The Magical Numbers Seven, Plus or Minus Two

8/10/2019 The Magical Numbers Seven, Plus or Minus Two

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From: http://psychclassics.yorku.ca/Miller/

[Classics Editor's Note: Footnotes are in square brackets; references in round brackets]

The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for

Processing Information[1]

eorge !" Miller #$%&'(Harvard University

First published inPsychological Review, !, 81-9.

My problem is that ! ha"e been persecuted by an inte#er. For se"en years this number has $ollo%ed me

around, has intruded in my most pri"ate data, and has assaulted me $rom the pa#es o$ our most public&ournals. 'his number assumes a "ariety o$ dis#uises, bein# sometimes a little lar#er and sometimes a

little smaller than usual, but ne"er chan#in# so much as to be unreco#ni(able. 'he persistence %ith%hich this number pla#ues me is $ar more than a random accident. 'here is, to )uote a $amous senator, adesi#n behind it, some pattern #o"ernin# its appearances. *ither there really is somethin# unusual about

the number or else ! am su$$erin# $rom delusions o$ persecution.

! shall be#in my case history by tellin# you about some e+periments that tested ho% accurately peoplecan assi#n numbers to the ma#nitudes o$ "arious aspects o$ a stimulus. !n the traditional lan#ua#e o$

psycholo#y these %ould be called e+periments in absolute &ud#ment. istorical accident, ho%e"er, has

decreed that they should ha"e another name. e no% call them e+periments on the capacity o$ people totransmit in$ormation. ince these e+periments %ould not ha"e been done %ithout the appearance o$

in$ormation theory on the psycholo#ical scene, and since the results are analy(ed in terms o$ the

concepts o$ in$ormation theory, ! shall ha"e to pre$ace my discussion %ith a $e% remarks about thistheory.

Information Measurement

'he amount o$ in$ormation is e+actly the same concept that %e ha"e talked about $or years under the

name o$ "ariance. 'he e)uations are di$$erent, but i$ %e hold ti#ht to the idea that anythin# that

increases the "ariance also increases the amount o$ in$ormation %e cannot #o $ar astray.

'he ad"anta#es o$ this ne% %ay o$ talkin# about "ariance are simple enou#h. 0ariance is al%ays stated

in terms o$ the unit o$ measurement - inches, pounds, "olts, etc. - %hereas the amount o$ in$ormation is a

dimensionless )uantity. ince the in$ormation in a discrete statistical distribution does not depend uponthe unit o$ measurement, %e can e+tend the concept to situations %here %e ha"e no metric and %e

%ould not ordinarily think o$ usin# [p. 8] the "ariance. 2nd it also enables us to compare results

obtained in )uite di$$erent e+perimental situations %here it %ould be meanin#less to compare "ariancesbased on di$$erent metrics. o there are some #ood reasons $or adoptin# the ne%er concept.

'he similarity o$ "ariance and amount o$ in$ormation mi#ht be e+plained this %ay: hen %e ha"e a

lar#e "ariance, %e are "ery i#norant about %hat is #oin# to happen. !$ %e are "ery i#norant, then %hen%e make the obser"ation it #i"es us a lot o$ in$ormation. 3n the other hand, i$ the "ariance is "ery small,

%e kno% in ad"ance ho% our obser"ation must come out, so %e #et little in$ormation $rom makin# the

obser"ation.


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!$ you %ill no% ima#ine a communication system, you %ill reali(e that there is a #reat deal o$ "ariability

about %hat #oes into the system and also a #reat deal o$ "ariability about %hat comes out. 'he input andthe output can there$ore be described in terms o$ their "ariance 4or their in$ormation5. !$ it is a #ood

communication system, ho%e"er, there must be some systematic relation bet%een %hat #oes in and %hat

comes out. 'hat is to say, the output %ill depend upon the input, or %ill be correlated %ith the input. !$%e measure this correlation, then %e can say ho% much o$ the output "ariance is attributable to the input

and ho% much is due to random $luctuations or noise introduced by the system durin# transmission.

o %e see that the measure o$ transmitted in$ormation is simply a measure o$ the input-output

correlation.

'here are t%o simple rules to $ollo%. hene"er ! re$er to amount o$ in$ormation, you %ill understand"ariance. 2nd %hene"er ! re$er to amount o$ transmitted in$ormation, you %ill understand

co"ariance or correlation.

