Repriuted. frorn SersNcn, Fel:ruary 2L, 1958, Yol. 727 , lio. 3295, pages 383*3S9.

In some parts of science we have re-duced the busiRess of measuring to sim-ple routires, chores to be done by tech-nicians-animate or inanimate. fn &eseareas the basic and challenging pr*blemsof measurement have been solved, andtJre only task left is to impleme*t, read,and record. In other parts of the disci-pline the problem of hcw and what tomeasure remains acute and real, Thetask is not simply to read a meter orgauge aa effectl it is to devise a pro-cedure by which to quantify som€ stub-bcr,n phea*menon-to reduce it to nu-merical order.

Much that pertains to man himselfposes kn*tty problems of just tJris sort.How shall $/e measure his capacities, hisattitudes, his sensatiors, or afry of themany aspects of man that cannot beweighed in a balance or marked off on astick? fs measurement possible here, andif so, to what degree? But first of all,what precisely do we mean by measure-ment and what are the forms it maytake?

Mathematics versus Measurement

"Probab,ly more nonsenser" said N. R.Campbell, "is talked about measurementthan about any other part of physics"( / ) . Crotchety as this remark maysound, Campbell did not intend therebyto belittle the power and beauty of physi-cal measurernent or the superlative in-genuity of laboratory practice, But theart of measurement is one thing; the un-

Dr. Steveas is director of the Psychological Lab-oratories of Harvard University. This *rticle isbased orr a paper that he presented duriag the In-dianapolis meeting of tfie AAAS, 26-30 Dec. 1957.

S. S. Stevens

derstanding of its fundamentals is an-other. And Campbell-the author ofPhysics: the Elemenrs-was trying toteach us the same truth that Whiteheadhad in mind when he cbserved that o'it isharder to discover the elements than todevelop the science" {2} . This is the wayit has been with measurement. Ffere as

elsewhere it has often taken our greatestminds to discover the simplest things.

One of these things is the relation be-

fween measurement and mathematics. ftseems clear to us ns\Ar that the process ofmeasurement is the process of mappingempirical facts and relations into a for-mal model-a model borrowed frommathematics. But this conception tookform only in very recent times. It is theproduct of long centuries of intellectualstruggle, to which many of the foremostmathematicians contributed. It is a con-ception that was imp*ssible, evea un-thinkable, until the nature of mathemat-ics as a postulational system becameclarified.

Once a basic and elementary notiondawns upcn us in ix full clarity, we often$/onder how our fathers could havemissed perceiving it. It is a curious factthat, although the posrulational methodHras applied to geometry some two rnil-lenniums ago, only in modern times werethe fundameatal assumptions of algebraexhumed from the hadgepodge af rulesthat govern algebraic practice. In thisserse, the modern postulates of algebrarepresent the distilled wisdom of morethan 3000 years of symbol juggling. Theyrepresent the outcome of our efforts topare away the nonessentials in order to'get a clear view of what constitutes theessence of a mathematical system.

And the essence is this: mathematics

Measurement and Man

is a game of signs and rules, man-madeand arbitrary like the game of chess. Itbegins with a set of undefined terms arda set of unproved assumptions regardingtheir interrelaticns. The mathematicianinr,rnts symbols, and at the same time he

Iays down rules to tell us how these syrn-bols shall' be allowed to combine andinteract. Nowhere in this process, as rue

now conceive it, is there any reference.toempirical objects-or any explicit con-cern for the world of sense and matter.Therein lies the revolutionary novelty,far aot long B$o, as human history Ssffi,it was argued that negative numbers were"absurd" and "fictitious." For how coulda*ything be less than ncthing? You see,

our ancestors thought it proper to testtheir mathematics by operation$ per-formed upon nature-upon actual' ob,jects-for th*y conceived arithmetic as

& system of concrete numerical -magr-

tudes whose relations should be verifiablein the empirical domain, and where inthe real world were the regative cbibcts?

The story of the slow and pai*fulgrowth of the number syste'm, th* storyof how the mathematicians, often againsttheir own better judgment, began towrite outlaadish symbols, such as - 3 and

\,iT is a fascinating tale. It could oc-cupy us at length, but we must forego it.fts bearing on our present concern re-lates mainly to its outccme. Wiih eachnew kind cf' number admitted to thenumber domain-the negatives, the ir-rationals, the imaginaries, and so on-itbecame more clearly impassible ta provearithmetic by appeal to experiment. Soin the end the formal, syntactical systemof mathematics achieved its full emanci-pation, its cornplete dec*uplirs fr*m em-pirical matters of fact. Thence it tack offinto the realm of pure abstraction, whereit properly belonged in the first place.

