16 ms·
What number is halfway between 1 and 9?
- deleted 14y ago[deleted]
- mratzloff 14y agoPerhaps the fact that human brains seem hardwired to perceive relative differences by orders of magnitude is an argument for graphing logarithmically by default, instead of linearly.
- Robin_Message 14y ago1.8 (harmonic mean) Seriously though, I thought we already knew that human perception is mostly logarithmic (for example, the eye performs well across 9 magnitudes of brightness – a linear system doesn't make sense). On the other hand, human perception of future value has generally found to be discounted hyperbolically, which is illogical (http://en.wikipedia.org/wiki/Hyperbolic_discounting http://en.wikipedia.org/wiki/Hyperbolic_discounting). I wonder is there is a correspondence with the harmonic mean but I can't get my head around it at the moment.
- pessimizer 14y agoNot illogical: http://en.wikipedia.org/wiki/Hyperbolic_discounting#Uncertain_risks http://en.wikipedia.org/wiki/Hyperbolic_discounting#Uncertai...
- davetannenbaum 14y agoThe reason why it is usually considered irrational/illogical is because hyperbolic discounting leads to dynamically inconsistent preferences. For example, under hyperbolic discounting I may prefer $100 today to $150 in one month, but prefer $150 in 61 months to $100 in 60 months. It's the same rate of discounting in both cases ($50 for an additional month of delay) but my preferences are not consistent. Of course, there are other criteria for rationality besides coherence/consistency (as the wikipedia article alludes to).
- pessimizer 14y agoBut if you view the payoff as not certain, $100 now is worth a lot more than $150 in a month - because you are here now with $100. The $150 is worth its normal discount rate multiplied by the strength of the probability that you will be here in a month with $150. The discount rate here is entirely due differences in risk of nonpayment - which are marginal when comparing 60 months to 61 months, but massive when comparing now to an hour from now. According to criteria that would label hyperbolic discounting irrational or illogical, debt collectors are equally confused for not accepting the promise of a $100,000 payoff 50 years from now rather than a settling of the debt today.
- LeBleu 14y agoThat's because the traditional economic definition of "rational" is an incomplete model of real world sanity that only works in incomplete models of the real world. In the real world, you have to deal with the uncertain risk that the offer will be unfulfilled due to external factors, which always increases in time relative to present. You, the entity offering you the choice, or the entire environment both exist in, may not still exist in 60 months. If the same offer is made 60 months from now, you should make a different choice, because the risk of non-existence between time of choice and time of reward has changed. If the amount of hyperbolic discounting is genetically determined, then is likely that we are still adjusting to the massive drop in risk of death over the past several centuries.
- smoyer 14y agoI'm wondering if somehow our predisposition to think logarithmically is related to Benford's law - http://en.wikipedia.org/wiki/Benfords_law http://en.wikipedia.org/wiki/Benfords_law.
- atamyrat 14y agoI don't think Benford's law has anything to do with human psychology, it's more of mathematical property that applies to random sample from certain distributions.
- baddox 14y agoI suspect only in the sense that both are related to http://en.wikipedia.org/wiki/Power_law http://en.wikipedia.org/wiki/Power_law.
- BenoitEssiambre 14y agoI think it is related to Benford's law in that they are both a manifestation of the fact that a log scale is the most natural (most uninformative, highest entropy) distribution (or prior) for things that have magnitude (having a parameter that can't go into the negatives such as height or length or volume as opposed to a positional parameter which can be negative). I do believe this is a property of the universe and not just a psychological phenomenon. Our mind here is simply following the natural distribution and Benford's law is just the observed manifestation of this pattern. This type of pattern is not odd if you are believer in the Bayesian interpretation of probability theory. The shape of Benford's distribution follows the shape of a maximum entropy, most uninformative distributions for a value of magnitude (as I mentioned, positions do not follow benford's law). Take street lengths for example. Assuming that they follow a log prior simply means that for a length of street L, if you pick another random street, you are as likely to pick a street within the length range L/2 to L than L to 2*L. From the original street length, to get a street twice as short, you need to subtract much less than you would have to add to get a street twice as long. That is why this distribution is not linear or rather it is linear on the multiplication and division operation, not on additions or subtractions. If instead you'd assume that a street x meter longer is as likely as a street x meter shorter you would end up with impossible probabilities. For example, for a street of 1 km, a 3 kilometer street would be as likely as a -1km street? Even if you'd assume probabilities were equal for all lengths between 0 and infinity that would mean you think there are as many streets measuring a tredecillion billion km long as there are street 5 km long. This is simply not how things are sized in the universe. Smaller things are in greater numbers. Log priors are one of these areas where the math predicts the universe logically and the universe is mirrored by the math beautifully iff you do your calculations properly (using Bayesianity). It seems evolution has made our psychology reflect this reality.
