Psychophysics is the study of physical stimuli and their relation to sensory reactions. Of the energy that strikes the sensory surfaces of man and the animals, only a restricted fraction is capable of eliciting a reaction. Thus, visual responses in man are triggered by a narrow band of the vast electromagnetic spectrum (wavelengths between about 400 and 750 millimicrons); and auditory responses result from periodic displacements of the eardrum in the frequency range from about 20 to 20,000 cycles per second (cps). Over these stimulus ranges, neither the eye nor the ear is uniformly responsive: to produce a sensation may require thousands of times more energy at one wavelength or frequency than at another. It is the goal of psychophysics to map out the relations between the physical events and the psychological responses of organisms, and thus to provide a basic, over-all description of the function of the senses.
Major problems. The traditional questions posed by psychophysics fall into four groups. For a given sense modality we may ask about (1) the smallest detectable energy (the measurement of sensitivity); (2) the smallest detectable change in energy (the measurement of resolving power); ( 3 ) the configurations of energy that produce an invariant sensory effect, such as a constant loudness or color (the measurement of static invariances); (4) the way in which the magnitude of a sensory effect depends functionally on the stimulus (the measurement of dynamic properties).
The problem of sensitivity involves the determination of the smallest detectable intensity of a stimulus (called the absolute threshold), often as a function of another stimulus dimension, such as wavelength, frequency, duration, or areal extent. The threshold of audibility, for example, depends on the frequency of the tone; sensitivity is greatest to frequencies between about 2,000 and 3,000 cps. In the vicinity of 20 and 20,000 cps (which are conveniently but arbitrarily called the “limits” of hearing), the threshold energy may rise to roughly 108 times the minimal value. A similar relation exists between the threshold of visibility and the wavelength of the stimulating light, except that the shape of the visibility curve depends on what part of the sensory surface is stimulated; foveal (cone) vision has maximum sensitivity at about 555 millimicrons; peripheral (rod) vision, at about 505 millimicrons; and the “limits” of visibility are also different for the two populations of receptors.
Variation in sensitivity
The absolute threshold is not a rigidly fixed value. Sensitivity fluctuates irregularly, so that a given stimulus level may trigger a response at one time but not at another. The threshold is usually defined statistically, e.g., as the energy level that is detected as often as not over a series of presentations.
Sensitivity is also subject to systematic variation, either of the permanent kind encountered in aging or in pathology of the sensory tissues, or of a temporary kind observed, for example, in the relatively rapid decline of visual sensitivity under exposure to light (light adaptation) and the subsequent gradual recovery of sensitivity in the dark (dark adaptation). These and many other systematic changes in sensitivity are frequently expressed as alterations of the absolute threshold.
The study of absolute thresholds reveals the exquisite sensitivity of the sense organs under optimal conditions. A periodic displacement of the eardrum through a distance equal to the diameter of a hydrogen molecule may suffice to produce an audible sound, and a couple of quanta of light absorbed at the retina may suffice to arouse a faint visual sensation.
Methods of measurement
Because it fluctuates, the threshold is difficult to measure, and the various methods that have been tried do not always yield the same value. The method of adjustment provides a rapid approximation; the observer is required to set the level of the stimulus so that it is just perceptible. The threshold may be defined as the average of several settings. In the method of limits, either the stimuli are presented in order of increasing magnitude until the observer reverses his response from “imperceptible” to “perceptible,” or they are presented in order of decreasing magnitude until the observer reverses his response from “perceptible” to “imperceptible.” The threshold may be defined as the average value that marks the reversal in response over several ascending and descending series. In the method of constant stimuli, fixed stimulus levels are presented several times, each in irregular order. The threshold may be defined as the stimulus value that is perceived on half the presentations. This value is interpolated from a plot relating the percentage of positive responses to the stimulus magnitude.
The methods of adjustment, limits, and constant stimuli are known as the classical psycho-physical methods because they have continued in widespread use ever since G. T. Fechner described them in his Elemente der Psychophysik (1860), the monumental work that marks the establishment of psychophysics. (Reviews of the classical methods are given by Urban 1908; Titchener 1905; and Boring 1942.)