Why did this decoupling take so long?Why so much travail to achieve some-thing so simple and obvious? The diffi-culty, it se€ms, was measurement. Inparticular, it was the fact that the earlymathematicians did not readily discernthe difference between measurernent andmathematics.: Man was usually more in-terested in empirical measurement thanin mathematics-as the scientist, no

doubt, still is-and it was the problem ofmeasurement that first gave rise to arith-metic. In the beginning mathematicsand measurement were so closely boundtogether that ro one seemed to suspect

that two quite difierent disciplines wereinvolved. The earliest scales of measure-ment were scales of numerosity-scalesfor the ccunting of pebbles or cattle orwarriors. fn some dirn era in the past,

somebody invented the system of natu-ral numbers precisely for the purpose ofrepresenting what he did with collectionsof objects. No doubt this forgotten genius

\ ras oblivious to the formal-empiricaldichotoffiy, which we now consider so

crucial. But that is beside the point.However he may have regarded it, thefact is that he built himself a formalmodel to stand for an aspect of theempirical world, much as an architectdraws a plan for a house. Kroneckeronce said, "God created the whole num-bers; all the others are the work of man."Passable theologT, perhalx, but surelybad history.

Since arithmetic l1ras invented formeasurement, it is not surprising thatthe isomorphic coffesl)ondence betureen

whole-number arithmetic and the empiri-cal numerosity of piles of pebbles is tightand complete. ft was, in fact, the verytightness of this isomorphism thatbliaded the ancients to the essential dif-ference between rnathematics and meas-

urement. But modern mathematics is nolonger constrained to serve only as a syn-

tax for quantitative discourse. Far fromlimiting itself to serving as a model fornumerosity, or even as a model for suchcontinuous dimensions as length, it has

become largely nonquantitative in soure

of its more abstract reaches. This out-come has suggested to Gtidel a startlingthought-namely, that is was purely anhistarical accident that mathematics de-veloped along quantitative lines (3). Inone $ense Gtidel is undoubtedly right,and his conjecture is a profound com-mertary on the nature of mathematics.But the story of measurement suggests

that this "accident" had about it a cer-tain inevitability. Striving somehow tocount his possessions, ancient man seems

destined in the nature of thirrg* to havehit upon &e concept of number and tohave made therein his first triumphantabstraction. Given the deeply humanneed to quantify, could mathematicsreally have begun elsewhere than inmeasurement?

It is not, however, only in history thatwe s€e the slow development of the for-mal-empirical dichotomy. An analogous

,

development takes place in the lives afall of us. Just as ontogeny to some extentrepeats phylogenl so in the life of each

maturing child the s*uggle of the ages is

reenacted in the child's attempt to graspthe abstraction of mathematics. Helearns his first arithmetic with the aidof fingers Gr buttons or beads, aad onlywith great labor does he finally, if ever,

achieve the reoriented view that mathe*matics is an abstract game having no*ecessary relation to solid objects. Eachof us has suffered through this process ofrevision" Even &ough you rnay haveshifted gears more smoothly than I, stillyou may well sympathize with my owndismay at my first eacounter with imagi-nary numbers.

:.i

The Nature oI a Scale

In its broadest sense, measurement isthe business of pinning nurnbers onthings. More specifically, it is the assigu-

ment of numbers to objects or events inaccordance with a rule of some sort. Thisprocess tums out to be a fruitful enter-prise oaly because some degree af iso-

morphism obtains between the empiricalrelations amcng the properties of *bjectsor events, on the one hand, and soure ofthe properties of the number system, oB

the other. Some of these properties, andtheir uses in .measurement, are these:

{i) Identity: numbers may serve as labelsto identify items or classes. (ii ) Order:numbers may serve to reflect the rankorder of items. (iii) fntervals: numbersmay se{ve to reflect differences amongitems. {i") Ratios: numbers m*y serve

to reflect ratios among items.These are war'rs in which we may depu-

tlr,e numbers to repr€sent one or anotheraspect a{ a state of affairs in nature. De-pending upon what kinds of empiricaloperations we are able to perform, oneor more of these aspects of the numbersystem may be used as a model to repre-sent the outcome. The empirical opera-tions are sornetimes a matter of choice;rrore often they are limited by o-ur ex-perimeatal ingenuitf. In any case, thenature of the operations determines thatthere may eveatuate one or another offour kinds of scales {4, 5}. These I havecalled "nominalr" t'ordinalr" t'intervalrt'

and "ratio." ?hey are listed and de-scribed in Table 1.

The key to the nature of these difrerentscales rests with the concept of invari-ance. How can we transform the num-bers on the scale with no loss of empiricalinformation? If all we can do about a

set of objects is identify or cla-ssify them,$/e have only a no.rninal scale, and thenumbers ure assign can be permuted atwill, for all that the numbers provide arelabels. If operations exist for determiningorder, and if we have assigned nurnbersto reflect this fact, then the perrnissiblescale transformation rnust be order-pre-serving. l{hen intervals have empiricalmeaning-as on the ordinary tempera-ture scale-we are limited to linear trans-fsrmations. We can multiply by a con-stant and add a constant. And finally, ifia additicn to all this we can give empiri-cal meaning to ratios, the only permissi-ble transformation is multiplicatio* by aconstant, as when \Me convert frorn feetto inches. Any more liberal transforrna-tion entails a loss of informatisn" In gen-eral, the richer the experimental opera-tions, the greater is the isomorphismbetween them and the formal model ofarithmetic, and the more restricted is theraage of invariant tra::sformations. [Fo:a possible fifth type af, scale having a

still different transformation group, see

(5) and {6).1Each of these scales has its uses, but it

is the more pcwerful' ratio scale thatserves us best in the search for nature'sregularities" On these ratio scales wemeasure basic things, like numerosity,length, and weight, and, depending onour artistry, w€ contrive more elusivemeasures, like the charge on tlte electronor the strength af a magnetic field.

