Three Psychophysical Laws

Louis L. Thurstone
University of Chicago

In a previous article[2] I have shown that Weber's law and Fechner's law are separate laws so that it is logically possible to have a set of data in which either one of these laws is verified and the other one not verified. This distinction comes about primarily by what I have called the discriminal dispersion of the stimulus. It is a vital factor in verifying Weber's law but it is not logically a necessary factor in the verification of Fechner's law. If the discriminal dispersions of all the stimuli are equal, then the two laws can be verified by the same set of data although even then they constitute statements about different aspects of the data.[3] In the same article I have developed the logic by which a psychological scale or continuum becomes a possibility. In that connection I formulated a third psychophysical law which I called the law of comparative judgment. It is my present purpose to explain the differences and the similarities among these three psychophysical laws.

Before making comparisons it may be well briefly to review the statement of each of these three laws.

Weber's law is sometimes carelessly stated as follows: The just noticeable increase in a stimulus is a constant fraction of the stimulus. When the ambiguity of the term `just noticeable' is removed, the law takes the following form: The stimulus increase which is correctly discriminated in any specified proportion of the attempts (except 0 and 100 per

( 425) cent) is a constant fraction of the stimulus magnitude. Since the experimental verification of the law practically always refers to the comparison of two stimuli as to relative magnitude, it is more accurate and satisfactory to state the law so that it really refers to this comparative situation involving differentiation of two compared stimuli rather than the perception of an increase in a single stimulus magnitude. Weber'slaw would then take the following form: Within the same modality and with constant experimental. conditions, the ratio between two stimulus magnitudes which are correctly discriminated in any specified proportion of the attempts, excepting zero and 100 per cent, is a constant. Restating the law in more convenient form we have

R2 / R1 = K or R2 = KR1 (1)

in which K is a constant and R1 and R2 are defined by the relation

P R1 > R2 = C (2)

From (x) we have

P R1 > KR1 = C.

in which C is any arbitrarily assigned constant between 0 and unity. This may be generalized into a complete algebraic statement of Weber's law, as follows:

P R > KR = C (Weber’s Law). (3)

Here the notation R refers to any stimulus magnitude, P is a proportion, P R > KR is therefore the proportion of judgments `R is greater than KR,' while K and C are constants. The constant C may be arbitrarily assigned. It is customary to specify .75 as the arbitrary value for C, while the constancy of K remains to be experimentally verified.

Another frequent but erroneous statement of the law is as follows: Sensations increase in arithmetical progression as the stimuli increase in geometrical progression. This is actually a statement of Fechner's law and not of Weber's law. It will be noticed that Weber's law says absolutely nothing about sensation intensities. These do not occur in the above

(426) statement of the law and it is brought into the discussion of Weber's law only when this law is confused with that of Fechner. None of the variables in equation (3) above say anything directly about sensation intensities. The principal variables are the stimulus magnitude and the factor K, the constancy of which is subject to experimental verification.

Fechner's law is usually and correctly stated in the form

S = K log R (Fechner's law), (4)

in which R again refers to the stimulus magnitude and S refers to the so-called sensation intensity. The factor K is a constant. The universality as well as the psychological acceptability of this law will probably be greatly increased if instead of interpreting S to mean the intensity of a sensation, we interpret it to mean a linear measurement along an abstract psychological continuum from the sensory process of a specified stimulus magnitude as an origin. The term S then refers to a measurement along a psychological continuum from an arbitrary but specified origin. These measurements can of course be objectively identified experimentally only in terms of their respective stimuli. But in either the literal interpretation of the term S, as an actually measured intensity of sensation or the more liberal interpretation of it as a measurement along an abstract psychological continuum, we have two variables, S and R which are related by a simple logarithmic function.

Notice that Fechner's law says absolutely nothing about any proportion of correct discriminatory judgments which constitutes an integral part of Weber's law. In fact, Fechner's law is not at all concerned with the phenomenon of con-fusion of stimulus magnitudes, a phenomenon the description . of which Weber's law is essentially aimed at. It is possible to test Fechner's law by using exclusively supraliminal stimulus differences, whereas the testing of Weber's law requires obviously the use of liminal and infraliminal stimulus differences because Weber's law is primarily concerned with the phenomenon of confusion of stimuli. Fechner's law is a statement concerning the rather common functional relation between

( 427) the stimulus magnitude and the corresponding psychological continuum.

