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tible positive and of the just imperceptible positive ciifiFerence
defines the threshold in the direction of increase, and the
arithmetical mean of the jusc perceptible and of the just im-
perceptible negative difference gives the threshold in the direc-
tion of decrease.*

The practical application of this method meets with two very
peculiar difficulties which impair its serviceability and necessi-
tate a change in the experimental procedure. It seems that
it is essential for the result, by which steps the threshold is ap-
proached. After the first rough determination is made, one tries
to get a more accurate one by using smaller intermediate steps.
In experimenting with this new series one notices very soon
that the subject is more apt to perceive smaller differences in
the determination of the just perceptible difference, and not to
perceive larger differences in the determination of the just im-
perceptible difference than he was before. Usually one explains
this fact by the influence of expectation, because the subject
knows that an imperceptible difference is to be increased and
a perceptible difference is to be diminished. In order to avoid
this inconvenience one has employed two means. The first was
to use only trained subjects who are free from the influence of
expectation, and the second consisted in appl3dng the stimuli
in irregular order. The first way cannot always be used, because
one frequently is obliged to experiment on untrained or half-
trained subjects, and this requirement of training on the part
of the subject, furthermore, disposes of the possibility of using
this method for practical purposes. The determination whether
the sensitivity of a patient is below or above the normal must
necessarily be made on untrained subjects, and until now no
other method but the method of just perceptible differences
could be used for this purpose. The method of presenting the
comparison stimuli in irregular order has the inconvenience that
the results cannot be worked out by the algorithm of the method

♦The threshold in the direction of increase and that in the direction of
decrease are frequently combined in order to obtain a generalized threshold
or to eliminate constant errors. This procedure, which was first suggested
by VoLKMANN, gives a result which seemed to be not entirely clear in its
signification; see Titchner, /. c. Part 2, p. 112 sqs.



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METHOD OF JUST PERCEPTIBLIQ DIFFERENCES 43

of just perceptible differences. This modification of the method
of just perceptible differences is called the method of irregular
variation. The results obtained in this way are usually worked
out by an algorithm similar to that of the error methods.*

The second difficulty is due to the fact that frequently a differ-
ence is not noticed after a smaller one has been perceived in a
series for the determination of the just perceptible difference,
or that a difference is noticed after a larger difference has been
imperceptible in a determination of the just imperceptible differ-
ence, f The question arises, wh^t must be done with such
results? Some investigators take the view that these series
must be ruled out and that only those series must be kept
for the final computation, where no such inversion occurs
and where the two classes of judgments are strictly sep-
arated. Another possibility consists in not going beyond the
first difference which is perceived or which fails to be perceived,
thus avoiding these dubious cades. It is obvious of course that
one may try to escape this difficulty by taking very large inter-
mediate steps, but usually one is afraid of doing this, because one
seems to renounce the hope of obtaining a determination of the
threshold the accuracy of which may compare favorably with
that of determinations by one of the error methods.

To these practical difficulties comes the theoretical problem
of finding the relation between the results of the method of just
perceptible differences and those of the error methods, especially
of the method of right and wrong cases. This question is a very
urgent one since the individual experiments are the same for

♦WuNDT, Physiologische Psychologies 6 cd., 1902, Vol. I, p. 478. Some
criticism of Wmidt's view on the relation of the method of just perceptible
differences to the method of irregular variation may be found in E. B. Holt,
Classification of Psychophysical Methods, Psych. Review, Vol. XI, Nov. 1904,
p. 348, who contends that the method of irregular variation does not ap-
proach the error methods, as Wundt says, but that it is identical with them.
He furthermore remarks that the method of just perceptible differences is
not a method, if this word is used in its proper sense, but the statement of
the intention to find a significant value for the threshold.

fSeveral examples of such series were published lately by WnjiBLif
Spbcht, Das VerhaUen von UnterschiedssckweUe und ReixschweUe im Gebiete
des Gehdrsinnes, Archiv. f. d. ges. Psychologie, Vol. 9, 1907, p. 207.



