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In psychophysics, the Weber–Fechner law combines two different laws of human perception, which both describe ways the resolution of perception diminishes for stimuli of greater magnitude. Ernst Heinrich Weber (1795–1878) was one of the first people to approach the study of the human response to a physical stimulus in a quantitative fashion.〔Ross, H.E. and Murray, D. J.(1996)(Ed. and Transl.) ''E.H.Weber on the tactile senses''. 2nd ed. Hove: Erlbaum (UK) Taylor & Francis.〕 Weber's law states that the just-noticeable difference between two stimuli is proportional to the magnitude of the stimuli, (and the subject's sensitivity), i.e. if you sense a change in weight of 0.5 lbs on a 5 pound dumbbell, you ought to feel the extra pound added to a ten pound dumbbell. Gustav Theodor Fechner (1801–1887), a scholar of Weber, later used Weber's findings to construct a psychophysical scale in which he described the relationship between the physical magnitude of a stimulus and its (subjectively) perceived intensity. Fechner's law (better referred to as Fechner's scale) states that subjective sensation is proportional to the logarithm of the stimulus intensity. Fechner scaling has been mathematically formalized. In fact, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to log of actual intensity measured with an accurate nonhuman instrument. Some authors use the term "Weber–Fechner law" to mean Weber's law, and others use it for Fechner's law. The use of the term "Weber–Fechner law" was criticised as a misnomer by Ewald Hering. == Derivation of Fechner's law for weight perception == Weber found that the just noticeable difference (jnd) between two weights was approximately proportional to the weights. Thus, if the weight of 105 g can (only just) be distinguished from that of 100 g, the jnd (or differential threshold) is 5 g, or in the SI system, a force or weight of 0.005 kg N. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight (any weight) seems to have to increase by 5% for someone to be able to reliably detect the increase, and this minimum required fractional increase (of 5/100 of the original weight) is referred to as the "Weber fraction" for detecting changes in weight. Other discrimination tasks, such as detecting changes in brightness, or in tone height (pure tone frequency), or in the length of a line shown on a screen, may have different Weber fractions, but they all obey Weber's law in that observed values need to change by at least some small but constant proportion of the current value to ensure human observers will reliably be able to detect that change. This kind of relationship can be described by the differential equation : where ''dp'' is the differential change in perception, ''dS'' is the differential increase in the stimulus, and ''S'' is the instantaneous stimulus. The parameter ''k'' is to be estimated using experimental data. Integrating the above equation gives : where is the constant of integration and ''ln'' is the natural logarithm. To solve for , put , i.e., no perception; then subtract from both sides and rearrange: : where is that threshold of stimulus below which it is not perceived at all. Substituting this value in for above and rearranging, our equation becomes: : The relationship between stimulus and perception is logarithmic. This logarithmic relationship means that if a stimulus varies as a geometric progression (i.e., multiplied by a fixed factor), the corresponding perception is altered in an arithmetic progression (i.e., in additive constant amounts). For example, if a stimulus is tripled in strength (i.e., 3 x 1), the corresponding perception may be two times as strong as its original value (i.e., 1 + 1). If the stimulus is again tripled in strength (i.e., 3 x 3 x 1), the corresponding perception will be three times as strong as its original value (i.e., 1 + 1 + 1). Hence, for multiplications in stimulus strength, the strength of perception only adds. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law (see (link1 ) or (link2 )). Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all jnds are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned.〔Heidelberger, M. (2004)''(Nature from within: Gustav Theodor Fechner and his psychophysical worldview )''. Transl. C. Klohr. Pittsburgh, USA: University of Pittsburgh Press.〕 Most researchers nowadays accept that a power law is a more realistic relationship, or that a logarithmic function is just one of a family of possible functions. Other sense modalities provide only mixed support for either Weber's law or Fechner's law. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weber–Fechner law」の詳細全文を読む スポンサード リンク
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