Why, it may be asked, do we botherwith the other types cf scales? Mmtlylve use the weaker forms of measurement'only faute de mteux. When strongerforms are discovered we are quick toseize thern. But science is an art. Thereare no ab initio princfples to tell us howto be clever in devising procedures ofmeasurement. The way to empirical dis-covery lies not through mathematics,even, but through the exercise of uncom-mon experimental sense and ingenuity.We invent mathematical models, but wediscover measures in the labaratory. AsNorbert Wiener {7 } said, "Things donot, in general, run around with theirmeasures stamped on thern like the ca-pacity of. a freight car; it requires a cer-tain amcunt of investigation t* discoverwhat their measures are."

Perhaps those who stand apart fromthe practice of the scientific art, andwho philosophize about the "scientificmethod," think there really is such a

thing, and that it can be captured in abook of rules. But the man or the labo-ratory stool is likety to agree with Hilde-brand that "there is no such thing as the

slr-}

*cientific rnethod" i8i " If you think sci-ence is a simple and unitary thi*g, tryasking several scientists to define it. Oaeof the entertaining thiugs abeut scieace

is that no one has succeeded ir: explain-ing precisely what it is.

Hawever you de*ne the scielrtific ac-tiviry, measurement pervades most o{ theenterprise . Measur*m*nt is essential tcthe deterntination el fuactional relations,to the disc*very of order alrd regularity.I aeed nct ext*l it further, for r,tre allknow ths reality of its power. fn fac! we

take it sa raxch for graated that it be-

comes alaast unthinkable that the pur-suit of mea$rreme:rt did nct always sta*din high regard.

I vividly recall Professor l,Yhitehead,peering over his lectem i& Ilarvard'sEmerson Hall and rasping aut r*risdcm inhis high-pitched voice: o'X only the

scho$knen ef the Middle Ages had meas-

ured in*tead af classifying, haw muchthey might have learned." tJnder the in-fluence of Aristotelian logic, with its eat-phasis o:t classi$catian, the schaolmenforsaok the Pythagorean tradition, whichtaught the primacy af number axd rfieas-

urernent. Classificati*a, to be sutre, is *first and essential step srl the road upthe hi*rarchy af scales. ft gets us to the

nosrinal level. Bat this is ao more thana qrrart€r-way house on the road tc meas-

urerrlerlt in its rnore por,verful forrns. Therevival of modern scieace i$ the 17th

ce*rury-the cffItury *t genius-was a

revival of the Pythagarean outlsokl * re-

vival of measurement. lSith Galileo,Newto*, and the rest, scienee becameprirnarily quantitative, 4nd sc it has re-urai:red.

In his diagaostic satire entitled S ci-#nG€ is a. Sacred Cow, Stande* perceivedcorrectly the rnadern order of thingswhea he put measuremer:t at tlre tap ofthe scientist's tstern pole lsee {9}1.

Measuremcnt in Psychcphysics

Ideasurement, as ure have seerr, is marethan the pedaatic pursuit of a decimalplace. Its vital and absorbing aspect

emerges mcst clearly perhaps urhea itbecomes a question cf measuring some-

thing thht has never beerr measured. Orbetter still, sornetlring that has been heldto be uala€asurable. Quantification is arespe*table enterprise in physics andchemistry, and eve$ in much of bialogy.But what about man, and the meas*re-ment of his higher Frocesses? Are we al-\irays abjective and emotionally neutralabout this prospect?

The economist Bdgewcrth {I0} once

wr*te, "There is an cld prejudice stillreviving, horlrever often slain, against thereign of law ir psychology, as incampati-ble with the higher feelings." Sorne thereare, f suppose, who still feel that quanti-

ficatio*, by some brutal rigor, r,vill s&atterthe human sprrit if we probe with the aidaf numbers. But man carl hardly fall iastature by understanding maa, sr everl

by quaatifying that understandi*g Thegreater beauty of discovered srder rsillsurely rrrore than comFensate for thenastalgic pain of a rcrnantic yearning toremain securely inscrutable.

However we regard this issue, the factrernains that ma$ is underg+ing rneas-

uremerrt. lrYe are all familiar with tlehighly developed busiress sf testir:g hu-rrlan perforrnance and ability, and withthe pi*aeering arork of Bi*et, whslaunched us an the raad ta the rleasure-ment of the IQ. This rneasure, d* ixapproximate invariance over the child'sgrowing years, stands as one sf the first-r:ate contributions to hurnan under*taad-iog. Interesti*g issues fcr the the*ry ofmeasurement arise almast daily in these

burgeanir:g fields of ability astessmeqt.

But since this is r:ot my olrrr area of i::-terest, let me turn to anotJrer quest: themeasurement of sensation.

Mcdern experimental pqychology hadits begir:nings in this inquiry, whichstarted just about a hundred years ag**in the 1850ns.

