A Mental Unit of Measurement
Louis L. Thurstone
The measurement scale of primary interest in psychophysics is the S-scale, which may be called the psychological continuum in contrast with the physical or stimulus-continuum, the R-scale. In the original formulation of Fechner's law the S-scale was described as sensation intensity, but this interpretation of the psychological continuum is now pretty much out of date. We now speak of the S-scale as though it measured so-called sense distances in relation to the corresponding differences or distances on the scale of stimulus magnitude. It is my present purpose to offer a revision of the sense-distance interpretation of the S-scale so that it will be independent of the j.n.d. and the difference limen and to propose a mental unit for mental measurement instead of the ambiguous physical unit, the j.n.d. I shall also try to show that this revised interpretation of the S-value makes it a continuous function of the stimulus magnitude because Fechner's paradox does not exist.
This continuum which is designated S in Fechner's law, S = k log R, represents, first of all, a gradation of qualitative processes. These processes may be described as subjective or physiological to suit individual and temperamental preferences. That distinction does not affect the psychophysical problem. The continuum implies qualitative variation, and it may or may not also possess quantitative aspects. It may be unidimensional or multi-dimensional.
A point on this continuum may be designated only by a corresponding stimulus. The stimulus may in turn be designated (1) by a quantitative stimulus attribute such as intensity or magnitude or (2) by a qualitatively varying stim-
( 416) -ulus attribute such as excellence of handwriting or the beauty of rug patterns or the degree of pacifism expressed in statements about peace and war. Any perceptual quality which may be allocated to a point on the psychological continuum is not itself a magnitude. It is not divisible into parts. It is not a sum of any mental or physical units. It is not twice, three times, or four times as strong, high, beautiful, or good as some other process on the same continuum. It is not a number. It is not a quantity. These are statements about the psychological continuum on which at least fair agreement among psychologists is to be expected.
But furthermore, the j.n.d. is not a reliable unit of measurement along the S-scale. Any point on the scale represents a unified indivisible experience. Any two such points represent two such experiences which may be qualitatively entirely different and in every way incommensurate, so that they may represent by their own immediate attributes perhaps no similarity by which any "distance" between them could be measured.
With these negations granted, just how do these qualitative entities or processes become a measurable continuum? They acquire conceptual linearity and measurability in the probability with which each of them may be expected to associate with any prescribed stimulus. This is the crucial characteristic of the psychological continuum in terms of which psychological measurement is possible. The S-continuum is constructed or defined in such a manner that the frequency distribution of the S-experiences for any given stimulus R is normal. Each modal Sk experience is that particular S-experience which is most frequently associated with the stimulus Rk. Hence if measurement begins with the stimulus R. for which S. is the most common or modal S-experience, then some other S-experience, Sb, will be spaced far away, along the imaginary continuum, if the probability is low that Sb will be experienced with the stimulus R.. It will be spaced close to S. if the probability is relatively high that Sb will be experienced with R. Fortunately, it is possible to verify experimentally the validity of this definition or construction of the S-scale be-
( 417) -cause the continuum may be constructed separately for each stimulus, and the attainment of internal consistency of the scale will indicate whether the correct form of frequency distribution has been used. Therefore the normal frequency distribution is not blindly assumed. It is tested for. If found incorrect, other forms of distribution may be tried in a similar manner. There is one assumption underlying this construction of the psycho-logical continuum, namely, that all the stimuli in a series project the same form of frequency distribution on the S-scale, but it is not assumed that their dispersions are equal.
Since the dispersions which the several stimuli project on the S-continuum are not assumed to be equal, the natural unit for psychological measurement becomes the dispersion of one of these stimuli measured on the psychological scale. This is what I have called the discriminal dispersion or, more specifically, the discriminal error of each stimulus, and it should not be confused with the customary error of observation. There are two fundamental differences, namely, (1) the discriminal dispersion or error is measured on the S-scale, whereas errors of observation are naturally measured on the R-scale; and (2) the discriminal dispersion is the dispersion projected on the S-scale by a single stimulus, whereas an observational error is naturally the pooled effect of two stimuli. The two stimuli are the two terms in the psychophysical judgment by the method of constant stimuli, or they are the presented and the reproduced stimuli in the method of reproduction.
It should be noted that an observational error can itself be objectively produced, and it can be directly measured on the R-scale. The discriminal dispersion, though more elemental in character than the observational error, cannot by itself be objectively produced. It can be measured only indirectly since it concerns the S-scale. Every judgment when objectively produced constitutes a single observational error which is loaded with at least two discriminal dispersions or errors. A single discriminal dispersion or discriminal deviation cannot by itself ever become an objective record, and consequently its measurement must necessarily be indirect.
