Journal of Experimental Psychology1971, Vol. 90, No. 1, 75-80

PROCESSING NUMERICAL INFORMATION;

A CHOICE TIME ANALYSIS 1

ROBERT SEKULER,2 ELLIOT RUBIN,3 AND ROBERT ARMSTRONGNorthwestern University

The time needed for human adults to identify the numerically larger of twodigits was studied. In Exp. I, the digits to be compared were presented in suc-cession at two different exposure durations. In Exp. II, a single visuallypresented digit was compared with another digit held in memory. In bothexperiments, the larger the numerical difference, the faster the identification.

The time needed to identify the numeri-cally larger of two digits depends upon theirnumerical difference (Moyer & Landauer,1967). In general, the larger the numericaldifference, the shorter the time requiredto make the discrimination. A variety ofreasons have compelled us to replicate theMoyer and Landauer finding. First, theresult is of considerable theoretical im-portance for what it may imply about thegeneral storage and use of numerical in-formation. Second, we were surprised thatcollege students, having had 15 or moreyears of experience with such decisions,would still take longer to decide that "5"is numerically larger than "4" than theydo to decide, for example, that "7" isnumerically larger than "4." Third, theerror rate in the Moyer and Landauer ex-periment covaried strongly with judgmenttime. Using their published graphs, wehave calculated a Spearman rank-order cor-relation of .80 between error rate andjudgment time. As a number of peoplehave suggested, such a strong relationshipbetween error rate and judgment timeraises some obstacles to the unambiguousinterpretation of judgment times (Egeth &Smith, 1967; Smith, 1968). Consequentlywe have tried to devise a situation wherethe error rates might not be so highly cor-

1 Computing time and equipment were made possi-ble by Grant EY-00321 from the National Institutesof Health.

2 Requests for reprints should be sent to RobertSekuler, Department of Psychology, NorthwesternUniversity, Evanston, Illinois 60201.

3 Now at the Department of Psychology, Univer-sity of Illinois, Chicago Circle Campus, Chicago,Illinois.

related with the expected judgment times.Fourth, we wanted to record judgmenttimes in an experimental situation whichcontrolled the viewing time associated witheach member in a pair of digits. Thiscontrol would permit an estimate of theprocessing time for a single digit. To meetthese requirements we used successiverather than the simultaneous presentationof each pair of digits used by Moyer andLandauer. We expected that successive,controlled exposures might permit theclearer analysis of the various componentsof overall processing time involved in theultimate, binary judgment of numericalinequality.

EXPERIMENT IMethod

Apparatus.—The single digits 1-9 were presentedin pairs using a single plane display (IEE Series10 rear projection display with #1813 lamps in-stalled). The digits, viewed binocularly, werepresented in a viewing box at a distance of 60 cm.from S's eyes. Each digit subtended 4°40' visualangle in the vertical by a maximum of 2°20' onthe horizontal. The luminance of the digits, pre-sented against a constant, dark background, wasabout 3.4 cd/m2.

The entire apparatus was under the control of ahigh-speed incremental paper tape reader and as-sociated programming system. On each trial thefollowing sequence of events took place. The firstdigit was illuminated for a predetermined interval.At the end of that interval, the digit was extin-guished and there followed immediately the illumina-tion of a second, different digit which appeared inthe same spatial location as the first. Simultaneouswith the presentation of the second digit, an elec-tronic buffer began to accumulate a count from acrystal clock at the rate of 1,000 pulses per second.When 5 had decided which digit, the first or second,was numerically larger, he threw a toggle switch in

75

76 R. SEKULER, E. RUBIN, AND R. ARMSTRONG

1000"

4OO-

1 2 3 4 5 6 7 8NUMERICAL DIFFERENCE

FIG. 1. Mean choice time for identifying thenumerically larger of two digits as a function oftheir numerical difference. (Starting from the top,curves are shown for Exp. I Cond. FDS and FDL,the Moyer and Landauer (1967) experiment(M & L), and Exp. II. The bottom curves showthe error rates for the two conditions of Exp. I.)

the assigned direction to signal that judgment. Theswitch throw had three effects: the second digit wasextinguished, the accumulation of a count in thebuffer stopped, and the contents of the buffer wereread out into a teleprinter. Following a constantintertrial interval of 4.5 sec., the entire sequencewas repeated with another pair of digits.