Let me pcse the problern in this $ray.

Suppose you loak at a photograph in thebright sunlight and then a#* in a dimlylighted room. The remarkable fact is

that t}le picture lcoks rnuch the sarn€

Table l. A clasgiEcation of scales of measurement. Measuremeat is the assigrrment of nuoberr to objeets or eveilts according to rule-The rules and the resulting kinds of scales are tabulated below. The basic operations leeded to create a givea scale are all tboscIisted in t&e secoad column, donrn to and including the operatiou listed opposite the scale. The third caluen gives tbe mathematicaltraasformations that leave the scale form invariant. Aay numbet x Qn a scale can be repl,aced by another number.t' where r/ & thefuactior of * Iisted in columa 2. The fourth column lists, curulatively dorrnward, example of statistica that show invarlance uoderthe trarsformations of column 3 (the mode, however, is invariant only for discrete variables).

Permi*sible statistics{invariantive}

Typical *xamplesScaleBasic empirical

operationsMathem*tical

group-structure

Nominal

Ordir:al

Interval

Determiaationof equality

Deterrainatienof greater orless

Determinationof the equdityof intervals orof differences

Determinationof the equalityof ratios

Permutation group{ =f t*}

where f{r} meansany one-to-onesubstitution

Isotonic SroupN, =f {r}

where f {*} mean$a*y increasingmcnotcnic function

Linear or affine groupj/= ax*b*7I

Similariry groupxn=GxclS

Number of eanies

Madeo'f nforrrration" mea$uresC ontin gency correlation

MedianPercentilesOrder carrelatioa {t}pe 0

interpreted as a testof order)

Mean$tandard deviatiaaOrder correlation ( type I

iaterpreted as r)Praduct mome*t {r}Geometric $rearrHarmonic raeanPercent variation

*'Numbrring" of football playersAssignment of type or model numhers to

classes

Hardness of rnireralsSrades of leather, lumber, woo},

and so forthIntelligence-test raw scores

Temperature { Fahre*heit and Celsiuc}Position on a }ineCalendar timePotential energyIntelligence-test "standard score*'n { ? }

Length, nurnerosity, density, rrcrk, timeintervals, and so forth

Temperature {Ketvin}Loudness { sones }Brightness {brils}

Ratin

under the two coaditions. Despite achange of illumination of perhaps several

thousaad-fold, the light parts of the pic'ture lo*k light and the dark parts dark.The perceived relation between light andshade within the picture remains highlystable, is subjectively constant. But justwhat is it that is subjectively constant,rrre may ask. There are at least two possi-

bilities. One is that the subjective difer-eftce between the light and shade remainsconstant as we go from outdoors to itr-doors" The other is that the subjectiveratio between the light and shade r€-rnains constanrt. If we could find outwhich of these relations holds, then wewould know, for these conditions, thelaw that relates subjective brightness tathe physical intensity of the stimulus.

Back in the 1850's two major figuresin science, Fechner and Plateau, bothconsidered the problem and reachedquite opposite conclusions ( a fact thatsuggests that you cannot settle the mat-ter ,merely by looking at pictures!) .

Fechner argued that the subjective dil-ference between light and shade rernainsconstant, and that therefore the subjec-tive brightness is a 'logarithrnic functionof stimulus intensity. That is the well-known Fechner's law. Plateau arguedthat thb ratio remains cbnstant, and thattherefore the subjeictive brightness is apower fuirction of stimulus intensity.

' Formula-wise we may state these twolalvs as the relation between psychologi-cal value {, and physical value 4 in thiswayi

Power larr: tI= &eO"

The exponent z is a constant whosevalue $Ia), vary with sense modality andurith conditions of stimulation.

Of course, the charnpions of these lawscited other facts and evidence, and for ahundred years tlis issue has stoad as akind of antinomy in psychophysics. Ifyou have heard only of Fechner in thisco#rectioa, it is because it was he whodefended his view inore fiercely, whomore tirelessly outargued his critics.Plateau's interest was only casual, and,as a matter of fact, h* later changed hismind-and for a reason that was notreally relevant {see 6}. So the field rryas

Ieft mainly to Fechner. But others re-vived the power law from time to time,and the contradiction persisted.

How can this conflict of opposing laws

-t}e logarithmic and the power law-be resolved? By measurement, of course.AII that is needed is a scale for the meas-urement of sensation. But that is

'easier

said than done.

4

The Operational Principle

At this point, let me try to clarify a

sricky issue. This questicn of sensatianand its measurement has often gottenitself bogged down in metaphysical de-bate. Ever since Descartes set mind apartfrorn matter, we have been trying in oneway or another ta put them back to-gether agaiq for if we accept the dual-istic view that mind is something apart,something inaccessible to science andmeasurement, the game is lost before thefirst move is made. To rescue sciencefrom tJris hopeless gambit, three moderrrdevelopments have converged on a com-mon solution. The three are behaviorismin psychologf, operationism in physics,

and logical positivism in philosophy

U I). Despite certaia differences in lan-guage and emphasis, all three of thesemovements have sought to clarify ourscientific discourse by ridding its con-cepts of metaphysical overtones and us-testable rneanings. IJnder the operationalview, Iength is what we measure withrods; time is what we measure withclocks. However well grounded in com-mon sense may seem the notions of Ab-solute Space and Absolute Tirae, thephysicist, as physicist, can know nothingabout them-for he can do nothingabout them.