- maratd 14y agoDid anyone permit for the possibility that some people may be wired to have an affinity for one and some for the other?
- andrewcooke 14y agoi wish it had described how people were asked. i guess it's through some graphical means. anyone know (original paper is paywalled)?
- takluyver 14y agoThat question only seems to appear in the press release - it may be a reference to an earlier study, but press releases don't tend to cite sources. The paper in question is theoretical, although it gives an example with the intensity of various human and animal sounds. (I have access through my university)
- derekp7 14y agoOne thing not mentioned in the paper is also music. Each octave increases exponentially (each is twice the frequency of the previous). However our ears perceive the difference between notes as linear (it is actually linear if you take the log of the frequency). One theory is due to the shape of the inner ear, where the individual hairs that hear sounds each octave apart are spaced lineally.
- cs702 14y agoAccording to the research discussed in this article, when the intensity of some sensory stimuli increases exponentially, our minds somehow unconsciously perceive the increase as linear. Our perception, in other words, is often logarithmic. What I find most fascinating about this is that our conscious minds are not very good at thinking about exponential growth. My favorite example is that of the twin brothers who invest with annual gains of 12% each: one twin saves $2,000 from age 19 to 26 (a total of $16,000) and then stops saving; the other twin doesn't start saving until age 27 but then puts away $2,000 every year until age 65 (a total of $78,000). Do the math, and you will see that the savings of the second twin, growing at 12% every year, never catch up with those of the first one. Our perception might often be logarithmic, and we might understand the math of exponentiation, but our brains are not very good at 'seeing' exponential growth!
- taejo 14y agoOTOH, sometimes our perception is multiplicative when it should be relative: charge me $100 extra on a house, I probably won't even notice; charge me $100 extra on a bar of chocolate, there's no way I'm going to buy it. But either way I lose $100!
- StavrosK 14y agoYou only buy a house once, you buy chocolate bars all the time. $100 on a chocolate bar every time compounds.
- RyanIyengar 14y agoGive me 30 years and 4% interest on a loan to pay off the chocolate bar and I probably wouldn't notice either.
- masterzora 14y agoOne-time coupon for $.50 off a chocolate bar: probably going to take the time to clip it. One-time coupon for $.50 off a laptop: probably won't bother. You can find silly faults with individual examples all you want, but studies indicate that people have a tendency to think percentage-based about savings rather than absolute terms.
- artursapek 14y agoRadiolab did a good piece about this. http://www.radiolab.org/2009/nov/30/innate-numbers/ http://www.radiolab.org/2009/nov/30/innate-numbers/
- kevindication 14y agoRadiolab's Numbers episode covered this fascinating topic back in 2009: http://www.radiolab.org/2009/nov/30/ http://www.radiolab.org/2009/nov/30/
- woodchuck64 14y ago"But pose the same question to small children, or people living in some traditional societies, and they're likely to answer 3." Citation needed! I mean, if "half" is not understood as mathematically half, how can traditional societies even have an economy?
- evincarofautumn 14y agoProbably because there is a difference in perception between half of an amount and half of a number.
- inemnitable 14y agoIt's not "half," though, it's "halfway between." This implies a mean, but which mean to use is not explicitly expressed. Choosing the arithmetic mean gives 5, choosing the geometric mean gives 3.