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44 PROBLEMS OF PSYCHOPHYSICS

both methods. The difiference merely consists in using several
pairs of comparison stimuli in a particular order in the method
of just perceptible differences, ot in any order in the method of
irregular variation, and only one pair in the method of right and
wrong cases. To this comes that the results of the error methods,
as well as those of the method of just perceptible differences
seem to confirm Weber's Law, in spite the fact that they are
not comparable among each other. Some investigators believed
that the results of the method oF right and wrong cases were
in no direct relation to those of the method of just perceptible
differences, but others tried to establish such a relation by math-
ematical formulae; this relation however proved to be of very
complicated nature. Of course one might take the view that
one of these methods is not legitimate, but this view, which is
somewhat narrow, would be justifiable only if the interpretation
of the results of either one of these methods were absolutely
clear. The difficulties of the method of just perceptible differ-
ences, which were mentioned above, are considered strong argu-
ments against the use of this method and at present many, if
not most psychologists would favor the error methods against
the method of just perceptible differences, if they had to choose
between them, in spite the fact that the foundations of the method
of right and wrong cases are little known and not very well under-
stood.

For the development of the psychophysical methods one
circumstance proved to be of fundamental importance: The
introduction of the theory of errors by Mbbius and Fechner,
which necessitates the division of the judgments into the two
classes of correct and wrong cases. The theory of errors of
observation gave some insight into the nature of the error meth-
ods and one could hope to find the relation of these methods to
the method of just perceptible differences, because the Gaussian
coefficient of precision (the **mensura ^redsionia") seemed to
give a measure of the accuracy of sensations similar to that
afforded by the smallest perceptible difference. It escaped
attention for a long time that the application of the theory of
errors of observation, though helpful for certain purposes, en-
tirely excludes the notion of a just perceptible difference. It



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METHOD OF JUST PERCEPTIBLE DIFFERENCES 45

is the merit of Jastrow and Cattell to have called attention to
this fact. The theory, indeed/ starts from the supposition that
the probability of every error is a function of the sixe of this er-
ror; the theory makes certain assumptions as to the nature of
this dependence, which imply that no error, no matter how large,
is impossible although its probability may be very small. One
also finds that the greatest error which is likely to be committed
in a certain series of observations depends on the number of
observations, so that the more extended the series is the greater
the largest error becomes which is likely to be committed. Cat-
tell, supposing that the errors in judgments on differences of
intensity of two stimuli follow the same law as the errors of ob-
servation,* concludes that there does not exist a just percep-
tible difference in any absolute sense of the term, because the
smallest difference beyond which all the judgments of a series are
correct depends on the number of observations in this series.
The supposition that there exists a difference which is always
perceived is, therefore, in contradiction with the fundamental
supposition of the method of right and wrong cases. Cattell
stands in his demonstration entirely on the ground of the theory
of errors of observation and he does not go beyond it. His
arguments prevail against the criticisms of his view, some of
which miss entirely the point of his argument, e. g. the jexperi-
mental demonstration that one may make the difference be-
tween two light intensities so small that it cannot be perceived.
It seems that the source of these difficulties is the introduction
of the distinction between correct and wrong judgments. This
is a logical category. What is immediately given are not right
and wrong judgments but the judgments '* greater", ''equal"
and 'smaller"; the correctness or incorrectness of the judgments
is a secondary feature. By introducing this distinction between
correct and incorrect judgments it becomes necessary to dispose
in some way of the equality cases, which as a matter of fact were
so troublesome a feature in the method of right and wrong cases,
that some investigators have tried to get rid of them by not
allowing the subject to pass the judgment "equal." The imme-

*G. S. FuLLBRTON and J. McKbEN Cattbll, On the Perception of Small
Differences, 1892, p. 12 sqs.



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46 PROBLEMS OF PSTCHOPHTSICS

diate data of experiments for the determination of the sensitivity
of a subject are the observed frequencies of the cases in which
the judgments "greater", "equal" and "smaller" were passed.
We will try to base our judgment on the sensitivity of a subject
on these percentages for various differences, without obliterating
one feature of the results by eliminating a class of judgments,
and without introducing the logical category of right and wrong
judgments. We shall use for this purpose the notion of the
probability of a judgment of certain type, which was introduced
in the preceding chapters.