One of the difficulties inherent in these methods is the observer’s awareness that a stimulus event actually takes place on each trial. When a “catch trial” (a feigned presentation of a stimulus) is given, observers will occasionally give an affirmative response (a “false alarm”). The knowledge that the observer’s expectations and motivations come into play has stimulated the invention of new methods that offer the hope of better understanding and controlling the observer’s response biases. An example is the forced choice method, in which at regular intervals a stimulus is presented or withheld and the observer must decide each time whether or not he detected it. Results obtained under this procedure reveal that detection may depend not only on the magnitude of the stimulus but also on the prearranged probability of a stimulus event. A high proportion of “no-stimulus” trials causes a relatively high incidence of “false alarms”; a low proportion of “no-stimulus” trials, on the other hand, causes a lower incidence of correct detections of actual stimulus events. The probability of a “Yes” or “No” response can also be systematically influenced by rewarding correct detections and punishing the false alarms.
The forced-choice experiments have done much to underscore and clarify the role of response variables in the measurement of thresholds. It is sometimes suggested that the “detection” model may actually do away with the conception of the threshold as a simple, determinable value marking the critical terminus of sensory experience. According to this view, the detection of a stimulus (the “signal”) has much in common with the mathematical process of statistical decision. The observer is confronted with two distributions: that of the persistent background noise and that of the signal added to the noise. He decides from which of the two distributions a sample is taken in much the way that a statistician tests a statistical hypothesis. The decision will depend on the overlap of the distributions and also on the “pay-off matrix”—the consequences of false detections and failures of detection (see Swets 1964; Luce et al. 1963).
Interesting technological advances have recently been made in the field of threshold measurement. The Bekesy audiometer, for example, uses the method of tracking for the efficient measurement of the just-audible intensity as a function of tonal frequency (Von Bekesy 1928-1958). The observer “tracks” his threshold by pressing a key whenever the tone is audible and releasing it whenever the tone becomes inaudible. While the key is pressed, the level of the tone steadily decreases; while the key is not pressed, the level steadily increases. The observer may continue to track the threshold while the tonal frequency changes from one end of the audible spectrum to the other. On a moving paper chart, the stimulus level, which weaves back and forth across the threshold, is recorded continuously as a function of the frequency.
The tracking method has also been used to determine the visual thresholds of human observers and has been adapted to mapping the sensitivity functions of animals.
The second major concern of psychophysics is to measure the smallest detectable change in a stimulus (the so-called difference threshold). The problem may be to measure the just-noticeable differences in intensity, e.g., in the brightness of a light or in the concentration of a sweet solution, or in quality, e.g., in the hue of a colored light.
The capacity for resolving stimulus differences is expressed in terms of the Weber fraction, ΔI/I, where ΔI stands for the increment that produces a just-noticeable change when added to the stimulus level I. The smaller the value of ΔI, the keener is the ability to discriminate.
Methods of measurement
Like the absolute threshold, the difference threshold is a fluctuating quantity, so that ΔI must be assessed by a statistical treatment of a series of measurements. Most of the methods used are versions of those used to measure absolute thresholds. In the method of adjustment, for example, the observer sets a comparison stimulus to match a standard fixed stimulus. The threshold, ΔI, may be defined as the average error or the standard deviation of several settings. The greater the variability of the settings, the grosser the discrimination and the larger the Weber fraction. A difference threshold may be regarded either as a measure of the precision or as a measure of the variability or “noisiness” of the sensory process.
In the method of constant stimuli, the observer judges whether each of a set of fixed discrete stimulus levels appears greater or smaller than a standard stimulus (a judgment of “equal” is also permitted by some experimenters). The difference threshold may be defined as the difference between a standard stimulus and a comparison stimulus that is perceived as being greater (or smaller) than the standard stimulus on a certain percentage of the trials. This value can be interpolated from a poikilitic (scatter) function.
In Figure 1 the ordinate represents the relative frequency with which the comparison stimulus is judged greater than the standard. The threshold, ΔI, is the difference between the stimulus magnitude that is perceived as being greater than the standard on 75 per cent of the trials (L) and the stimulus magnitude that is so perceived on 50 per cent of the trials (E), i.e., the stimulus that appears to match the standard stimulus. Often there is a small difference, called the time error, between the standard stimulus (S) and the stimulus value associated with the 50 per cent point. Urban (1908) provides a detailed discussion of poikilitic functions.
