What is the name of the law that states that the size of the just noticeable difference remains constant?

Learning Outcomes

• Explain the concept of just-noticeable difference in sensory perception

In the branch of experimental psychology focused on sense, sensation, and perception, which is called psychophysics, a just-noticeable difference (JND) is the amount something must be changed in order for a difference to be noticeable, or detectable at least half the time (absolute threshold). This limen (another word for threshold) is also known as the difference limen, differential threshold, or least perceptible difference.

For many sensory modalities, over a wide range of stimulus magnitudes sufficiently far from the upper and lower limits of perception, the JND is a fixed proportion of the reference sensory level, and so the ratio of the JND/reference is roughly constant (that is the JND is a constant proportion/percentage of the reference level). Measured in physical units, we have:

[latex]\displaystyle\frac{\Delta{I}}{I}=k[/latex]

where I is the original intensity of the particular stimulation, ΔI is the addition to it required for the change to be perceived (the JND), and k is a constant. This rule was first discovered by Ernst Heinrich Weber (1795–1878), an anatomist and physiologist, in experiments on the thresholds of perception of lifted weights. A theoretical rationale (not universally accepted) was subsequently provided by Gustav Fechner, so the rule is therefore known either as the Weber Law or as the Weber–Fechner law; the constant k is called the Weber constant. It is true, at least to a good approximation, of many but not all sensory dimensions, for example the brightness of lights, and the intensity and the pitch of sounds. It is not true, however, of the wavelength of light. Stanley Smith Stevens argued that it would hold only for what he called prothetic sensory continua, where change of input takes the form of increase in intensity or something obviously analogous; it would not hold for metathetic continua, where change of input produces a qualitative rather than a quantitative change of the percept. Stevens developed his own law, called Stevens’ Power Law, that raises the stimulus to a constant power while, like Weber, also multiplying it by a constant factor in order to achieve the perceived stimulus.

The JND is a statistical, rather than an exact quantity: from trial to trial, the difference that a given person notices will vary somewhat, and it is therefore necessary to conduct many trials in order to determine the threshold. The JND usually reported is the difference that a person notices on 50% of trials. If a different proportion is used, this would be included in the description—for example a study might report the value of the 75 percent JND.

Modern approaches to psychophysics, for example signal detection theory, imply that the observed JND is not an absolute quantity, but will depend on situational and motivational as well as perceptual factors. For example, when a researcher flashes a very dim light, a participant may report seeing it on some trials but not on others.

It is easy to differentiate between a one-pound bag of rice and a two-pound bag of rice. There is a one-pound difference, and one bag is twice as heavy as the other. However, would it be as easy to differentiate between a 20- and a 21-pound bag?

Question: What is the smallest detectible weight difference between a one-pound bag of rice and a larger bag? What is the smallest detectible difference between a 20-pound bag and a larger bag? In both cases, at what weights are the differences detected? This smallest detectible difference in stimuli is known as the just-noticeable difference (JND).

Background: Research background literature on JND and on Weber’s Law, a description of a proposed mathematical relationship between the overall magnitude of the stimulus and the JND. You will be testing JND of different weights of rice in bags. Choose a convenient increment that is to be stepped through while testing. For example, you could choose 10 percent increments between one and two pounds (1.1, 1.2, 1.3, 1.4, and so on) or 20 percent increments (1.2, 1.4, 1.6, and 1.8).

Hypothesis: Develop a hypothesis about JND in terms of percentage of the whole weight being tested (such as “the JND between the two small bags and between the two large bags is proportionally the same,” or “. . . is not proportionally the same.”) So, for the first hypothesis, if the JND between the one-pound bag and a larger bag is 0.2 pounds (that is, 20 percent; 1.0 pound feels the same as 1.1 pounds, but 1.0 pound feels less than 1.2 pounds), then the JND between the 20-pound bag and a larger bag will also be 20 percent. (So, 20 pounds feels the same as 22 pounds or 23 pounds, but 20 pounds feels less than 24 pounds.)