Equally inaccessible are the nonopera-tional aspecti of sensation. What we canget at in the study of living thi*gs arethe responses of organisms, not somehyperphysical mental stuff, which, bydefinition, eludes objective test. Conse-

quently, verifiable statements about sen-

sation become statements about re-sponses-about differential reactions oforganisms. In psychology, perhaps even

more than in physics, this operationalstance is indispensable to scientific sense

and meaning. fn. line with this necessity,let us agree that the term seflsatistt. de-notes a construct that derives its mean-ing from the reactions, verbal or other-wise, made by an organism in response

tc stimuli. I know nothing about yoursensations except what your behaviortells me. But what is equally true, weknow nothing

'tbout the charge on the

electron except for what its behaviordiscloses. We must be thoroughly opera-tional in both instances.

Now, some will object that there is adifference here: that electrons do notstudy themselves, whereas men do. Thisis true enough. But if the scieace of manis to contain public, repeatable, verifiablegeneralizations, we must always in effectstudy the other fellow-we must pursue

"the psycholosy qf the other one.o' The

psychologrst as experime*ter may lookin upon himself if he cares to, and hemay often thereby gain insight into fruit-ful hypofheses, But these hypotheses carr

lead to valid general laws only after th*yhave been verified under experimentalcontrol on other people. If the experi-menter serves as an observer in his ownexperiment, as I often do, he must pro-ceed to treat his own respomes as objec-tive data, on a par with those cf otherobservers. This manner of working, itseerx tc me, is the anly sound, objective,operational approach" fn what follows,therefare, I hope it will be taken forgranted that I mean no more by seusa-

tion &an what experiment tells us. Ourgoal is to make quantitative order of thereactions of sensory systems to the ener-getic configurations of the environment.

;.]Configtiag Laun

Let us return row to our problem.Fechner, as I have said, uron the firstround, and for almost a century it lookedas though the logarithmic law wouldprevail over the power law" Two rathercoavinciag kinds of evidence seemed tofavor it, First, there was the argumentbased on differential sensitivity, which$re measure by noting how large an in-crement must be added to a stimulus inorder for a person to detect the differ-errce a certain percentage of the time.These just noticeable differences tumout to be roughly propartional to t}lemagnitude of the original stimulus(I{eber's law}. There is a kind ef re}a-tivity here, You can detect a candleadded to a candle, but not a candleadded to the light of the noonday surr.

Fechner noted this principle and thenproceeded to postulate that each jwtnoticeable difference corresponds to a

canstant increment in sensation.At this point we are reminded of what

Bertrand Russell said in aaether connec-tion about postulation: "The method of'postulatiog' w,hat we \#ant has many ad.vantages; they are the same as the ad-vantages of theft over honest toil" U2).

Be that as it fr?y, if we grant Fech-rer's postulate, and if Weber's law is

true, it follows that sensation grosrs as

the logarithm of the stimulus.The other line of evidence is exempli-

fied in the astronomer's scale o{ stellarmagnitude, which appears to date fromHipparchus (about 150 g.c,). Before thedays of photometry, men looked at thestars and judged their apparent bright-ness on a scale from 1 to 6, where Istands for the brightest stars and 6 for

i]

the faintest. Successive numbers on thescale rArere assigued to successive equal-appearing intervals of stellar magnitude.Then arl interesting thits happened.Men finally learned to measure thebrightness of the stars by photometricmethods, and, much to Fechner's de-light, it turned out that the magnitudesassigned by the simple process of lookingand judging u/ere spaced by approxi-mately equal steps on a logarithmicscale of photometric value, In keepingwith this fact, the step on the modernscale of stellar magnitude has now been

standardized at 4 decibels {0,4 log unit}(J3). flActually, the early hstronomers'scales differed among themselves, andmost of thenr were slightly, but systemati-cally, differeut from the logarithmicscale U4).1

So here we have two classes of sensorymeasures lending some degree of cr€-dence to the logarithmic law: the resultsof measuring differential sensitivity andt}re results of partitioning a sensory cou-tinuum into equal-appearing intervals.

Then what about Plateau's view? Isthere any experimental evidence thatsupports the power !aw? Actually, Pla-teau appears to have been tJre first ex-perimenter to bring the partitiooi*gmethod out of the heavens and into theIaboratory; or, more preciseln into thestudio, for he asked eight artists to painta gray that wauld appear halfway be-tween extreme black and white. Theeight grays, independently produced,turned out to be " presque identiques."Furthermore, the goodness of the parti-tion into equal interyals-black to grayto white*appeared to rernain stableunder different degrees of illuminatioa.Starting from this latter fact, Plateauconjectured his power law.

