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Article Dans Une Revue Journal of Mathematical Psychology Année : 2014

Expressing relative differences (in percent) by the difference of natural logarithms

Résumé

Most psychophysical investigations measure stimuli or performance in Système International units and use relative differences between them for comparison. In this theoretical note, we propose the ratio’s natural logarithm or the difference between the Napierian logarithms, as a desirable measure of relative differences between two psychophysical quantities. It challenges the more frequently used (x2−x1)/x1(x2−x1)/x1, (x2−x1)/x2(x2−x1)/x2, as well as (x2−x1)/xM(x2−x1)/xM, where x1,x2x1,x2, and xMxM are the initial value in a change, the larger value, and a mean value between x1x1 and x2x2, respectively. As for the three aforementioned expressions, it can be conveniently expressed as a percentage. For two physical measures, x1x1 and View the MathML sourcex2(x1>0;x2>0), the difference between natural logarithms View the MathML sourceDNL=Ln(x2)−Ln(x1)=Ln(x2/x1) sits between (x2−x1)/x2(x2−x1)/x2 and (x2−x1)/x1(x2−x1)/x1; it is actually the mean value of (x2−x1)/x(x2−x1)/x for all xx values between x1x1 and x2x2. Unlike other estimates, it satisfies all three of the following properties: symmetry, i.e. Δ(x1;x2)=−Δ(x2;x1)Δ(x1;x2)=−Δ(x2;x1); agreement between inverted units, such as hertz and second, i.e. Δ(x1;x2)=−Δ(k/x1;k/x2)Δ(x1;x2)=−Δ(k/x1;k/x2) thus |Δ(x1;x2)|=|Δ(k/x1;k/x2)||Δ(x1;x2)|=|Δ(k/x1;k/x2)|; and additivity “à la Chasles”, i.e. Δ(x1;x2)+Δ(x2;x3)=Δ(x1;x3)Δ(x1;x2)+Δ(x2;x3)=Δ(x1;x3). Finally, it complies with the Weber–Fechner and Stevens laws.
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Dates et versions

hal-01419360 , version 1 (19-12-2016)

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Christian Graff. Expressing relative differences (in percent) by the difference of natural logarithms. Journal of Mathematical Psychology, 2014, 60, pp.82 - 85. ⟨10.1016/j.jmp.2014.02.001⟩. ⟨hal-01419360⟩
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