Test the hypothesis: Enlist 24 participants, and split them into two groups of 12. To set up the demonstration, assuming a 10 percent increment was selected, have the first group be the one-pound group. As a counter-balancing measure against a systematic error, however, six of the first group will compare one pound to two pounds, and step down in weight (1.0 to 2.0, 1.0 to 1.9, and so on.), while the other six will step up (1.0 to 1.1, 1.0 to 1.2, and so on). Apply the same principle to the 20-pound group (20 to 40, 20 to 38, and so on, and 20 to 22, 20 to 24, and so on). Given the large difference between 20 and 40 pounds, you may wish to use 30 pounds as your larger weight. In any case, use two weights that are easily detectable as different.

Record the observations: Record the data in a table similar to the table below. For the one-pound and 20-pound groups (base weights) record a plus sign (+) for each participant that detects a difference between the base weight and the step weight. Record a minus sign (−) for each participant that finds no difference. If one-tenth steps were not used, then replace the steps in the “Step Weight” columns with the step you are using.

Analyze the data/report the results: What step weight did all participants find to be equal with one-pound base weight? What about the 20-pound group?

Draw a conclusion: Did the data support the hypothesis? Are the final weights proportionally the same? If not, why not? Do the findings adhere to Weber’s Law? Weber’s Law states that the concept that a just-noticeable difference in a stimulus is proportional to the magnitude of the original stimulus.

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October 22, 1850 Gustav Fechner's Law "S = k Log R"

• Philosophers had argued for centuries that sensation ("conscious experience") doesn't take up space and since it doesn't you can't measure it.
  • Physical quantities can be be subdivided and not be qualitatively changed.
    • 1 qt of water = 1 cup of water + 1 cup of water + 1 cup of water + 1 cup of water
  • But how do you do that with psychological phenomenon. How can you break down a red experience and talk about adding together 4 pinks (each is red-experience but less not much of it) to produce red (obviously a lot of red experience)?
• Fechner woke up one day and realized there was a way to measure the magnitude of sensations.
  • He built on the work of others who had been measuring stimulus thresholds (what is the weakest stimulus you can detect? what is the smallest difference in intensity between two stimuli that you can detecct?).
  • Given the information on stimulus thresholds, Fechner realized there was an equation that would allow you to describe changes in the magnitude of psychological events ("How much sweeter is it now") in terms of changes in the stimulus ("I'll dissolve another teaspoon of sugar in the coffee.")
  • Sensation increases as the logrithm of stimulus intensity or "S = k log R".
  • So what you say? Well this is as revolutionary as E = mc2.
    • Einstein's equation says that energy and matter are interchangable. They are the same thing and you can convert a given amount of matter into a given amount of energy.
    • Fechner's law says mind (conscious experience) and matter (stimulus intensity) are the same thing as well.
• This was a BIG DEAL.
  • If you can measure sensation then you can do science on the mind, on conscious experience. WOW!.

But first we need some basic concepts.

Psychophysical Problems Introduced

• Detection Problem
  • What is the weakest stimulus you can detect
  • That stimulus is known as the Absolute Threshold (RL)
• Discrimination Problem
  • What is the smallest difference between two stimuli you can detect?
  • That difference is known as the just noticeable difference (jnd)
  • The stimulus just-noticeably-different from another stimulus is the Difference Limen (DL).
• Scaling Problem
  • As stimulus intensity changes how does psychological intensity change?
    • Stimulus change=The amount of electromagnetic radiant energy in the room; psychological intensity change=How bright the room looks.
    • Stimulus change=dissolve more sugar in the water; psychological intensity change=the water tastes sweeter.