This sequence was repeated 432 times for each S.The absolute value of the differences between thefirst and second digits ranged from 1 to 8. Allof these absolute values of the difference betweendigits occurred with equal frequency during thecourse of 432 trials in a block-randomized order.For any of the absolute values of the difference be-tween digits, on half of the trials the larger digit ap-peared first. With one exception, any givennumerical difference between the two digits couldbe constituted by several different pairs of particulardigits. The difference 1, for example, could begenerated by the presentation of many differentpairs: 4 and 3, S and 4, 9 and 8, etc. A differenceof 8 could be generated only by the digits 1 and9. The stimulus sequence was constructed sothat each of the possible pairs of digits which con-stituted a given numerical difference occurredwith approximately equal frequency. This con-straint, taken with the constraint that all numeri-cal differences occurred with equal frequency, meantthat particular pairs of digits did not occur with

equal frequency. For example, 1 and 9, the onlypair constituting a difference of 8, occurred manytimes more often than did 9 and 8, which wasonly one of the many pairs that constituted adifference of 1. All 5s were tested with the samesequence of digits.

Subjects.—The 5s were nine volunteer introduc-tory psychology students and one of the authors(RS). Each of the 5s was assigned, in alternation,to one of two conditions. In Cond. First DigitLong (FDL) the first digit in each pair was pre-sented for 2.0 sec.4 In Cond. First Digit Short(FDS) the first digit was presented for only SOmsec. In each group, three of the five 5s indicatedthat the second digit was numerically larger by aswitch-throw to the right, that the first was largerwith a throw to the left. The other two 5s in eachgroup used the opposite S-R mapping. Since thepresent data showed no effect of direction of switch-throw, in all analyses the two assignments of re-sponse to direction of switch-throw have beencombined.

Each 5 was told to make and signal his judgmentsas quickly as possible while making sure that hemade as few errors as possible. Every 5 served inone session which lasted about 35 min. for membersof the FDS group and 45-50 min. for members of theFDL group.

ResultsFigure 1 shows mean time required for

the correct identification of the larger oftwo digits plotted as a function of thenumerical difference between the digits.Data from the Moyer and Landauer(1967) experiment (M & L) are alsoplotted to facilitate comparisons. Notethat for both FDS and FDL conditions,judgment time decreased systematically asthe difference between the first and seconddigit increased. An analysis of varianceindicated that the absoulte value of the dif-ference between digits in a pair was asignificant source of variance, F (7, 56)= 14.82, p < .01. The analysis alsoshowed First Digit Duration X NumericalDifference between Digits interaction,F (7, 56) = 2.96, p < .05. This interaction

4 These intervals are defined by the duration forwhich the reed-relay controlling the display wasclosed, and do not take into account the delay addedby the onset times for the display lamps. We haveestimated this delay by determining for how longthe reed-relay controlling power to the display lampshad to be closed in order for an observer to cor-rectly recognize what digit was presented. Aduration of approximately 30 msec, was neededfor recognition.

PROCESSING NUMERICAL INFORMATION 77

arose from the difference between the twodigit duration groups at the extremes of thenumerical difference continuum. Figure 1shows that over the range of numericaldifferences from 2-7, there was an approxi-mately constant 150-msec. difference be-tween the FDS and FDL conditions. Theanalysis of variance also indicated nosignificant effect (p = .50) of the sign of thedifference between first and second digits;i.e., as far as judgment times are concerned,it does not matter whether the first digitis larger or smaller than the second as longas the absolute value of the numericaldifference is the same. The error rateshown at the bottom of Fig. 1 was lessthan 2.2% in all cases and was not sys-tematically correlated with judgment times.