Unfortunateln for reasons we,wilt con-sider shortln the method of partiticningis not capable of verifying the pollier la$r.lt was because Plateau did not knorrthis fact that he later felt obliged tochange his mind about the law. Actuallphowever, he never should have changedit, for he was right in his basic conjec-ture. The correct law is the polrer law.

Ratio Scale of Sensory Magnrtude

fn our struggle to discover the meas-ures of things, w€ do not always hit uponthe simplest and easiest procedure. firstoff.tFechner's method of constnrcting ascale by the tedious process of measuriugjust noticeable differences and countingthem off was involved and indirect-andeven included one of Russell's larcenous

50

80 90LUMINANCE IN OECIBELS RE IO-O LAMBEHT

Fis. 1. Direct magnitude estimations ofthe apparent brightness of a small targetsubtending an angle of about 1.5 minutesof asc, The obser"rer uras first shown aluminance of 92 decibels and told to callit "10." Relative to this modulus he thenestimated the other brightnesses, whichwere presented twice each in irregularorder, Points are medians for 15 ob-serv€rs. The straight line in this log-Iogplot determines a power function with anexponent of A,47.

postulates in the bargain. Plateau'smethod was more direct, certainly, but itaimed, at best, only at the construction ofan interval scale-one on which the zeropoint would be arbitrary and on whichratios could have no meauing.

Clearln if a ratio scale was to be

achieved, judgments of subjective ratioswould have to be made. In the early1930's tJre first serious efforts to get peo-ple to respond to ratios of sensory mag-nitude finally got under we1t, and overthe past few years a swelling tide of ratioscaling procedures has given this wholesubject an exciting new look. It turnsout that the ordinary thoughtful observerc&n make quantitative estimates of sen-

sory events. He can adjust a light so thatit appears half as bright as another, or afifth as bright, or a teath as bright. llecan also set it ta a given multiple of theapparent brightness of a standard light.Furthermore, given some standardbrightness, to which is assigued aa arbi-

::il,:*:-:#"tr"13;*."-?::*-*nesses proportioaal to their apparentlevel, as he sees them. These and severalothers are the procedures used.

Oa 17 different perceptual continuathe application of these methods has re-sulted in power functioas. To a fair ap-proximation, estirnated zubjective magni-tude is proportional to the stimulusmagnitude raised to a po$rer. The expo-nents, experimentally determined, have

ranged from about 0.3 for loudness to3.5 for the subjective intensity af electricshock applied to the fingers. The funda-mental psychophysical law that emerges

from these findings is simply &is: equalstimulus ratios produce equal subjectiveratios. That is all there is to it. The pro-portionality between stimulus ratios andsubjective ratios is a pervasive first-orderrelation, observed in empirical studies on

numerous perceptual continua. Second-order departures from this law are sureto exist (** already know about some ofthem), but the wide invariance of thefirst-order relation is a matter of primeimportance.

I was particularly interested to see

what form the ratio scale of subjectivemagnitude would take for small luminoustargets resembling a star, for the astron-omers' estimates of stellar magnitudesgave us the first psychological scale,

though it was not a ratio scale. Fifteensubjects were asked to assign numbersproportional to the apparent brightnessof a small spot of light resembling a star,whose intensity was varied over a rangeof 30 decibels (I5). The median esti-mates gave a close approximation to a

power function with an exponent of 0.47.Thus, the apparent subjective magnitudeof the "star" grows approximately as thesquare root of the photometric level (see

Fig. 1). (The exponent here is greaterthan that for larger lurninous targets,where the exponent is close to one-third. )

Now the question arises, why did theearly astronomers' scale approximate a

Iogarithmic function, whereas direct esti-mations of apparent brightness give a

po\nier function? This stubbcrn question,which has long been a ptruzle, actuallyturns out to have a very simple anslyer.It hinges oa the fact that a person's sen-sitivity to differences ( measured in sub-jective units) is not uniform over thescale-a fact related to Weber's law. Agiven difference that is large and ob-vious in the lower part of the range ismuch less impressive in the upper partof tle scale. This asymmetry in the ob-seler's sensitivity to differences producesa systematic bias whenever he tries topartition a cCIntinuum into equal-appear-ing intervals, On all continua of tJre class

I have called "prothetic" {6}, of whichbrightness is one, we observe that thescale constructed by partitioniag intocategories is a convex function of theratio scale obtained by direct estimation

-that is, the category scale plottedagainst the ratio scale grves a curye thati$ concave downward {see the uppercurve in Fig. 21 .

azoEzo=;alrI lotrJ(}tGs=

2

BRIGHTNESS OF "STAR"

6

The systematic bias that warps ourjudgments whenever u/e try to divide a

segment of a prothetic continuum intoequal-appearing intervals was presumablyoperating, of course, when the early as-

tronomers arranged their scale of stellargragmtudes. The bias was apparentlystrong enough to make this scale approxi-mate ? logarithmic function of photo-metric intensity. But this roughly logar-ithmic outcome really helps Fechner'sargument not at all, for when we lookmore carefully at the processes involved,we find that the form of the scale of stel-lar magnitudes is merely another ex-ample of the fact that man exhibits abuilt-in bias whenever he tries to parti-tion a segment of a prothetic continuum.It is too bad that Plateau, when con-fronted with the results of another ex-periment on partitioning (conducted byDelboe,rf ), let himself be persuaded torenounce the porrer law.