- chinpokomon 14y agoInteresting... this has been something I've postulated for several years now. My reasoning had most to do with our vision system. As we perceive distances, the distance between something 10 feet, 100 feet, 1000 feet away, etc, becomes less about knowing the absolute difference and more about relative distances; someone is more likely to determine that something is 110 feet away than they are to realize something is 1010... but realizing something is 1100 rather than 1000 feet away will feel about as off as suggesting something 110 feet away is really 100. Our vision measurements are further reinforced as we observe parallax scrolling as we are driven around the country in our youth. I've also felt that teaching a number system that is linear means that it is harder to understand the difference between 100 and 1000000000. Even many adults find it difficult to understand the magnitude of numbers when you approach a billion or a trillion. I think this is largely because we are taught to think in a relatively small range of the number line and perceiving numbers outside that range, we try to relate it back to what we were taught in schools. I'd love to see schools teach a logarithmical number line instead of the linear system kids are taught today.
- chinpokomon 14y agoInteresting... this has been something I've postulated for several years now. My reasoning had most to do with our vision system. As we perceive distances, the distance between something 10 feet, 100 feet, 1000 feet away, etc, becomes less about knowing the absolute difference and more about relative distances; someone is more likely to determine that something is 110 feet away than they are to realize something is 1010... but realizing something is 1100 rather than 1000 feet away will feel about as off as suggesting something 110 feet away is really 100. Our vision measurements are further reinforced as we observe parallax scrolling as we are driven around the country in our youth. I've also felt that teaching a number system that is linear means that it is harder to understand the difference between 100 and 1000000000. Even many adults find it difficult to understand the magnitude of numbers when you approach a billion or a trillion. I think this is largely because we are taught to think in a relatively small range of the number line and perceiving numbers outside that range, we try to relate it back to what we were taught in schools. I'd love to see schools teach a logarithmical number line instead of the linear system kids are taught today.
- hcarvalhoalves 14y agoFor anyone who hikes or goes outdoor, this might be obvious. If you notice how we perceive distances, it's logarithmic too. It's really hard to judge accurately the time to arrive over long distances. Like the child's example on the OP, people often perceive 1/4 of the distance as being "halfway there". It's some kind of psychological phenomenon.
- malingo 14y agoCould this be related to the recent post about software estimates?
- saraid216 14y agoSoftware estimates are just another unit of distance. That's a valid application of this general idea: Halfway between a day and a month is a week.
- jamesbressi 14y agoWhat does it mean when you can't answer the question properly and said "4.5"?
- sukuriant 14y agoYou're either a math major or an engineer. We can't do simple math anymore. Please give it to us in the form of an integral. Also, there's the ambiguity of "inclusive or exclusive? And, which side?" [1,9], (1,9], [1,9) or (1,9)?
- kylebrown 14y agoThose ambiguities make no difference to the answer. Answering 4.5 just means you forgot that starting a count from 0 gives a different total than starting from 1.
- sukuriant 14y agoIf you're using the set of integers, then [1,9) is equivalent to [1,8]; and that would have a different answer from 5 for "what the center is"; so, it does affect the answer.
- kylebrown 14y ago[1,9) is not equivalent to [1,8]. The center of [1,9) is infinitesimally close to the center of [1,9]. Its exactly equal when you take the limit.
- sukuriant 14y ago... I was using that nomenclature for the short hand of "inclusive/exclusive". 1 to 9 exclusive is 1 to 8 in the integer set. Sorry for the lack of clarity. Perhaps now we can get to my original point?
- deleted 14y ago[deleted]
- jarel 14y ago"Hi, quick question: Which number is halfway between 100 and 10,000?" "1000. No, wait..." (I might actually say that)
- gilgoomesh 14y agoWhere is the study that showed "But pose the same question to small children, or people living in some traditional societies, and they're likely to answer 3." Without a source on this, it's a difficult article to take the article seriously. It offers no support for its most prominent claim.
- cr4zy 14y agoMusical scales are exponential as well. (octaves are frequency doublings) There's also this experimental evidence about sensing loudness, weight, vibration, lightness, saltiness, thickness and several dozen other things. From: http://en.wikipedia.org/wiki/Stevens%27_power_law http://en.wikipedia.org/wiki/Stevens%27_power_law ... The general form of the law is ψ(I) = kI^a where I is the magnitude of the physical stimulus, ψ(I) is the psychophysical function relating to the subjective magnitude of the sensation evoked by the stimulus, a is an exponent that depends on the type of stimulation and k is a proportionality constant that depends on the type of stimulation and the units used.