To make a more detailed analysis pos-sible, we can consider only those trials onwhich the numerical difference betweenfirst and second digits was 1. The onlysignificant term in our analysis of varianceon mean judgment times was the inter-action between the order of digits (largerfirst or second) and the digits in eachpair, regardless of order, F (7, 56) = 2.34,p < .05. This relationship can be seenin Fig. 2. Judgment times are plottedagainst the magnitude of the lesser digit ofthe pair. The parameter of the curves isthe sign of the difference between thedigits. When the second digit is numeric-ally larger (solid line), choice times in-crease with the size of the lesser digit.When the first digit is numerically larger(dashed line), choice times decrease withthe magnitude of the smaller digit. Ofparticular interest in Fig. 2 are the fourdata points labeled 2-1, 1-2, 8-9, and 9-8.The exceptionally short choice times as-sociated with 1-2 and 9-8 indicate 5scorrectly recognized, while the first digitwas presented, the inevitability of the cor-rect response. For example, in the caseof 1-2, the first digit was 1 and con-sequently it was a certainty that the seconddigit would be numerically larger. In thecase of 9-8, the first digit was 9, making itcertain that the second digit would besmaller. The lack of significant interactionbetween duration of the first digit, the sign

I I I I1 2 3 4 5 6 7 8

L E S S E R D I G I TFIG. 2. Mean choice time for trials on which

numerical difference between digits was unity as afunction of the lesser digit in a pair. (The solid lineconnects data from conditions in which the lesserdigit was presented first; the dashed line connectsdata from conditions in which the numericallylarger digit was presented first.)

of the difference between digits, and thepairs involved indicated that the relation-ships shown in Fig. 2 are essentially un-changed by the duration of the first digit.This means that the effects of expectancyreflected in the short times for Pairs 1-2and 9-8 are found even with the shortfirst digit duration. This resembles ex-pectancy effects found in somewhat dif-ferent choice time situations by other in-vestigators (Bernstein & Reese, 1965).The other pairs of data points of particularinterest in Fig. 2 are those labeled 2-1 and8-9. With these pairs too, 5s could beexpected to strongly anticipate, after seeingonly the first digit, what the relationshipbetween the two digits would be. In thecase 2-1 they would have expected, afterthe first digit, that the second would belarger; in the case 8-9, they would haveexpected the second digit to be smaller.Of course these anticipations would haveproved wrong. The unusually long judg-ment times associated with these pairs may

78 R. SEKULER, E. RUBIN, AND R. ARMSTRONG

reflect the fact that 5s, after seeing the firstdigit, prepared a response that wouldhave been the most likely response, andthen, upon seeing the second digit, had toabort the already prepared response, makeready and then release the correct one(see Smith, 1968).

Figure 2 shows then, that with numericaldifference between digits in a pair heldconstant, which digit is presented first doesinfluence the choice time. This effect seemsnot to be controlled by the sign of the dif-ference between digits in a pair (i.e.,whether the numerically larger is first orsecond), but rather seems to depend uponthe certainty with which the correct re-sponse can be anticipated given only thefirst digit. When the first digit makesone of the responses "larger" or "smaller"far more certain than the other, the choicetime is low; when the correct response ismore difficult to anticipate only on thebasis of the first digit, choice time lengthens.The relationship between the first digit andthe probability of anticipating the correctresponse, given only that digit, renders therelationships shown in Fig. 1 somewhatambiguous. The decrease in choice timewith increasing numerical difference be-tween digits in a pair could have been pro-duced not by the numerical difference, butby the fact that increasing numerical dif-ferences makes it more possible to antici-pate the correct response, given only thefirst digit in a pair. Figure 2 suggests thatexpectancies based on the first digit alonemight mediate a decrease in choice timewith increasing numerical difference. Asecond experiment was performed to meas-ure the effect of numerical difference un-contaminated by associated changes in thepredictive value of the first digit in a pair.

EXPERIMENT IIIn order to eliminate the differential

predictive value of various digits when eachis presented first in a pair, only one digitwas presented on each trial in Exp. II.The S was instructed to compare thenumerical magnitude of that single digitwith the magnitude of a single referencedigit (5). Moreover, for each 5, the refer-

ence digit was the larger on half of thetrials and smaller on the other half.