Our confidence in the view that somekinds of partitioning are subject to biasgathers strength from the findiag thatnot all partitioning is distorted in thismanner. On another class of continua,

called "metatheticr" where sensitiviqy isnot asymmetrical, the process *f parti-tioning may produce an unbiased, linearscale tI1). Pitch is arl example of ametathetic continuurn, whereas loudnessis prothetic. With loudness, the physio-logical process underlying our discrimina-tions seems to involve the addition of ex-citation to excitation. W'ith pitgh, theprocess is believed to be the substitutionof excitation for excitation, a change inthe locus of the excitation. It is indeedinteresting that the difrerence betweenthese two basic classes of physiologicalmechanisms reflects itself in the behaviorof the psychotogical scales which we con-struct from the sensory responses in-volved.

Tke Ear as a Connpressor

Since scales of measurement bear littlefruit if thuy do not serve to predict orexplain anything it is fair to ask whatother insights into natural phenomenamay stem from this boom in sensory

measurement. I do not pretend to know

where it all will lead, but I would liketo cite sne final example of its bearing onan interesting question.

One of the amazing properties of a

sensory system like hearing is the almostincredible dynamic range of its oFera-tion. Energy ranges of billions to oae aretaken easily in stride U6). In order toencompass such dynamic ranges, h orderto detect sound vibrations whose ampli-tudes are less than the diameter of a

hydrogen molecule and, at the sarne

time, respond adequately to a thunderousraat, the senscry system must behave insome sense as a "compressor." The inter-esting question is, where does the com-pression take place-in the end orgar orin the central nervous system?

First, it is to be noted that the de-gree of the compression we are con-cemed with is given by the expCInent ofthe polver function relating loudness tosound intensity (16). This exponent ofabout 0.3 tells us that in order to doublethe apparent loudness we must multiplythe energy by a factor of abcut ten (otthe sound pressure by the square root often) . Contrast this relatioa with thegrawth of the subjective intensity ofelectric shock, which shoots up as the3.5 polver of the current applied to thefingers U7). Here, when we double thecurrent, the typical obsenrer judges theshock to be some nine or ten times as

great as it was previously. There is no

compression under this direct electricalstimulation. On the cs.ntrary, the system

behaves as though it contained an "ex-pander" of some sort. Through the directmeasurement of sensory magaitudes, a

striking difference is revealed betweenthe behaviors of twa sensory mechan-isms,

Now the question rs, what *vould h*p-pen if we Hrere to stimulate the auditorynerve directly with an electric current?Some of us once explored this problemin a group of clinical patients whose mid-dle ears had been opened, for one reasonor another, so that an electrode could beplaced inside the open cavity (r8). Since

other nerves, such as the facial and thevestibular, were readily stimulated underthese circumstancesr w€ had reason tobelieve that electrical stimulation also

reached the auditory nerve, as indeed itmust have done in those ears that heardonly a noise whose character bore nosystematic relation to the frequency ofthe stimulating current. A random, un-pattemed excitation of the auditorynerve fibers would be expected to resultfrom a current applied to the middle ear,

and an unpatterned excitation of fibers

7

lrjJ(}{,

E,o(,UT

k(}

6

5

4

3

TRIANGLES-> o toSUBJECTIVE

30 40 50 60BRIGHTNESS (RELATIVE BRILS)

cncus+ 8O 90 IOO llO

LUMTNANcE tN DEcIBELS RE to-lo LAMBERTFig. 2. Judgments of brightness on a category scale from I to 7. A lu-inance of 80decibels was presented and called "1r" and one of 110 decibels was presented and called"7." The ob,server then judged thc various levels twice each in irre€ular order. Pointsale averages for 15 observers. The rcsults are plotted against two different abscissa scales.The triangles are plotted against t}e magnitude scale obtained from the liae in Fig. l.The circles are plotted against tle luminaace scale in decibels. Note that tle triaaglerdetermine a curve that is concave downward. lbe lower curve (circles) suggestr thatpartitioning iato a finite number of categories produces a function that is roughly loga-rithmic, but aot precisely so.

2

BRIGHTNESS OF .,STAR,'

should lead ts the perceptioa cf noiserather than tore.

The interesti*g thing frarn cur pre$entpaint of view, was the rapid grswth ofthe loudness of the noise as the currert$ras i*creased. The patient was asked tocsmpare the noise with a saurrd prCI-

duced by aa acousfic stimulus led to hisnormal, unoperated ear. He adjusted theI**drless in his normal ear ts match theloud*ess af the noise in the aperated ear.This simptre proeedure disclcsed a star-tlirg fact. The growth of lsudness wasrrrany times steeper under electrical t}anunder acoustical stirnulation. The exps-nent of the polrrer fur:ctioa under electri-cal stimulation trras! in fact, $f about thesame order of magnitude as that ob-served when a S$-eycle current was ap-plied to the fingers.

