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# Weber–Fechner law

Def: The Weber-Fechner law states that the amount of change needed for sensory detection to occur increases with the initial intensity of stimulus, and is proportional to it. The change in stimulus that will be just noticeable is a constant ratio of the original stimulus. Applied to vision, the amount of change in brightness detected by the visual senses is dependent on and proportional to the present brightness.

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. 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 .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 psycho-physical 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.

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If you lift up and hold a weight of 2.0 kg, you will notice that it takes some effort. If you add to this weight another 0.05 kg and lift, you may not notice any difference between the apparent or subjective weight between the 2.0 kg and the 2.1 kg weights. If you keep adding weight, you may find that you will only notice the difference when the additional weight is equal to 0.2 kg. The increment threshold for detecting the difference from a 2.0 kg weight is 0.2 kg. The just noticeable difference (jnd) is 0.2 kg.

Now start with a 5.0 kg weight. If you add weight to this, you will find that the just noticeable difference is 0.5 kg. It takes 0.5 kg added to the 5.0 kg weight for you to notice an apparent difference.

For the weight of magnitude, I, of 2.0 kg, the increment threshold for detecting a difference was a I (pronounces, delta I) of 0.2 kg.

For the weight of magnitude, I = 5.0 kg, the increment threshold I = 0.5 kg.

The ratio of I/I for both instances (0.2/2.0 = 0.5/5.0 = 0.1) is the same. This is Weber’s Law.

Weber’s Law states that the ratio of the increment threshold to the background intensity is a constant. So when you are in a noisy environment you must shout to be heard while a whisper works in a quiet room. And when you measure increment thresholds on various intensity backgrounds, the thresholds increase in proportion to the background.

The fraction I/I is known as the Weber fraction (aka Fechner fraction). If we rearrange the equation to I=IK, you can see that Weber’s Law predicts a linear relationship between the increment threshold and the background intensity. Below is a plot of some hypothetical data showing Weber’s Law. The slope of the line is the Weber fraction.