Let us suppose a subject compares n pairs of stimuli which
have one stimulus, the standard, in common. Let the order
in which the comparison stimuli are presented be the same for
all the pairs, e. g. let the standard be the first stimulus of every
pair Let us call the different stimuli with which the standard
is compared *ri, r,, .... r„ and let us suppose that the pairs are
presented in such an order that

n<^< < V

Keeping all the conditions which might possibly influence the
judgment as constant as possible we give this series repeatedly
to a subject. The judgments will vary between "greater,"
"lighter" and "equal" and there exists under the conditions
of the experiments a definite probability for every comparison
resulting in a judgment of one of the three types. The proba-
bilities that a "greater "-judgment will be given may be called



Pif Vtf



where p^ is the probability that the judgment "greater" will
be given on the comparison of the standard with the comparison
stimulus of the k*^ pair. The probabilities that a "greater"-
judgment will not be given are correspondingly



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METHOD OF JUST PERCEPTIBLE DIFFERENCES 47

The cases where a "greater "-judgment is not given comprise
those cases where the comparison stimulus was judged equal to
the standard, and those cases where it seemed to be smaller than
the standard. The comparison stimuli may be chosen in such
a way that the probability of a "greater "-judgment is small for
the stimuli at the beginning of the series, and large (close to the
unit) for the stimuli at the end of it. Presenting this series to
the subject we will obtain a determination of the just perceptible
positive difference in the comparison stimulus of the first pair
on which the judgment "greater" is given, all the preceding
pairs being judged "smaller" or "equal." The probability
that the k*^ pair is the first to be judged ''greater" is the comr
pound probability that the comparison stimulus of this pair is
judged "greater" and that the judgment "greater" is not passed
on any one of the pairs with smaller comparison stimuli. The
probability, therefore, that rj^ will be obtained as a determination
of the just perceptible difference is given by

i =k.i

^k=^i92 ^v^iPk^Pk n 9i (I.)

This quantity is different for every pair. In a series in which
the probabilities of "greater "-judgments increase from very
small values at the beginning of the series to large values, the P's
increase at first and decrease after having attained a maximum,
if certain conditions are fulfilled which will be enumerated later.
The results of N determinations of the just perceptible positive
difference after being brought in proper order will have this
form:
The stimulus Tj was the first to be judged "greater" Nj times,

ct It . (( (( n ti It tt vr tt

"2 ^i



The stimulus r^ was the first to be judged "greater" N^ times,
where

The algorithm of the method of just perceptible differences pre-
scribes to take the arithmetical mean of all these determinations.



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48 PROBLEMS OF PSYCHOPHY8ICS

for the final determination of the just perceptible positive differ-
ence. This average is given by

In a considerable number of determinations every stimulus r^
will tend to occur in a number of times proportional to its prob-
ability Pi£. The most probable result for a great number of
determinations of the just perceptible positive difference is there-
fore

T« r,P,-\-r^P^+... -{-r^P^ (II.)

This relation shows that the result of the method of just percep-
tible differences depends on the probabilities of the judgments
of different types and that, therefore, its basis is identical with
that of the error methods. The second remark which we have
to make regards equation I. The value of P does not change

if the order of the terms q^, q,, qj^., Pi^ is changed, because

a product is independent of the order of its factors. It does not
matter either if a stimulus Tj^^., is given before rj^, because
Pk + . does not enter into relation I. This means that the prob-
ability of a stimulus being the smallest on which a " greater "-
judgment is given does not depend on the order in which the
pairs are presented. The method of just perceptible differences,
therefore, is not tied down to the rule of presenting the stimuli
in the order of their magnitude.