Many interesting questions are raisedby these xreasurements, but sae implica-tion is clear. The "compr€ssiolr" *bseryedin the aormal response of the auditorysystem to a saund stimulus is appareutlynot an affair of the central neryous sys-

temo for if we bypass the ear aad stimu-late tlre auditory ilerve directly, r,ne de-tect $c c$mpression. Rather, fhere re-sults an "expan:sion" in the subjectiverespcnse. Apparently, therefcre, it is tathe end organ itself that we rsust laok

?

far the rnechaaism af compressian thatgsveil$ the strow growth of louduess withaccsstic intensity,

So it appear$ that, with the aid ofscales constructed for the measurernentof sensation, w€ may have disclosed a.

fundarnental di$erence betureea twctransducer mecharrisms. The transduc.tion cf souad er:ergy into $ervous energyis by nray af an "operating characteristic"that somehaw coffipresse$ the aver*allsenssry r€str)onse. The trarsduction ofelectrical energy into nsrrlous energyseems to f*llaw q*ite a different rule. Tobe sure, this outcome is but a trifle in thevast ard relentless co:rtest to unwind thetangle of *ature, but it testifies, in sim-ple example, to the prcfit that may ac-crue from measuring the "$nmeasur-able" {J9}.

Reftrerce* asd Nots

1. N, R. Campbetrl in Aristcteliar Saei*ty, 'Sym-pasia;ttz: .*Isa*r,re*te*l e*d lts Imporiaxca lorP*ilosopky iHarrisar, London, 1938), vol. 17,5uppl.

7. A. N. l{hitshead, Sciexee end the klodardW*rId. {Macmil}aa, l$ew Yark, 1925}, rsprinted by Pelican Meator tNew YorL, t34S).

3. R, Oppeaheirrrsl, *'Aaalogy in scie*ce r" l&.Psyeholog*l ll, 1?7 { 195$i,

4. S, S. Stevers, "Or tLe theory of scales ofEleasursme&tr" Scie*ce tr83, 677 1t9{6};Mathematics, *easurement a*d prychaphys-icq" irr l{a*dbaok o{ &xperimcntal Psyckal-o€1., S. S. Steveas, Ed. {l{iley, New Yo"k,1951), FF. 1-49,

5. S. S. Stevens, "Mea$rrement, psychophysics

and utility," in Syrxp*sittrnt a*, Me*sxre*tat:I{etd by iha AAAS, Deca*ber 1956 {l{ilcy,Nerr York, in press).

6. S, S. Stevene, o'0tr the psychophysical larr,"Psychal. Rcu, #, 153 {1957}"

7 , I{. I{ieaer, "A new f}eory of re*asrrramett:a st*dy in the logic sf mat&ematis," Prcc^Londo* fr{ath..foc. 19, lSI {1920}.

8. J. I{. Hildebrand, Scieace in the Ma*ir,g(Calunbia {Iaiv. Prcss, Ncw Y*rk, 1957} "9. f'. S. Silsbee, "Measure for measnre: sarncproblerns and parad*xes sf precisioart, I.W*sh- Aoad.,fci. ,Ser. 2 41, tl$ i1951).

tS. f'. Y. Edgeworth, Matl*ematical ?sychits{Kegan Paul, London, I8S1},

11. S" S. $tevens, "Psychalagy ard the scicace olscie*ee," Psyeftol. Bult, 3S, ?21 {1933}, rc-prirted in Psychologirul Tke*ry, C*xte*So-rary Readings, M. H. Marx, Ed (Macmillan,New York, 195llr pp- ?l*54; P. P" I{iener,Ed-, llecdi*gs i* Philosa*hy of Sair'*ce {$crib-iler's, Nerar Yord 1953), pp. 158*1S4.

l?. 3. Russelt l*trod,*etion ts Mat*eaatia*lPkil*sophy {Macmillar, New Yark, ed. 2,1920) , ?. 17.

f 3. ?. Maan, "Photomstrics i& astrcaaayr', I.Fronkli* Inli. ?58, 4SI i1$54); S- $. Stevers,

B, Nc. 18, lZ {IS55}.14. J" ]astra*,, "The pycha*physie law aad sbr

magnittrdesr" A*?. t" ?sychaX, 1, 1I2 tlSST).15. S. $. Stevens and E. II. Galan*er, "Ratio

scales aad stegory scales for a dozen perc€p-tual co&ti*uar" l. ExSfl. ?sy*kol. 3*, 377{IS57}. The experiments ou the apparentbrightaess of the "star" lrere coaducted byJ. C. Steveff.

16. $. S- $teve*s, "The measurem*nt ol loud-rer$," t. Acausi. Sos. Arn.ZTr 915 {1955}.

17. S. $. $tevens, A. S. Cartoa, G. *{. Shickman,"A smle of apparent intensity of. electricshocd" t. Exptt. Psyehol., in press.

I8. R" C. ]o*es, S. S, Stevens, M. H. Lurie,"Three mechanisms o{ hearirg by electricalstimulationr" t. Acoust. Soo. An.. 12, 281{ 134S} .

19, This research r,cas aided by the {}&cs of ffavalResearch *ad the Natien*l Scieace Foa*da*tioa {reynt PNR-210}"

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