Weber-Fechner Law

The Difference Threshold or Just Noticeable Difference

When quantifying a difference threshold, the reason for doing so is to determine the minimum difference between two stimuli that can be detected. Researchers in the field of classical psychophysics pose the question this way: What is the just noticeable difference (JND) required to perceive that a comparison stimulus is different from a standard (or reference) stimulus?

Weber's Law

Ernest Heinrich Weber was an early pioneer in the field of psychophysics, and it was Weber who developed the concept of the difference threshold or just noticeable difference. Weber published the results of experiments in which he asked observers first to lift a standard weight and then a comparison weight and judge whether the comparison weight was greater than, equal to, or less than the standard weight. By having observers compare a large number of different standard and comparison weights, Weber was able to determine the smallest difference between two weights that could be detected reliably (i.e., the difference threshold). He found that the difference threshold or just noticeable difference was dependent on the weight of the standard (reference) stimulus. For example, if an observer can just notice the difference between a 100 g standard weight and a 103 g comparison weight, the JND in this example would be 3 g. Weber found, however, that if the weight of the standard was increased to 1,000 g, the JND was no longer 3 g but had increased to 30 g (i.e., the comparison weight must be heavier than 1,030 g to perceive a just noticeable difference). Weber investigated further and found that the size of the JND for most human senses (e.g., sight, sound, taste, and touch) is a constant fraction of the size of the standard stimulus. Expressed mathematically, this is known as Weber's Law:

where k is a constant called the Weber fraction and S is the value of the standard stimulus. This equation is usually expressed in the form

Fechner's Law

Gustav Fechner derived a relationship between the intensity of a specific stimulus and the perceived (estimated) magnitude. To derive this relationship, Fechner made two important assumptions:

• 1.
  that the JND is a constant fraction of the stimulus (i.e., Weber's Law holds), and
• 2.
  that the JND is the basic unit of perceived magnitude, so that one JND is perceptually equal to another JND.

By accepting these assumptions, Fechner hypothesized that the magnitude of a stimulus can be determined by starting at the detection threshold (JND) and then adding JNDs. From this, Fechner derived the following mathematical relationship between perceived magnitude (P) and stimulus intensity

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