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In the psychology of perception (psychophysics), for two stimuli to be just-discernible (just-noticeable difference, JND), the difference between their magnitudes (e.g. the pitch or the loudness of two tones) needs to be greater if their individual absolute values are high, a relationship known as the Weber-Fechner law.

Similarly, in numerical cognition (the psychology of numbers), it becomes increasingly difficult to discriminate among two numbers separated by a constant amount, as their absolute values increase (by the same amount).

In other words, the JND depends not (only) on the difference between the stimuli but (also) on their ratio.

It is tempting to consider whether Weber's law also applies to the statistical significance of the difference between two means, as depicted for instance in the plot below:

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But is it not in fact the case that, when the relevant statistics are computed, e.g. independent samples t-test and its associated p-value, it turns out that it is only the difference between the means (plus, of course, the variance that goes into each one of them, and the respective sample sizes) and not their ratio that predicts the p-value that in turn tells us about the significance of the difference between them?

z8080
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    I'd argue that such comparison is too abstract to be meaningful... – Tim Oct 20 '16 at 12:45
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    I don't have any answer, but you may find it interesting (if you haven't already) to read Chapter 4 of Jaynes' Probability Theory. While the whole book is interesting, he makes a similar comparison between Weber's law and Bayesian evidence. – Greg Oct 20 '16 at 12:52
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    Very interesting suggestion Greg, thank you. Jaynes says: "When probabilities approach one or zero, our intuition doesn’t work very well. Does the difference between the probability of 0.999 and 0.9999 mean a great deal to you? It certainly doesn’t to the writer. But after living with this for only a short while, the difference between evidence of plus 30 db and plus 40 db does have a clear meaning to us. It is now in a scale which our minds comprehend naturally. This is just another example of the Weber–Fechner law; intuitive human sensations tend to be logarithmic functions of the stimulus" – z8080 Oct 20 '16 at 12:58
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    @z8080: Long before Jaynes, CS Peirce observed the same! See my answer at https://stats.stackexchange.com/questions/87182/what-is-the-role-of-the-logarithm-in-shannons-entropy/463828#463828 – kjetil b halvorsen Jul 02 '22 at 18:40
  • @Greg: See my other comment ..,. – kjetil b halvorsen Jul 02 '22 at 18:40

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