The law of comparative judgment[4] can be stated without approximations in the following form:

S1– S2 = x12√(σ12 + σ22 - 2r σ1 σ2)
(Law of comparative judgment)

in which the two terms Sl and S2 define a linear distance on the psychological continuum, x12 is the sigma value of the observed proportion, PR1 > R2 of judgments `Rl is greater than R2,' σ1 and σ2 refer to the relative ambiguities or discriminal dispersions of the two stimuli while r is the coefficient of correlation for the two discriminal deviations involved in each of the comparative judgments.

Note that this law is entirely independent of the stimulus magnitudes. Hence it is directly applicable for measuring the psychological continuum corresponding to any qualitative series, to any psychological values which are perceived as a continuum, even though the objective counterparts of these values may not be themselves measurable. Since this law deals with the discriminatory process independently of the objective stimulus magnitudes, there can be no objective criterion for the `correctness' of each discriminatory judgment. The proportions which enter into the law of comparative judgment are not proportions of `correct' judgments. They are merely proportions of similar judgments, preferably when each judgment is a choice between only two stimuli, omitting the alternatives of `equal' and `doubtful,' and when the procedure follows otherwise the method of constant stimuli. It does not depend on the phi-gamma hypothesis, because the stimulus magnitudes do not enter into this law.

In order to summarize the similarities and the differences between these three psychophysical laws the following tabular statement may be useful. In the first column are listed the variables that are involved in these laws. ` The `plus sign'

( 428) refers to the fact that the law in question does include this variable, while the minus sign indicates that the law does not say anything regarding it. The variables are as follows:

S = a linear measurement on the abstract psychological continuum.

R = stimulus magnitude

p = proportion of judgments `Rl greater than R2.'

ΔR = (R1— R2) corresponding to the proportion of judgments,

σ = discriminal dispersion.

Variables Weber Fechner Comparative
S – + +
R + + –
p + – +
ΔR + – –
σ – – +

The first row of comparison shows that measurement alonga truly psychological continuum is not involved in Weber's law but that it is involved in Fechner's law and in the law of comparative judgment. . This maybe seen by direct reference to the explicit statements of the three laws. The second row of comparison shows that the stimulus magnitude is involved in both Weber's law and Fechner's law but that stimulus measurement is not involved in the law of comparative judgment. The psychological continuum is defined according to this law in terms of the proportion of similar discriminatory judgments, and the validity of the continuum can be established only in terms of experimental consistency of all the observed proportions.

With the factor P goes also the phenomenon of confusion of stimuli, which is not involved in Fechner's law while it is involved in the other two. Still another way of describing the same difference is to point out that Fechner's law might be experimentally verified by a single set of observations of the stimuli selected so that they appear equally distant from each other. When the physical stimulus magnitudes are plotted

( 429) against a scale of equal appearing intervals, the logarithmic .law should be at least roughly verified. Such a demonstration is not possible for either of the other two laws. For them it is necessary to have a rather long series of separate judgments with a proportion of judgments for each pair of stimuli. This is but another way of showing that Fechner's law is, strictly speaking, not explicitly concerned with errors of observation, whereas the other two laws deal primarily with the magnitudes of errors of judgment.

The stimulus increase (RI— R2) which yields a prescribed proportion of similar discriminatory judgments is involved in Weber's law but it is not involved in either of the other two laws. The stimulus increase is involved in the proportion which is a principal factor in Weber's law, but the proportion involved in the law of comparative judgment defines- the psychological continuum without reference to the stimulus magnitudes. Hence the stimulus increase becomes a part of Weber's law but not of the law of comparative judgment.

The last row of the comparative table shows that the dis-criminal dispersion, the relative ambiguity with which each stimulus is perceived, is a part of the law of comparative judgment,' but that it isnot explict in either Weber's law or Fechner's law. It is in the possibility of variation in the discriminal dispersions of the stimuli that the separation between Weber's law' and Fechner's law' appears. I -have' previously shown that when the discriminal dispersions can be, assumed or shown to be constant for all the stimuli in an experiment, then the same set of data will be found to verify both Weber's and Fechner's laws, but that if, the discriminal . dispersions are not constant, then one of these laws may be verified when the other one is not.