The nature of the difference threshold has often been studied by the method of quanta! increments. From time to time, a small increment, ΔI, is added briefly to a steady stimulus, I. The task is to indicate whether the increment was detected. Of theoretical concern is the mathematical form of the poikilitic function that relates the proportion of detections to the size of the increment. If the precision were ultimately limited by nothing but the
random “noisiness” of the sensory process, then the function would be expected to have the sigmoid shape (the integral of a bell-shaped distribution) that is predicted by the theory of random error. The usual result approximates this form. When pains are taken to aid the attention and to eliminate the extraneous sources of error, the obtained function may assume a linear rather than a sigmoid form and conform to a predictable slope. According to the neural-quantum hypothesis, a linear function of the appropriate slope demonstrates that discrimination is basically all-or-none and that sensation grows by the addition of minute but finite (or quantal) steps (see Von Bekesy 1928-1958; S. S. Stevens 1961; Luce et al. 1963; Swets 1964).
A major aim in threshold measurement has been to test the famous generalization, credited to Ernst H. Weber but formalized and promoted by Fechner, that the Weber fraction is constant along a given sensory continuum, or that ΔI/I = k. In other words, a just-noticeable change should occur when a constant fractional increment is added to a stimulus of any magnitude. Under good conditions the increment is about 1 per cent for brightness, 2 per cent for loudness, and 20 per cent for saltiness. The sensory systems differ greatly in their resolving power, but for any one system it is the percentage change that matters most.
On most continua Weber’s law holds over a substantial portion of the stimulus range (for loudness and brightness over at least 99.9 per cent of the range), but the law fails near the absolute threshold, where discrimination is relatively gross. Nevertheless, Weber’s law stands as one of the oldest and broadest empirical generalizations of psychophysics—psychology’s “law of relativity,” as one writer put it (S. S. Stevens 1951).
If discrimination has seemed to receive more than reasonable attention among students of the senses, the explanation is likely to be found in the significance that the founder of psychophysics attached to the subject. For Fechner, discrimination provided the key to the measurement of sensory magnitude. He began with the postulate that on a given sensory continuum all just-noticeable differences (jnd) represent subjectively equal units. Subjective equality of jnds is a powerful (if questionable) assumption, because the integration of such units would provide a true scale of subjective magnitude. Since by Weber’s law a jnd corresponds to a constant fractional increase in the stimulus, it follows that the number of jnds grows in an arithmetic series when the stimulus intensity grows in a geometric series. Fechner concluded, therefore, that the magnitude of sensation is a logarithmic function of the stimulus. The logarithmic function implies that equal ratios of stimulus magnitude give rise to equal differences in subjective magnitude.
Plateau’s power function
In contrast to the indirectness of the Fechnerian approach to the measurement of sensory magnitudes was an early experiment by the Belgian physicist Joseph A. F. Plateau, who asked a group of artists each to paint a gray that seemed to lie midway between a white sample and a gray sample (a version of a scaling method later termed equisection). Of historical interest is Plateau’s conclusion that sensation grows as a power function rather than a logarithmic function of the stimulus. But the power function was subsequently given up by Plateau and virtually forgotten until the 1950s. With new techniques for the direct assessment of sensory magnitude, it was shown that Plateau’s early conjecture about the form of the psychophysical function happened to be correct. The current approach to the problem is generally to regard as separate properties the resolving capacity of the sensory system and the functional dependence of sensory magnitude on the stimulus.
A third major problem of psychophysics is to determine those arrangements of stimuli that produce responses that are equivalent in some respect. The goal of this kind of measurement is to specify all the energy configurations in the environment that produce an invariant or equivalent sensory response.
For example, the goal may be to determine the combinations of intensity and duration of a flash target that produce the same apparent brightness. The level or the duration of a comparison flash is adjusted to match the brightness of a standard flash of fixed intensity and duration. The judgment requires a degree of abstraction because the task is to match for brightness without regard to a difference in apparent duration. A plot relating the duration and intensity that produce a constant (standard) brightness provides an example of an equal sensation function. Usually it is desirable to map the family of these equal sensation functions for a pair of parameters. In the present example this means that a function is obtained for each of a set of representative standard brightnesses along the brightness continuum. We learn from this family that, up to a critical duration (roughly 150 milliseconds), a decrease in the stimulus level can be offset by lengthening the flash. Moreover, the critical duration gets systematically shorter as the brightness is increased.
