# 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

R_{2} / R_{1} = K or R_{2}
= KR_{1} (1)

in which *K *is a constant and *R _{1} *and

*R*are defined by the relation

_{2}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 (Webers 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:

S_{1} S_{2} = x_{12}√(σ_{1}^{2}
+ σ_{2}^{2} - 2r σ_{1} σ_{2})

(Law of comparative judgment)

in which the two terms S_{l} and S_{2} define a linear
distance on the psychological continuum, x_{12 }is the sigma value of
the observed proportion, P_{R1 > R2}* *of judgments `R_{l}
is greater than R_{2},'_{ }σ_{1 }and σ_{2 }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 *`R _{l} *greater than

*R*

_{2}

*.'*

ΔR* = (R*_{1}*— R*_{2}*) *corresponding to the
proportion of judgments,

σ = discriminal dispersion.

Variables | Weber | Fechner | Comparative Judgment |
---|---|---|---|

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 may*be 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 (R_{I}— R_{2}) 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 i*snot 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

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:

*E*_{l}*= M √ R*_{1}

in which *E*_{l}* *is the standard observational error for
the stimulus magnitude *R*_{1}*, *while M is a constant.
Restating it more generally without committing ourselves to the square root
function, we have

*E*_{l}*= M . f (R*_{1})

which brings out clearly the two variables, *E*_{1}* *and *
R*_{1}* *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* *R*_{1})^{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, σ_{l}, of single stimuli.

If Fechner's law, or a substitute for it which involves the *SR *
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)

*N=C*.* (Δ*S/S),

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.