In brief, Fechner's law deals with the apparent interval in relation to the stimulus interval, while Weber's law deals with the frequency of correct discrimination of an interval in relation to the stimulus interval. Since apparently equal intervals are not necessarily equally often discriminated, the two laws become logically separated .5 Fechner's law con-

( 430) -cerns the nature .of the S—R function, while Weber's law concerns the frequency with which adjacent stimuli are confused. It would be logically possible for Fechner's law to be applicableto a set of data even if errors of observation or confusion should never occur, but in such a state of discriminatory perfection Weber's law could not exist


Cattell's formulation of Weber's law, or a substitute for it, makes it a relation between (t) the magnitude of the stimulus and (2) the magnitude of the standard observational error, expressed, also tin terms of the stimulus scale. His literal statement is as follows: [6] "The error of observation tends to increase as the square root of the magnitude, the increase being subject to variation . . . ." Whether the exact relation is that of the square root function is not for the moment our primary concern. We are noting here primarily the variables involved in these laws. Cattell's formulation may be re-stated explicitly as follows:

El= M √ R1

in which El is the standard observational error for the stimulus magnitude R1, while M is a constant. Restating it more generally without committing ourselves to the square root function, we have

El= M . f (R1)

which brings out clearly the two variables, E1 and R1 in terms of which Cattell has cast Weber's law. This is consistent with our tabular analysis in that the law involves the stimulus magnitude R explicitly. It involves OR which is synonymous with the observational error E, and it involves the proportion, p, in the definition of the observational error. The standard error, E, can be defined as follows:

E = √ ((Δ R1)2)/n)

( 431) for the method of reproduction so that p is about 2/3, or in some other equivalent manner. No matter how the observational error, E, may be defined, it automatically locks the proportion, p, of the reproduced stimuli which are counted within the range (R + ΔR) or (R + E). Conversely, a prescribed proportion, p, for the method of reproduction automatically determines the error of observation, E. According to Cattell, Weber's law does not involve the sensory or psychological continuum, S. Speaking of Weber's law he says: (p.23) "All the experiments made by the first three methods which we\have described seem to us to determine the error of observation under varying circumstances, and not to measure at all the quantity of sensation." Nor does Webers, law recognize the subjective standard error of observation, σl, of single stimuli.

If Fechner's law, or a substitute for it which involves the S–R relation, can be assumed, it is possible to express Cattell's objective observational error, E, in terms of our discriminal dispersion, σ, for single stimuli. The objective observational error, E, is measured 'on the stimulus continuum, whereas the discriminal dispersion, a, is measured on the psychological continuum.

In attempting to make an algebraic statement of Weber's law Cattell writes (p. 21)


in which he defines S as his stimulus, ΔS is the increase in the stimulus S which can be just noticed, while N is also defined as a least difference, with insistence that it be a physical quantity also. I fail to see any meaning in it since there are three physical quantities, N, S, and ΔS and it seems that N and ΔS are the same thing. This equation is not a statement of Weber's law. His verbal formulation is clearer. The main point we get from Cattell in the present analysis is that Weber's law does not directly concern the measurement of sensation intensity or the psychological continuum.

( 432)

These three psychophysical laws have been stated algebraically in equations (3) (’) and (5) respectively. The principal variables are (i) the stimulus magnitude, (2) the psychological S-value, and (3) the degree of confusion of stimuli. It has been shown that each of the three laws relates two of these variables and ignores the third.


  1. This is one of a series of articles by members of the staff of the Behavior Research Fund, Illinois Institute for Juvenile Research, Chicago. Series B, No. 108.
  2. Psychophysical Analysis,' Amer. J. Psychol., 1927, 38, 368—389.
  3. In the previous article cited I made the statement (p. 386) that the two laws become identical when the discriminal dispersions are equal. That was not quite accurate. The two laws are then verified by the same set of data but they describe different aspects of the data.
  4. For a more complete statement of this law with different degrees of approximation for its experimental verification, see my article, 'A Law of Comparative Judgment,' PSYCHOL. REV., 1927, 34, 273 - 286
  5. Thurstone, L. L., `Equally often Noticed Differences,' J. Educ. Psychol., 1927, 18, 289-293.
  6. Fullerton and Cattell, 'On the Perception of Small Differences,' University of Pennsylvania Philosophical Series, 1892, p. 25.

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