'he situation can be described #raphically by t%o partially o"erlappin# circles. 'hen the le$t circle canbe taken to represent the "ariance o$ the input, the ri#ht circle the "ariance o$ the output, and the o"erlap

the co"ariance o$ input and output. ! shall speak o$ the le$t circle as the amount o$ input in$ormation, the

ri#ht circle as the amount o$ output in$ormation, and the o"erlap as the amount o$ transmitted

in$ormation.

!n the e+periments on absolute &ud#ment, the obser"er is considered to be a communication channel.

'hen the le$t circle %ould represent the amount o$ in$ormation in the stimuli, the ri#ht circle the amounto$ in$ormation in his responses, and the o"erlap the stimulus-response correlation as measured by the

amount o$ transmitted in$ormation. 'he e+perimental problem is to increase the amount o$ input

in$ormation and to measure the amount o$ transmitted in$ormation. !$ the obser"er6s absolute &ud#mentsare )uite accurate, then nearly all o$ the input in$ormation %ill be transmitted and %ill be reco"erable

$rom his responses. !$ he makes errors, then the transmitted in$ormation may be considerably less than

the input. e e+pect that, as %e increase the amount o$ input in$ormation, the obser"er %ill be#in tomake more and more errors7 %e can test the limits o$ accuracy o$ his absolute &ud#ments. !$ the human

obser"er is a reasonable kind o$ communication system, then %hen %e increase the amount o$ input

in$ormation the transmitted in$ormation %ill increase at $irst and %ill e"entually le"el o$$ at someasymptotic "alue. 'his asymptotic "alue %e take to be the channel ca"acityo$ the obser"er: it representsthe #reatest amount o$ in$ormation that he can #i"e us about the stimulus on the basis o$ an absolute

&ud#ment. 'he channel capacity is the upper limit on the e+tent to %hich the obser"er can match his

responses to the stimuli %e #i"e him.

o% &ust a brie$ %ord about the bit [p. 8] and %e can be#in to look at some data. 3ne bit o$ in$ormation

is the amount o$ in$ormation that %e need to make a decision bet%een t%o e)ually likely alternati"es. !$

%e must decide %hether a man is less than si+ $eet tall or more than si+ $eet tall and i$ %e kno% that thechances are ;-;, then %e need one bit o$ in$ormation. otice that this unit o$ in$ormation does not

re$er in any %ay to the unit o$ len#th that %e use - $eet, inches, centimeters, etc. o%e"er you measure

the man6s hei#ht, %e still need &ust one bit o$ in$ormation.

'%o bits o$ in$ormation enables us to decide amon# $our e)ually likely alternati"es. 'hree bits o$

in$ormation enable us to decide amon# ei#ht e)ually likely alternati"es. Four bits o$ in$ormation decideamon# 1< alternati"es, $i"e amon# , and so on. 'hat is to say, i$ there are e)ually likely

alternati"es, %e must make $i"e successi"e binary decisions, %orth one bit each, be$ore %e kno% %hich

alternati"e is correct. o the #eneral rule is simple: e"ery time the number o$ alternati"es is increased by

a $actor o$ t%o, one bit o$ in$ormation is added.

'here are t%o %ays %e mi#ht increase the amount o$ input in$ormation. e could increase the rate at

%hich %e #i"e in$ormation to the obser"er, so that the amount o$ in$ormation per unit time %ould


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increase. 3r %e could i#nore the time "ariable completely and increase the amount o$ input in$ormation

by increasin# the number o$ alternati"e stimuli. !n the absolute &ud#ment e+periment %e are interested inthe second alternati"e. e #i"e the obser"er as much time as he %ants to make his response7 %e simply

increase the number o$ alternati"e stimuli amon# %hich he must discriminate and look to see %here

con$usions be#in to occur. =on$usions %ill appear near the point that %e are callin# his channelcapacity.

!bsolute )u*gments of +ni*imensional Stimuli

o% let us consider %hat happens %hen %e make absolute &ud#ments o$ tones. >ollack 415 asked

listeners to identi$y tones by assi#nin# numerals to them. 'he tones %ere di$$erent %ith respect to

$re)uency, and co"ered the ran#e $rom 1;; to 8;;; cps in e)ual lo#arithmic steps. 2 tone %as soundedand the listener responded by #i"in# a numeral. 2$ter the listener had made his response he %as told the

correct identi$ication o$ the tone.

hen only t%o or three tones %ere used the listeners ne"er con$used them. ith $our di$$erent tones

con$usions %ere )uite rare, but %ith $i"e or more tones con$usions %ere $re)uent. ith $ourteen

di$$erent tones the listeners made many mistakes.