Measurement of the static invariances is common in psychophysics. Examples include the equal brightness functions relating energy and wavelength, the equal loudness functions relating sound pressure and tonal frequency, the equal pitch functions relating frequency and sound pressure (within limits, the apparent pitch of a tone can be altered by a change in sound pressure level), and the equal hue functions relating wavelength and light intensity. (The change in hue when intensity is altered has long been known as the Bezold-Brucke phenomenon.)
The measurement of invariance may call for complete equivalence. An example is the concept of metamerism in color vision. The measurement of metameric pairs (sample lights of identical appearance but different wavelength compositions) has made it possible to state the laws of color mixtures and to predict the color of a sample of any spectral composition.
Methods of measurement
Because of its speed and immediacy, the method of adjustment usually recommends itself for the mapping of equivalents. Other usable procedures, however, include constant stimuli, limits, and tracking. The measurement of equivalence is straightforward in principle, but any procedure is usually beset by constant errors, such as the “time error” (see Figure 1).
It is one thing to know the stimulus conditions that produce an invariant sensory effect and another thing to know how much larger one sensory effect is than another—e.g., how much brighter one luminance level appears than another or how much two tones seem to differ in pitch. A major problem of psychophysics is to learn how much the magnitude of the sensory response grows when the stimulus intensity increases.
The direct scaling methods
In the 1930s it became apparent to students of hearing that the logarithmic function fails to agree with the reports of observers who are asked to judge the relative loudness of stimuli. Attempts were made to measure the loudness function by a variety of direct methods. Subsequently, the direct methods were expanded and refined and finally applied to the study of all the major sensory continua.
The main feature of the direct methods is the attempt to match segments of the number continuum directly to segments of the sensory continuum. In the method of magnitude estimation, various fixed levels of the stimulus are presented one by one in irregular order, and the observer attempts to assign numbers to these levels in proportion to their subjective magnitude. The inverse of this procedure is magnitude production: a set of numbers is called out one by one to the observer, who adjusts the level of the stimulus so as to produce subjective magnitudes that are proportional to the numbers. In ratio production, a comparison stimulus is adjusted to appear in some fractional or multiplicative relation to a standard stimulus, and in ratio estimation, the observer estimates numerically the apparent ratio that corresponds to a pair of stimulus magnitudes. Variations on these procedures are numerous (S. S. Stevens 1958).
Two classes of continua
For a few continua, of which pitch is a noteworthy example, a scale of integrated jnds turns out to agree well with direct judgment. S. S. Stevens (1957) called these continua metathetic and distinguished them from the large class of prothetic continua on all of which the jnd does not afford a constant unit of subjective magnitude. (Table 1 provides a partial list of prothetic continua.)
The psychophysical power functions
S. S. Stevens has also proposed a general psychophysical relation pertaining to all prothetic continua (1957). Equal stimulus ratios are held to correspond to equal sensation ratios (rather than to equal sensation differences, as Fechner had conjectured). In other words, the apparent magnitude ψ grows as a power function of the stimulus magnitude, or ψ= fe$0, where k is a constant of proportionality and β is the exponent. The size of β varies from one continuum to another. In Figure 2A are plotted the power functions in linear coordinates for three continua: apparent length (β = 1), brightness (β = 0.33), and the apparent intensity of an electric current passed through the fingers (β = 3.5). When plotted in log-log coordinates, as in Figure 2B, these same functions become straight lines whose slopes equal the values of the exponents. This is true because the logarithmic form of the power function is logψ = log k + β log logø.
Table 1 shows that the size of the exponent may depend not only on the sense organ stimulated but also on the conditions of the stimulation. Note the difference between the monaural and the binaural loudness functions and the exponent’s dependence on frequency for vibration magnitude.
Although the simple equation ψ = kϕ2 holds for large stimulus values, in the neighborhood of the absolute threshold a more precise form is needed. The power equation can be written as “= k(ϕ — ϕ0 y, where #” approximates the absolute threshold. The
|Table 1 — Representative exponents of the power functions relating psychological magnitude to stimulus magnitude on prothetic continua|
|Adapted from S. S. Stevens 1957.|
|Brightness||0.33||5° target—dark-adapted eye|
|Brightness||0.5||point source—dark-adapted eye|
|Lightness||1.2||reflectance of gray papers|
|Vibration||0.95||60 cps—on finger|
|Vibration||0.6||250 cps—on finger|
|Repetition rate||1.0||light, sound, touch, and shocks|
|Finger span||1.3||thickness of wood blocks|
|Pressure on palm||1.1||static force on skin|
|Force of handgrip||1.7||precision hand dynamometer|
|Vocal effort||1.1||sound pressure of vocalization|
|Electric shock||3.5||60 cps—through fingers|
|Tactual roughness||1.5||felt diameter of emery grits|
|Tactual hardness||0.8||rubber squeezed between fingers|
|Viscosity||0.5||stirring silicone fluids|
|Visual velocity||1.2||moving spot of light|
|Visual length||1.0||projected line of light|
|Visual area||0.7||projected square of light|
correction for threshold brings into coincidence the zero of the stimulus scale and the zero of the sensation scale (Luce et al. 1963).