Two interpretations may be given to relation II. The first
is based on the notion of the mathematical expectation; its psychol-
ogical bearing is less obvious but it is more serviceable for cer-
tain practical purposes. Multiplying each stimulus with the
probability that it will be obtained as a result of the method
of just perceptible differences gives what is called the mathematical
expectation for this stimulus being the threshold, and the sum
of these products for all the pairs of the series gives the mathe-
matical expectation for the entire series. The mathematical
expectation for N repetitions of the same event is N times that
for a single event and taking the results of many observations
for its determination gives the result a greater exactitude. This
interpretation of the algorithm of the method of just percep-



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METHOD OP JtJST PERCEPTIBLE DIFFERENCES 49

tible differences is independent from any particular hypothesis
as to the law of distribution. One may try to make its meaning
clear in this way. Let us suppose that A pays to B a sum of
money proportional to the intensity of the comparison stimulus
which is obtained as a determination of the jiist perceptible
positive difference, how much must B pay to A in order to in-
duce him to make this agreement and to make it a fair wager?
Relation II gives the answer that B's payment must be propor-
tional to T, because in this case the expectation of A is equal
to that of B.

The second interpretation is based on the signification of the
arithmetical mean for symmetrical distributions. Taking the
average of a series of observations means that one tries to
determine the most probable value, if the distribution is sym-
metrical. The most probable result of one determination by the

series r^, rj, r^ is the one for which the product of formula

I is a maximum. P^ is a maximum if

or introducing the corresponding expressions

i=A?-2 i—k-i i—k

/>k.x n q,<p^ n qi>p^^, n 9, ....(iii.)

i — i i = i i = i

By splitting up this relation into two and eliminating the com-
mon factors we obtain the conditions

Pk-i<^k.iPk and />k>9k^k + i
which are identical with

^</.,and ^^>P... (IV.)

It is, furthermore, a fact that the probability of a "heavier "-
judgment becomes the greater the smaller negativa differences,
and the greater positive differences of the stimuli become, so that
relation IV must be simultaneous with

Pk.i<Pk<Pk + i (IVa.)

This relation shows that the position of the maximum of P^^
depends on the probability of a '*h3avier"-judgment for this



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50 PROBLEMS OP PSYCHOPHY8IC8

pair and on those for the pairs immediately preceding and im-
mediately following. The formulae IV and IVa give the condi-
tions, that the value of a certain Pj^ is greater than those in its
neighborhood. If these conditions are fulfilled it does not fol-
low that P]c is the absolute maximum (i. e. greater than any
other value of the series), but if Pj^ is the absolute maximum
the conditions IV and IVa must be fulfilled. Let us suppose
that the first k stimuli r„ r,, t^ are chosen in such a way that

Pi<Pa<.. .<Pk

and let us consider the possible effect of our choice of the next
stimuli. Does it depend on our choice of the stimuli Tj^ + ly rj^ 4. s

r„ that Pij is the maximum, or is the peculiarity of rj^ having

the greatest probability of being observed as the just perceptible
difference based on some particular quality of this stimulus?
In the latter case we will have in our result a determination of
the just perceptible difference irrespectively of the way in which
we approach it, but in the first case this quantity would be differ-
ent for different series of comparison stimuli. We suppose that
the probabilities of "greater "-judgments are analjrtic func-
tions of the intensities of the comparison stimuli, so that we have

/',='r(r,)



and we cboose the stimulus r^+i only slightly different from
rij, so that the higher powers of the difference

can be neglected. The second part of relation IV has the form



l-'F('-k)



>"¥ (r^+a).



The value of ^ (x) is, by its nature of being a mathematical
probability, smaller than 1, and the term on the left side of the



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METHOD OF JUST PERCEPTIBLE DIFFERENCES



51



above relation is the sum of a geometric series. Developing
^(rk+*) 1^ ^ power series we find

or



Neglecting the powers of d and summing up the series we find



*<;



^(r^)'



.(V.)



as the condition with which we must comply in our choice
of the stimulus rj^^, in order to make Pk+i smaller than Pjj.
The terms ^(r,^) and l-^^Crj^) are necessarily positive num-
bers and ^"'(rij) is also positive, because V(r) is an increasing
monoton function; d is, therefore, positive. From this it fol-
lows that it is always possible to choose a stimulus rj^ ^ , greater
than rj^ in such a way that r,^ is more likely to come out as the
result of the determination of the just perceptible difference than
Fi^ + j. This may be done by choosing r^.^.! only slightly diff-
erent from T^,

It is obvious that we may confine our considerations of the
conditions for Pk + i>Pk ^^ ^^^ case ^(rj^X^, because Pk + i
must necessarily be smaller than Pj^ if ^^(rij) is greater than J.
We put

^(^k)=i-^
and find that F^ 4. ^ will be greater than Pj^ if

or

which determines the relation

l-2e



^(^k + x)>



l + 2«



.(V.)