I have attempted to state briefly a definition of the psychological continuum and to show that a truly psychological unit of measurement may be established for the psychological scale. The psychological S-scale becomes, then, in effect a frequency scale as far as its experimental identification is concerned. It is an imaginary scale on which we allocate and space out the psychological counterparts of the several stimuli in the stimulus series. Since the stimulus series is regarded as strictly continuous, we define the corresponding psychological scale similarly, so that any stimulus magnitude or quality may be allocated to a point on the continuous S-scale. Similarly, since the psychological continuum is not directly or physically accessible or controllable, we identify any point on the S-scale by a stimulus magnitude.
The customary definition of the S-scale as the measurement of so-called sense distance is not here entirely rejected, but it is made more definite. As long as we define the S-scale as the measurement of sense distance and still deny that it measures sensation intensity or any other quantitative characteristic of sensation, there remains an unsatisfactory vagueness about the nature of the psychological continuum. If we insist that the sense quality is not itself an intensity or magnitude of any sort, how does it happen that we get quantitative measurement in the form of measured distances between these sense qualities which are themselves denied measurable and quantitative attributes? What constitutes the sense distance that is measured between two qualitative entities? That is a question concerning the very nature of the psychological continuum which, as far as I am aware, has not hitherto been answered. It is the answer to just this question that I have attempted in formulating a revised definition of the S-scale so that it may also fit the experimental facts.
Let the S-scale consist in a gradation of qualities by means of which we perceive any specified stimulus continuum. Allow that a given stimulus is not always perceived by the same process on the S-continuum. Let the quality most commonly perceived in the given stimulus be designated the
( 419) modal quality or process for that stimulus. Then we can assign numerical values to other qualities in the S-scale in accordance with the frequency with which they are perceived in the given stimulus. It does not matter whether the S-qualities or processes are in any real sense actually spaced out in a continuum so long as they behave as though they were so spaced out. Mental measurement depends according to the present interpretation on the frequency with which each of the processes constitutes the response to a given stimulus. It is reasonable to assume that two perceptual sense qualities or processes which are close together on the psychological continuum are qualitatively similar and that therefore either one of them may more or less readily be perceived in the same stimulus. To the extent that two perceptual processes are qualitatively similar, to that extent will their probabilities of association with the same stimulus be nearly the same, and to that extent will they tend to be adjacently spaced on the imaginary psychological continuum. It is sufficient for the purposes of mental measurement that the qualitative perceptual processes behave as though their respective probabilities of association with a given stimulus were a normal frequency distribution. The natural psychological unit of measurement becomes, then, the standard deviation of the frequency distribution for a specified stimulus. This unit of mental measurement I have called the standard discriminal error for the specified stimulus, and it is, of course, measured directly on the psychological continuum. It is entirely independent of stimulus measurement. It is independent of the validity of Fechner's law. It is also independent of the validity of Weber's law. It is a valid unit of measurement even when the objective stimulus cannot itself be quantitatively measured.
It should be noted that the unit of mental measurement that I have pro-posed is not in any sense a j.n.d. The just noticeable difference is in every case a stimulus measurement. It is measured on the R-scale. Hence it is in reality a physical unit which in some situations can serve indirectly the purposes of mental measurement. The discriminal error is a mental unit of measurement since it is defined on the psychological
(420) continuum. Its physical equivalent will vary from one situation to another, depending primarily on the validity of Fechner's law or some other S—R relation for the particular perceptual function under consideration. This proposed mental unit for mental measurement may be defined as the standard deviation of the frequency distribution projected by a standard stimulus on the psychological continuum. I propose to call this mental unit of measurement the standard discriminal error. The assumption, the correctness of which will determine the validity of this unit of measurement, is that an S-scale with internal consistency will be obtained by spacing the perceptual qualities on it so that their probabilities of association with any given stimulus will be Gaussian. With the psychological continuum so defined, the standard discriminal error, as a mental unit of measurement, will be the standard deviation of the perceptual qualities perceived in a standard stimulus.