The properties of sensory systems may reveal themselves as parametric shifts in the values of, k, and β. The changes in visual sensitivity that occur under light adaptation provide an example (Stevens & Stevens 1963). Light adaptation causes (1) an elevation in the absolute threshold (i.e., f0 increases), (2) an increase in the luminance necessary to produce a given subjective brightness (i.e., k decreases), and (3) a slight increase in the exponentβ. The mapping of these parametric changes has made it possible to write the power function that pertains to any given level of adaptation and consequently to predict the subjective brightness produced by any luminance level when viewed by an eye adapted to any other luminance level.
A method has been devised that circumvents the need for the observer to make numerical estimates of his sensation but leads to the same psychophysical power function (see S. S. Stevens 1961; and Luce et al. 1963). In cross-modality matching the task is to make the sensations in two different sense modalities appear equal in strength. The pairs of physical intensities that produce equal apparent intensities can be plotted as an equal sensation function. It turns out that the equal sensation function relating any two prothetic continua a and b is itself a power function of the form , where ɸa and ɸb stand for physical intensity. The size of the exponent γ depends on which two continua are matched. Within the experimental error, γ is predictable from the psychophysical function governing the two continua. Given that and (with suitable units of measurement), and given that ɸa= ɸb the equation of the equal sensation function becomes . The exponent γ thus turns out to be the ratio of the exponents α and β.
Any continuum could be substituted for the number continuum and used as a “yardstick” to measure sensory magnitudes on all of the other sensory continua. In one set of experiments, for example, force of handgrip as registered on a dynamometer was used to assess subjective magnitudes on nine other prothetic continua (S. S. Stevens 1961). Many other examples could be cited to show that the psychophysical power law is able to predict both the form and the exponent of the equal sensation function obtained by cross-modality matching.
Joseph C. Stevens
[See alsoScaling. Other relevant material may be found inAttention; Hearing; Information Theory; Pain; Senses; Skin Senses And Kinesthesis; Taste And Smell; Vision; and in the biographies ofFechner; Helmholtz; Titchener; Weber, Ernst Heinrich; Wundt. Statistical techniques applicable to psychophysical methods are described inQuantal Response.]
Boring, Edwin G. 1942 Sensation and Perception in the History of Experimental Psychology. New York: Appleton.
Fechner, Gustav T. (1860) 1907 Elemente der Psychophysik. 3d ed. 2 vols. Leipzig: Breitkopf & Hartel.
Luce, R. Duncan; Bush, Robert R.; and Galanter, Eugene (editors) 1963 Handbook of Mathematical Psychology. Volume 1. New York: Wiley.
Pieron, Henri (1945) 1952 The Sensations: Their Functions, Processes, and Mechanisms. New Haven: Yale Univ. Press; London: Miiller. → First published as Aux sources de la connaissance: La sensation, guide de vie.
Stevens, J. C; and Stevens, S. S. 1963 Brightness Function: Effects of Adaptation. Journal of the Optical Society of America 53:375-385.
Stevens, S. S. (editor) 1951 Handbook of Experimental Psychology. New York: Wiley.
Stevens, S. S. 1957 On the Psychophysical Law. Psychological Review 64:153-181.
Stevens, S. S. 1958 Problems and Methods of Psycho-physics. Psychological Bulletin 55.177—196.
Stevens, S. S. 1961 To Honor Fechner and Repeal His Law. Science 133:80-86.
Stevens, S. S. 1966 A Metric for the Social Consensus. Science 151:530-541.
Stevens, S. S.; and Galanter, Eugene 1957 Ratio Scales and Category Scales for a Dozen Perceptual Continua. Journal of Experimental Psychology 54: 377-411.