'hese data are plotted in Fi#. 1. 2lon# the bottom is theamount o$ input in$ormation in bits per stimulus. 2s the

number o$ alternati"e tones %as increased $rom to 1?, theinput in$ormation increased $rom 1 to .8 bits. 3n the

ordinate is plotted the amount o$ [p. 8?] transmitted

in$ormation. 'he amount o$ transmitted in$ormation beha"esin much the %ay %e %ould e+pect a communication channel

to beha"e7 the transmitted in$ormation increases linearly up

to about bits and then bends o$$ to%ard an asymptote atabout . bits. 'his "alue, . bits, there$ore, is %hat %e are

callin# the channel capacity o$ the listener $or absolute

&ud#ments o$ pitch.

o no% %e ha"e the number . bits. hat does it mean@

First, note that . bits corresponds to about si+ e)ually

likely alternati"es. 'he result means that %e cannot pick

more than si+ di$$erent pitches that the listener %ill ne"ercon$use. 3r, stated sli#htly di$$erently, no matter ho% many

alternati"e tones %e ask him to &ud#e, the best %e can e+pect him to do is to assi#n them to about si+

di$$erent classes %ithout error. 3r, a#ain, i$ %e kno% that there %ereNalternati"e stimuli, then his&ud#ment enables us to narro% do%n the particular stimulus to one out o$N / ollack6s results indicate that, at least $or pitches, this intuition is $airly sound.


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e+t you can ask ho% reproducible this result is. Aoes it

depend on the spacin# o$ the tones or the "arious conditionso$ &ud#ment@ >ollack "aried these conditions in a number o$

%ays. 'he ran#e o$ $re)uencies can be chan#ed by a $actor o$

about ; %ithout chan#in# the amount o$ in$ormationtransmitted more than a small percenta#e. Ai$$erent

#roupin#s o$ the pitches decreased the transmission, but the

loss %as small. For e+ample, i$ you can discriminate $i"e

hi#h-pitched tones in one series and $i"e lo%-pitched tones inanother series, it is reasonable to e+pect that you could

combine all ten into a sin#le series and still tell them all apart

%ithout error. hen you try it, ho%e"er, it does not %ork.'he channel capacity $or pitch seems to be about si+ and that

is the best you can do.

hile %e are on tones, let us look ne+t at Barner6s 45 %ork on loudness. Barner6s data $or loudness are

summari(ed in Fi#. . Barner %ent to some trouble to #et the best possible spacin# o$ his tones o"er the

intensity ran#e $rom 1 to 11; db. e used ?, ,


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'he ake-Barner e+periment has been repeated by =oonanand Elemmer. 2lthou#h they ha"e not yet published their

results, they ha"e #i"en me permission to say that they

obtained channel capacities ran#in# $rom . bits $or [p. 8ollack has been kind enou#h to $urnish me %ith the results o$ their measurements $or se"eral aspects o$

"isual displays. 'hey made measurements $or area and $or the cur"ature, len#th, and direction o$ lines.!n one set o$ e+periments they used a "ery short e+posure o$ the stimulus - 1 / ?; second - and then they

repeated the measurements %ith a -second e+posure. For area they #ot .< bits %ith the short e+posure

and . bits %ith the lon# e+posure. For the len#th o$ a line they #ot about .< bits %ith the short

e+posure and about .; bits %ith the lon# e+posure. Airection, or an#le o$ inclination, #a"e .8 bits $orthe short e+posure and . bits $or the lon# e+posure. =ur"ature %as apparently harder to &ud#e. hen

the len#th o$ the arc %as constant, the result at the short e+posure duration %as . bits, but %hen thelen#th o$ the chord %as constant, the result %as only 1.< bits. 'his last "alue is the lo%est that anyonehas measured to date. ! should add, ho%e"er, that these "alues are apt to be sli#htly too lo% because the

data $rom all sub&ects %ere pooled be$ore the transmitted in$ormation %as computed.

o% let us see %here %e are. First, the channel capacity does seem to be a "alid notion $or describin#

human obser"ers. econd, the channel capacities measured $or these unidimensional "ariables ran#e

$rom 1.< bits $or cur"ature to .9 bits $or positions in an inter"al. 2lthou#h there is no )uestion that thedi$$erences amon# the "ariables are real and meanin#$ul, the more impressi"e $act to me is their

considerable similarity. !$ ! take the best estimates ! can #et o$ the channel capacities $or all the stimulus