The Continuity of the S-Scale
We have so far taken for granted the continuity of the S-scale, but the descriptions of the psychophysical S-R relation so frequently give the idea of a discrete series of steps or jerks that the error of such a notion needs clearly to be corrected. The typical description of the psychophysical S-R relation starts with stimuli R1, R2, R3, so nearly similar that they differ successively by one "j.n.d." Corresponding to these stimulus magnitudes are postulated S-values Si, S2, S3. Then the implication is that any stimulus difference less than (R1— R2) cannot be discriminated at all and that as soon as the stimulus difference becomes as large as (R1— R2), then suddenly the difference jumps into perceptibility, and it is designated "a least perceptible difference," a sense minimum, the unit of mental measurement, the j.n.d. Even so clear a writer as Keyser describes the S-R relation essentially in this way as an example of the data on which we built our conceptual continuities. Even Titchener does not seem to avoid this possible misinterpretation. The following quotations from
( 421) Titchener illustrate the experimental discreteness which is apparently read into the S-continuum. "Suppose that a brightness or a noise is given, and that we seek to determine the just noticeably brighter brightness, or the just noticeably louder noise. The experiment is identical with one of our `friction' experiments; and its result is the ascertainment of a just noticeable sense-distance. Let us perform it at various points of the sense scale. . . . Now these are all least distances, minima of sensible dis-tance." And similarly on page xxxvii we read: "The facts that a stimulus must attain a certain magnitude in order to arouse a sensation at all, and that it must attain a certain magnitude in order to effect a noticeable change in sensation, are facts of the same order."
Another description of the S-R relation which is just as misleading as the quotations from Titchener begins with what is known as Fechner's paradox. This description runs usually as follows: let R1, R2, R3 be three stimuli of decreasing order of magnitude. Let the difference (R1— R2) be so slight that it cannot be noticed, i.e., less than the difference limen. Let the difference (R2— R3) also be less than the difference limen. Then it might still happen that the difference (R1— R3) would be a little larger than the difference limen so that it would be noticed. Then, so the argument goes, it would be possible to have the following psychological equalities:
which would be inconsistent with the fact that
This is known as Fechner's paradox. The truth is that no one has yet found a just noticeable difference. The least sense distance has never been experimentally demonstrated, and Fechner's paradox does not exist.
Now as a matter of fact, everyone who works at all seriously in psycho-physics knows that just noticeable differ
( 422) -ences have never been found, that is necessary to specify quite arbitrarily a stipulated frequency of discrimination in order to put any sense in the j.n.d., that the phi-gamma hypothesis or its variants assume continuity of the psychometric function, and that experiments such as those of Brown on very small stimulus differences indicate experimentally that sense minima in any genuine sense have not been found. It would seem best therefore to avoid these entirely misleading descriptions of the functional relation between the stimulus magnitude R and the psychological value S. These ambiguities of description disappear entirely if description of the S-R relation be kept true to the experimental facts so that the psychological magnitude, or sense distance, between two stimuli be expressed in terms of the observed frequency, or the probability, of its discrimination. As the stimulus difference increases by gradations ever so slight, at least as far as they are experimentally controllable, the probability becomes higher and higher that the stimulus difference will be discriminated. In this sense, then, we may arbitrarily define the difference limen or the j.n.d. as that stimulus difference which has a probability of .75 of being correctly discriminated. This is the manner in which we actually do determine the limen, and it is consequently inaccurate to allow the description of the S-R relation to remain as though it consisted in a series of jerks with a constant friction load for every jerk. The description of Fechner's paradox may still stand as a curious and intriguing distortion of truth. The argument that the two perceptual qualities are psychologically equal because they are equal in a single perceptual judgment simply ignores the fact that the quality perceived in the stimulus is fluctuating or varying. True, they are equal in the judgment in which they are perceived to be equal, but the S-value of a stimulus is the mean perceptual value of many repeated perceptions of the same stimulus. My point here concerning the continuity of the S-scale is nothing new. It is merely a plea for consistency so that throughout the description of psychophysical
( 423) effects the continuity of the S-scale may be insured, since it is everywhere experimentally indicated.
This consistency is readily obtained if we will recognize that there is a
difference between the S-value perceived in a stimulus on any particular
occasion and the mean of the S-values perceived in the stimulus on many
occasions. It is in the dispersion which a constant stimulus projects on the
S-scale that we have the opportunity to establish a truly mental unit of
measurement. It might seem that the average S-separation projected by a pair
of stimuli would be a better unit of measurement, but such a unit would suffer
from the logical disadvantage that the mental unit, so defined, would be loaded
with two discriminal dispersions which would in any event have to be solved for.
Since the dispersion projected by a single stimulus is more elemental than the
pooled effect of two stimuli, it seems more natural to adopt the projected
dispersion of a standard stimulus as the basis for a unit of measurement for the
psychological continuum. This unit I propose to call the standard discriminal
error. Its numerical treatment consists merely in assigning the value of unity
to one of the two discriminal errors, σ1 and σ2, in the
law of comparative judgment.