Swets, John A. (editor) 1964 Signal Detection and Recognition by Human Observers: Contemporary Readings. New York: Wiley.
Symposium On Principles Of Sensory Communication, Endicott House, 1959 1961 Sensory Communication: Contributions. Cambridge, Mass.: M.I.T. Press.
Titchener, Edward B. 1905 Experimental Psychology: A Manual of Laboratory Practice. Volume 2: Quantitative Experiments. London and New York: Macmillan.
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"Psychophysics." International Encyclopedia of the Social Sciences. . Encyclopedia.com. (October 23, 2017). http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/psychophysics
"Psychophysics." International Encyclopedia of the Social Sciences. . Retrieved October 23, 2017 from Encyclopedia.com: http://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/psychophysics
The subfield of psychology that deals with the transformation from the physical to the psychological through detection, identification, discrimination, and scaling.
Psychophysics originated with the research of Gustav Fechner (1801-1887), who first studied the relationship between incoming physical stimuli and the responses to them. Psychophysicists have generally used two approaches in studying our sensitivity to stimuli around us: measuring the absolute threshold or discovering the difference threshold. In studying the absolute threshold using the method of constant stimuli, an experimenter will, for example, produce an extremely faint tone which the listener cannot hear, then gradually increase the intensity until the person can just hear it; on the next trial, the experimenter will play a sound that is clearly heard, then reduce its intensity until the listener can no longer hear it. Thresholds can also be ascertained through the method of constant stimuli. In this approach, stimuli of varying intensity are randomly presented. Although an observer's measured threshold will change depending on methodology, this technique gives an estimate of an individual's sensitivity.
A different psychophysical approach combines the concept of sensory abilities with the decisions and strategies that an observer uses to maximize performance in a difficult task. Rather than try to identify a single point for the threshold, psychophysicists who employ the signal detection theory have developed ways to measure an observer's sensitivity to stimuli in ways that go beyond the simple concept of the threshold. Some psychophysical research involves the identification of stimuli. There may be no question as to whether we can detect a stimulus, but sometimes we cannot identify it. For example, people can often detect odors but cannot identify them. Research in this area has centered on determining how much information is needed to allow a person to identify a stimulus. Identification constitutes a relatively small part of psychophysical research, although such research has important practical applications. For example, in the development of useful telephones, researchers had to assess how much "noise" or unwanted sound could accompany speech in a phone conversation so that a listener could understand what was said—that is, identify the spoken words accurately.
A third area of psychophysics involves discrimination of different stimuli, or difference thresholds. No two physical stimuli are absolutely identical, although they may seem to be. The question of interest here is how large must the difference be between two stimuli in order for us to detect it. The amount by which two stimuli must differ in order for us to detect the difference is referred to as the JND, or just noticeable difference . Research has indicated that for stimuli of low intensity, we can detect a difference that is small, as the intensity increases, we need a larger difference. Sometimes psychophysicists use reaction time as a measure of how different two stimuli are from one another. When two stimuli are very similar, it takes a longer time to decide if they are different, whereas large differences lead to fast reaction times.
CONCEPTS IN PSYCHOPHYSICS
Absolute threshold: as the stimulus strengthens from the undetectable, the point at which the person first detects it.
Signal detection theory: theory pertaining to the interaction of the sensory capabilities and the decision making factors in detecting a stimulus.
Difference thresholds: at which point can one differentiate between two stimuli. This point is termed just-noticeable difference.
Scaling: using rating scales to assign relative values (for example, rating on a scale of one to ten) to sensory experiences.
The final area of interest to psychophysicists is scaling, the activity of deciding how large or small something is or how much of it is present. Any sensory experience can be scaled. For instance, if the attractiveness of a painting is rated on a scale of one to ten, it is being scaled. If the painting is rated nine, it is considered more attractive than a painting rated eight. This simple example gives the concept underlying scaling, but psychologists have developed more complicated techniques and sophisticated mathematical approaches to scaling.
"Psychophysics." Gale Encyclopedia of Psychology. . Encyclopedia.com. (October 23, 2017). http://www.encyclopedia.com/medicine/encyclopedias-almanacs-transcripts-and-maps/psychophysics
"Psychophysics." Gale Encyclopedia of Psychology. . Retrieved October 23, 2017 from Encyclopedia.com: http://www.encyclopedia.com/medicine/encyclopedias-almanacs-transcripts-and-maps/psychophysics