Fechner's idea was to markoff equally different sensations in terms of the stimuli that produce them. In the picture below:

The World of Stimulation--the Physical World

• S0 is the absolute threshold (RL) stimulus.
  • This stimulus is the weakest stimulus you can detect.
  • What is the least sucrose that can be dissolved in 1 liter of water so you can taste it.
    • How this misleads: There isn't a "weakest" even though it sounds reasonable. On any presentation a stimulus may be detected or not. Weaker stimuli have a lower probability of detection. Detection is "all-or-none" but there is no stimulus which is always detected and then weakened a tad and never detected.
• S1 is the stimulus just-noticeably-different from the RL stimulus.
  • If the difference were any smaller, then you would say there is no difference between the stimuli.
  • How this misleads: Again, there's no such stimulus. On any presentation a pair of stimuli either feel different or they don't. We can measure the proportion of time two stimuli feel different but again there is no difference that is never detected and then made a tad larger and always felt as different.
• The remaining stimuli each is just-noticeably-more-intense than the one before.

• R0 is the least conscious experience of a stimulus you can have. Any less experience is no experience! If the water were any less sweet you'd say it wasn't sweet at all.
• R1 is the experience of sweet that is the least possible increase in sweetness. Any smaller increase in the experience is no different than the experience R0.
• Likewise R2 and R1 differ by the least possible difference in experience. The two liquids would be "equally sweet" if the experienced sweetness of R1 were any less.

• There are no lines connecting stimuli with sensations. Its important that you realize there is no mechanism telling how energy is converted into sensation.
  • We can investigate and describe, in considerable detail, the mechanisms by which radiant energy impinges on the retinal receptors and cause changes in neural activity.
  • We can NOT specify how changes in neural activity are related to changes in sensation.

• In the 1830s a German physiologist discovered that for mid-range stimuli, two stimuli will be just-noticeably different when you change one stimulus by k% of the other stimulus. K is known as the Weber Constant.
  • "jnd" is a confustion measure. Two stimuli a jnd apart will be confused 50% of the time (in a commonly-used measurement method).
  • For small lifted weights, k is about .04. A 4% change will be just noticeable.
    • A 100 and 104 gms are a jnd apart; so are 1000 and 1040 gms.
• As the stimulus you are measuring the jnd from gets larger the amount of stimulation needed for a jnd gets larger in terms of the measured units of the stimulus. 4 grams at 100; 8 gm at 200; 20 gm at 500; etc.
• Here is a table k for several sensory modalities.

• S = k log R
• Equal sensed differences are produced by equal stimulus ratios.
  • The table illustrates the "log" relationship. Each stimulus is 8% larger than the one before.
  • The figure shows that equal size steps in sensation are produced by stimuli that increase in a constant proportion (aka a geometic ratio).

Sensation Level Stimulus Intensity jnd Log S Log jnd 0 10.00 1 1 10.80 0.8 1.033 0.0334 2 11.66 0.864 1.067 0.0334 3 12.59 0.933 1.1 0.0334 4 13.60 1.008 1.134 0.0334 5 14.69 1.088 1.167 0.0334 6 15.87 1.175 1.201 0.0334 7 17.14 1.269 1.234 0.0334 8 18.51 1.371 1.267 0.0334 9 19.99 1.481 1.301 0.0334 10 21.59 1.599 1.334 0.0334 Note: Each stimulus is 1.08 times the stimulus before it. For example, 13.60 = 1.08 X 12.59. The 1.08 stimulus ratio is somewhat obscured because the stimulus intensity is rounded to two decimals.

Stimulus 10 10.8 11.66 12.6 13.6 14.69 15.87 17.14 18.51 19.99 21.59 Stimulus Ratio x 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 1.08 Sensation Level 0 1 2 3 4 5 6 7 8 9 10 Sensation Increment x 1 1 1 1 1 1 1 1 1 1

• Fechner's law argues that all jnds are the same perceived size.

• Fechner says, the 4 gm difference between 100 gm and 104 gm feels the same difference as the jnd of 40 gm between 1000 and 1040.
• Fechner is wrong--While those two differences are equally confusable, they don't feel the same. The 40 gm difference feels larger.

©2002 by Burrton Woodruff. All rights reserved. Modified Friday, June 7, 2002