We'll start by reviewing some of what we know about sensation. In particular, we'll be focusing on sensory thresholds.
If you've ever had a standard hearing test, you've experienced the testing for absolute thresholds. Typically this involves listening to various pitches of tone through earphones. You are given a button to hold and are told to press the button until you hear a tone, then release the button until the tone fades away, then press the button until you hear it again, and so on. The intensity at which you "lose" and regain the tone is your absolute threshold for that particular tone.
Weber defined the absolute threshold as the intensity at which the stimulus was detected on 50% of trials. A stimulus detected on, say, 20% of trials is by this definition below the absolute threshold for detection, although of course it is sometimes detected nevertheless. Such a stimulus is termed subliminal (below threshold; the German word for threshold is limen).
As with the absolute threshold, Weber defined the difference threshold statistically. Starting with a standard stimulus intensity, one can increase or decrease the intensity until one can just barely tell that changed (comparison) stimulus is either more intense or less intense than the standard. The lower difference threshold is reached when the comparison stimulus is judged to be more intense than the standard on 25% of trials. The upper difference threshold is reached when the comparison stimulus is judged to be more intense than the standard on 75% of trials. The difference threshold is the average of the two differences between the comparison stimuli and the standard.
Weber noticed that the difference threshold is a constant proportion of the initial stimulus intensity. He expressed this relationship in a formula now called Weber's Law:
where "delta I" is the difference threshold, "I" is the initial intensity before the change, and "k" is the Weber fraction or Weber constant. For lifted weight, the Weber fraction is about 1/50 or 2%, meaning that the stimulus intensity must be changed by only one part in 50 or 2% of its initial value before you can tell that it is different. (Note that the smaller the number, the better able you are to discriminate small differences, or in other words, the more sensitive you are to a change in intensity.
About 20 years after Weber's pioneering work, another professor at the University of Leipsig, Gustav Fechner, decided to determine the relationship between the physical intensity of a stimulus (for example, the sound pressure level of a sound wave) and the psychologically perceived intensity (in this case, the loudness of the resulting sound). Fechner called the field of study than examines the relationship between the physical stimulus and its psychological representation Psychophysics.
After independently rediscovering Weber's Law, Fechner went Weber one better by showing that, with Weber's Law and the addition of a simple and reasonable assumption, one could determine the relationship between physical and psychological intensity he was seeking. Fechner called the difference threshold by a different name: the just noticeable difference, or JND. Fechner simply assumed that, psychologically speaking, all JNDs seem like the same amount of change in stimulus intensity. For example when lifting a 100 gram weight, one JND = 1/50th of 100 = 2 grams. When lifting a 200 gram weight, one JND = 1/50th of 200 or 4 grams. Fechner assumed that an increase of 2 grams from 100 seems like the same increase in weight as an increase of 4 grams from 200 grams. It follows from this assumption and Weber's Law that the perceived intensity of a stimulus will change in proportion to the logarithm of the physical stimulus intensity, and not in direct proportion to the physical stimulus intensity as might have been expected. This implies that at the high end of the intensity scale, we become almost (but not quite) insensitive to changes in the intensity of a stimulus, while retaining a high sensitivity to changes in stimulus intensity at the low end of the intensity scale.
This logarithmic relationship, which is known as Fechner's Law, was used to develop the scale of loudness called the decibel scale. Equal increments along the decibel scale reflect equal increments in loudness, as humans perceive it, and not equal increments in the intensity of the physical stimulus, the sound wave.