This fraction is smaller than 1 and it may represent, therefore,
a mathematical probability. It remains to show that it may



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52 PROBLEMS OF P8YCH0PHYSICS

represent the probability of a ''greater "-judgment on one of
the stimuli which may be chosen as the k+1 pair of our series.
These stimuli have to satisfy the relation

We find for the greatest value of the difference V(rk)-V(rjg^.,)
1-2^ (1-2^)?



i-e-



l+2e 2(14-2e)



which is always negative. From this it follows that it is always
possible to find a stimulus rjg^.,>r,^ which satisfies the relation
V. as long as p<i. The stimulus determined in this way is
more likely to be observed as a result of a determination of the
threshold by the method of just perceptible differences than the
preceding stimulus.

These considerations show the importance of the choice of the
comparison stimuli. The generality of our discussion is not
impaired by the supposition that there is only one maximum
of the P's, because if there are several values which satisfy rela-
tion IV we will have to consider an intermediate value (the mean)
and the influence of our choice of the comparison stimuli is equal
to the sum of the disturbances in the position of the single maxima.
The practical bearing of this part of the demonstration is that
one must not pick out the comparison stimuli at random, but
one will choose them according to a principle. The most obvious
rule will be to choose equidistant values

r, = ri + d
r3-ri+2d



= rj4-(n-l)d.



The choice of d will depend on the accuracy aimed at in the de-
termination of the threshold.

The outcome of a definite series of experiments for the deter-
mination of the just perceptible difference is a well defined quant-
ity, but whether the final average has the signification of a most
probable result depends on the distribution of the observed values.



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METHOD OF JUST PERCEPTIBLE DIFFERENCES 53

The discussion of this question will show that this suppos^ion
may be made under certain conditions. The result of the method
of just perceptible differences, therefore, may have a significa-
tion independent from the particular series of experiments by
which it is reached.

The arithmetical mean is the most probable value of a set of
observations, if the distribution is symmetrical i.e. if positive and
negative deviations have the same probability. The <I>(7')-law
is only a special ease of symmetrical distributions. Under what
conditions will the values r^Nj^ be distributed symmetrically
around their mean? The nature of this distribution obviously
depends on the values of the P's. The P's are constituted of
the p's and of the q's of the pairs and since it depends on our
choice which pairs we will use, no a priori statement is possible
in regard to any particular series unless one knows the pairs and
the respective probabilities of *'h" judgments. Generally the
distribution will not be symmetrical, but it may very well be
that a particular series has a symmetrical distribution or
one which approaches this type. A priori one even cannot make
the supposition that the distribution is regular i. e. that it shows
an- uninterrupted increase at first and, after having attained a
maximum, an uninterrupted decrease. Indeed, if the compar
ison stimuli are picked out entirely at random it may very well
be, that the P's increase in value at first, then after having reached
a secondary maximum decrease and increase again later on. This
will be the case if there are in the series two or more comparison
stimuli with only slightly different probabilities, which are
smaller than i, for the appearance of a ''greater- "judg-
ment. By taking the stimuli in equal intervals one is to
some extent guarded against this eventuality, but if one was
unfortunate in the choice of the comparison stimuli one cannot
eliminate this influence by any amount of care in the perform-
ance of the experiments, or by the combination of any number
of observations. It is not possible either to eliminate the influ-
ence of a skew distribution by taking the arithmetical mean of
a great number of experiments. On account of the fact that one


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Online LibraryFriedrich Maria UrbanThe application of statistical methods to the problems of psychophysics → online text (page 5 of 18)