MethodSubjects.—The 5s were 20 undergraduates, 10 in

each of two groups.Apparatus and method.—The apparatus was like

that of Exp. I with the following exceptions. Theviewing distance was increased to 125 cm. and thecomparison digit remained on for 1.92 sec. insteadof terminating with 5's response. The intervalbetween the onset of stimuli on successive trailswas constant at 4.75 sec. The response switch waschanged to one with a shorter handle. In addition,the switch was remounted so that it was centered in5's midsagittal plane with the handle protrudingtoward the 5 and the possible switch-throw direc-tions being left and right. As the data suggest,remounting the switch from the original positionat 5's right-hand side produced a significant asym-metry in the difficulty with which the switch couldbe thrown in the two directions.

Each 5 was tested in one session consisting of 30practice trials, a short rest, and an unbroken stringof 192 trials. Stimuli were block randomized sothat each of the possible digits from 1-9, excepting5, appeared twice in each block of 16 trials. Tendifferent sets of 192 stimuli were prepared, eachbeing used with one 5 from each of the twogroups. The 5s in Group 1 were told to throw theswitch to the right when the digit presented wasnumerically larger than the digit 5 and to the leftif the digit was numerically smaller. The 5s inGroup 2 were given the opposite S-R mapping. Inorder to control the error rate and response times,5s were informed that their base pay for the sessionwould be competitively scaled from $1.50 to $4.50depending upon their speed while $.25 could bededucted for each error.

ResultsTo avoid the effects of warm-up on re-

sponse times, only data from the last 160of the total 192 experimental trials areconsidered. Errors occurred on 1.3% ofthe trials and did so in a manner that wasnot systematically related to either testdigit or group. The mean response timeson correct trials are shown in Fig. 3 as afunction of the comparison digit. Datafor the two groups are plotted separately.The significant effects, according to ananalysis of variance, are digits, F (7, 126)= 7.63, p < .01, and the interaction be-tween groups and digits, F (7, 126) = 8.33,p < .01. The main effect of groups was notstatistically significant, F (1, 18) = 2.81,p > .10. The Groups X Digits interaction

PROCESSING NUMERICAL INFORMATION 79

can be interpreted as the result of the dif-ferential difficulty with which 5s couldthrow the switch in the two response direc-tions. It will be recalled that the switch,for this experiment, was changed and, moreimportantly, its position altered. Ap-parently it was easier for 5s to throw theswitch to the right than to the left. Al-though the main effect of groups was notstatistically significant, the appearance ofa consistent separation between the groupsin Fig. 3 is somewhat disquieting. Sucha group difference would reflect the dif-ferential ease with which 5s can operateunder the two S-R mappings used in thisexperiment. Each 5 was asked whichS-R mapping seemed the more "natural"to him. All indicated that the naturalmapping was the switch thrown to theright being indicative of a larger digit,the switch thrown to the left being in-dicative of a smaller digit. Figure 3shows, however, that this natural mapping,used by Group 1, surprisingly produceduniformly slower times.

The middle curve segments in Fig. 3represent the mean response times for bothgroups. They show that as the comparisondigit approaches the numerical value ofthe standard (5) from either side, the re-sponse time lengthens. This is the samerelationship that was found in Exp. I.To facilitate comparison between this ex-periment and Exp. I, we have calculatedthe mean times for each absolute value ofthe difference between standard and com-parison digits. As an example, we haveaveraged the times for comparison digits1 and 9 since both represent numericaldifferences of 4. These means have beenplotted in Fig. 1, and the line connectingthem is labeled Exp. II. Although thisexperiment differs in several respects fromExp. I, as well as from the Moyer andLandauer (1967) study, the curves shownin Fig. 1 all show the same general trends,a decrease in response time with increasingnumerical difference between standard andcomparison digits. The absolute heightsof the various curves are somewhat moredifficult to compare since the instructions

STD. DIGIT =5

2 3 4 6

Comparison Digits

FIG. 3. Mean choice time in Exp. II for Group 1,Group 2, and over all means as a function of thecomparison digit. (Group 1 threw a switch to theright to indicate that the comparison digit was largerthan 5; Group 2 threw a switch to the left for thatcondition.)

(or lack of them) concerning the speed-accuracy trade-off differ.

