Wikipedia article:

In the future, the United States has converted to an "electronic democracy" where the computer Multivac selects a single person to answer a number of questions. Multivac will then use the answers and other data to determine what the results of an election would be, avoiding the need for an actual election to be held.

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    Lol at "Multivac". – Carcigenicate Sep 2 '15 at 18:31
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    @Carcigenicate Multivac is nothing to lol at. – Mr Lister Sep 3 '15 at 7:38
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    One does not simply 'lol' at Multivac. It is obeying Zero-th Law, and you merely fall under First Law. – Eight-Bit Guru Sep 3 '15 at 11:22

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