Our number sense is logarithmic, the first number we hear drags us along, and we trust ourselves far too much.
If you're asked to place the number 150 on a line running from 0 to 1,000, you'll put it roughly where it belongs. A seven-year-old won't: they put it much further right than it belongs. It's not that they can't count. Their sense of number works like every human's does before school, and that sense isn't linear.
Psychologists Robert Siegler and John Opfer measured this with that very line: young children stretch out the small numbers and squeeze the big ones, as on a logarithmic scale. Between 1 and 10 there's plenty of room; between 900 and 1,000, almost none. Neuroscientist Stanislas Dehaene found the same in Munduruku adults, an Amazonian people with no words for large numbers: they arranged quantities by ratios, not differences.
It's the old Weber's law: we notice the difference between 1 and 2 pounds, but not between 50 and 51, even though both differ by one pound. What we perceive is the ratio. School teaches us to think on a linear ruler, but the other one is still underneath, and it shows up as soon as numbers get out of hand.
Past a million, language doesn't help either. A million, a billion and a trillion all sound like "a whole lot", even though each is a thousand times the one before. And other languages muddle it further: in Spanish or German, a billón or Billion is what English calls a trillion. Translated news stories regularly multiply a figure by a thousand without anyone noticing.
A trick to keep your bearings: count zeros in threes. Thousand (3), million (6), billion (9), trillion (12). Every jump of three zeros is the same thousand-fold jump, even if the names hide it.
In 1974, Amos Tversky and Daniel Kahneman spun a rigged wheel of fortune in front of people that could only land on 10 or 65. Then they asked them for a percentage about African countries. Those who had seen 65 gave much higher answers than those who had seen 10, even though they knew perfectly well the wheel had nothing to do with the question.
It's called anchoring, and it's one of the most studied traps in psychology. Any number in your head when you start estimating becomes your starting point, and you then move only a little from it. That's why it pays to reason the answer from scratch, from things you know, instead of adjusting the first number that pops into your head.
Marc Alpert and Howard Raiffa asked Harvard students to give, for questions about quantities, a range they were almost certain (98%) contained the answer. If the ranges had been honest, the answer would have fallen outside only 2% of the time. It fell outside around 40% of the time. The ranges were far too narrow: we think we know more than we do.
The practical lesson is simple: when you estimate, open your range wider than feels natural, then find the middle in times, as explained in how to estimate any number.
GutGuesser's ruler is logarithmic for all these reasons: it's how the mind works. Being off by double costs the same whether the answer is 10 or 10 million, which lets the same game ask about the weight of an ant and the weight of a ship. The hard part isn't knowing the figure: it's not getting dragged along and not being off by ten.
Four questions a day, one minute. Nobody knows the number; whoever gets closest wins.
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