DISCUSSIONThe present experiments have replicated in

all major details the earlier findings of Moyerand Landauer (1967). With successive presen-tations of the digits in each pair, time re-quired for judgment of numerical inequalitydeclines with increasing differences betweenthe digits. Even when only a single digit ispresented to be compared against the memorialrepresentation of a standard digit, we findthat the larger the numerical difference be-tween the two digits, the shorter the responsetime. As Moyer and Landauer suggest, thisfinding argues against what is probably apriori the most likely system for generatingjudgments of numerical inequality. Assumethat each visually presented digit activatesa process which searches out the psychologicalrepresentation of the numerical magnitudeassociated with the digit. If the psychologicalrepresentations are themselves arranged ac-cording to magnitude, an 5 having alreadylocated the representation of one digit shouldrequire an amount of time to find the repre-sentation of a second digit that is a nondecreas-ing function of the distance between storagelocations of both representations. This sys-

80 R. SEKULER, E. RUBIN, AND R. ARMSTRONG

tern for storing numerical magnitudes shouldgenerate data that are essentially the oppositeof our own; i.e., it implies that time requiredfor judgment of numerical inequality shouldincrease with increasing differences betweenthe two digits which are to be compared. Wethink that the alternative system suggestedby Moyer and Landauer (1967) should beaccepted at least as a working hypothesis.This hypothesis is that some analog representa-tion is generated for each digit which is to becompared. The two analog representationsare then compared to produce a judgment ofnumerical inequality. This places the psy-chological response to digits and pairs ofdigits in the same general theoretical frame-work as that used for other psychophysicaljudgments.

The average difference of ISO msec, be-tween the two conditions, FDS and FDL, inExp. I is of some theoretical importance. Itwill be recalled that in Cond. FDS the firstdigit was presented for only 50 msec, andwas followed immediately by the second digit.In Cond. FDL the first digit remained on for aful l 2 sec. The shorter response times in thislatter condition probably reflect, among otherthings, the fact that the 2-sec. exposure of thefirst stimulus permitted the stimulus to beentered into whatever form of short-term store(visual or other) serves in this task. We maytake the difference between Cond. FDS andFDL as a rough estimate of the time requiredto process the first digit. The validity of thisestimate needs to be examined in a study in-volving parametric manipulation of the ex-posure duration of the first digit in thesequence.

Recently, Lovelace and Snodgrass (1970)repeated the experiment of Moyer and Land-auer (1967) using letters of the alphabet in-stead of digits as stimuli. Lovelace and Snod-grass found that when the alphabetic distancebetween two letters increases, 5s require lesstime to identify which of the two comes firstin the alphabet. This is, of course, consistent

with both the present findings and those ofMoyer and Landauer (1967). Unlike the ex-periments with numerical stimuli, however, theLovelace and Snodgrass experiment found astrong "order" effect: when the left-hand letteroccurred earlier in the alphabet than did itsright-hand mate, 5s took significantly lesstime to identify the alphabetic order of thetwo than they did when the left-hand stimulusoccurred later in the alphabet. As indicatedabove, we found no analogous effect. For thepresent, we assume that this difference reflectsthe differential practice 5s have with back-ward number series and backward alphabets.Because of space-launch countdowns andSesame Street and other common situations,we have considerable practice with the num-ber series in backwards order. We do nothave nearly so much practice with the alphabetin its backward order and indeed a normal 5finds that it is quite hard to recite the alphabetbackward. We assume that such differentialpractice, rather than an inherent alphabetic-numeric difference, caused the alphabeticstimuli to show much stronger order effectsthan did their numerical stimuli counterparts.

REFERENCES

BERNSTEIN, I. H., & REESE, C., Behavioralhypotheses and choice reaction time. Psy-chonomic Science, 1965, 3, 259-260.

EGETH, H., & SMITH, E. E. On the nature of errorsin a choice reaction task. Psychonomic Science,1967, 8, 345-46.

LOVELACE, E. A., & SNODGRASS, R. D. Decisiontimes for alphabetic order of letter pairs. Paperpresented at meetings of Midwestern Psy-chological Association, Cincinnati, Ohio, May1970.

MOVER, R. S., & LANDAUER, T. K. Time requiredfor judgments of numerical inequality. Nature,1967, 215, 1519-1520.

SMITH, E. E. Choice reaction time: An analysis ofthe major theoretical positions. PsychologicalBulletin, 1968, 69, 77-110.

(Received February 22, 1971)