"ariables ! ha"e mentioned, the mean is .< bits and the standard de"iation is only ;.< bit. !n terms o$

distin#uishable alternati"es, this mean corresponds to about


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Gou may ha"e noticed that ! ha"e been care$ul to say that this ma#ical number se"en applies to one-

dimensional &ud#ments. *"eryday e+perience teaches us that %e can identi$y accurately any one o$se"eral hundred $aces, any one o$ se"eral thousand %ords, any one o$ se"eral thousand ob&ects, etc. 'he

story certainly %ould not be complete i$ %e stopped at this point. e must ha"e some understandin# o$

%hy the one-dimensional "ariables %e &ud#e in the laboratory #i"e results so $ar out o$ line %ith %hat%e do constantly in our beha"ior outside the laboratory. 2 possible e+planation lies in the number o$

independently "ariable attributes o$ the stimuli that are bein# &ud#ed. 3b&ects, $aces, %ords, and the like

di$$er $rom one another in many %ays, %hereas the simple stimuli %e ha"e considered thus $ar di$$er

$rom one another in only one respect.

Fortunately, there are a $e% data on %hat happens %hen %e

make absolute &ud#ments o$ stimuli that di$$er $rom oneanother in se"eral %ays. et us look $irst at the results

Elemmer and Frick 415 ha"e reported $or the absolute

&ud#ment o$ the position o$ a dot in a s)uare. !n Fi#. %e seetheir results. o% the channel capacity seems to ha"e

increased to ?.< bits, %hich means that people can identi$y

accurately any one o$ ? positions in the s)uare.

'he position o$ a dot in a s)uare is clearly a t%o-dimensionalproposition. Coth its hori(ontal and its "ertical position mustbe identi$ied. 'hus it seems natural to compare the ?.ollack 4185, %ho asked listeners to &ud#e both the loudness and the

pitch o$ pure tones. ince pitch #i"es . bits and loudness #i"es . bits, %e mi#ht hope to #et as much

as ?.8 bits $or pitch and loudness to#ether. >ollack obtained .1 bits, %hich a#ain indicates that thesecond dimension au#ments the channel capacity but not so much as it mi#ht.

2 $ourth e+ample can be dra%n $rom the %ork o$ alsey and =hapanis 495 on con$usions amon# colors

o$ e)ual [p. 88] luminance. 2lthou#h they did not analy(e their results in in$ormational terms, theyestimate that there are about 11 to 1 identi$iable colors, or, in our terms, about .< bits. ince these

colors "aried in both hue and saturation, it is probably correct to re#ard this as a t%o-dimensional

&ud#ment. !$ %e compare this %ith *riksen6s .1 bits $or hue 4%hich is a )uestionable comparison to

dra%5, %e a#ain ha"e somethin# less than per$ect addition %hen a second dimension is added.

!t is still a lon# %ay, ho%e"er, $rom these t%o-dimensional e+amples to the multidimensional stimuli

pro"ided by $aces, %ords, etc. 'o $ill this #ap %e ha"e only one e+periment, an auditory study done by>ollack and Ficks 4195. 'hey mana#ed to #et si+ di$$erent acoustic "ariables that they could chan#e:

$re)uency, intensity, rate o$ interruption, on-time $raction, total duration, and spatial location. *ach one


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o$ these si+ "ariables could assume any one o$ $i"e di$$erent "alues, so alto#ether there %ere


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!n human speech there is clearly a limit to the number o$ dimensions that %e use. !n this instance,

ho%e"er, it is not kno%n %hether the limit is imposed by the nature o$ the perceptual machinery thatmust reco#ni(e the sounds or by the nature o$ the speech machinery that must produce them. omebody

%ill ha"e to do the e+periment to $ind out. 'here is a limit, ho%e"er, at about ei#ht or nine distincti"e

$eatures in e"ery lan#ua#e that has been studied, and so %hen %e talk %e must resort to still anothertrick $or increasin# our channel capacity. an#ua#e uses se)uences o$ phonemes, so %e make se"eral

&ud#ments successi"ely %hen %e listen to %ords and sentences. 'hat is to say, %e use both simultaneous

and successi"e discriminations in order to e+pand the rather ri#id limits imposed by the inaccuracy o$

our absolute &ud#ments o$ simple ma#nitudes.

'hese multidimensional &ud#ments are stron#ly reminiscent o$ the abstraction e+periment o$ EIlpe 41?5.2s you may remember, EIlpe sho%ed that obser"ers report more accurately on an attribute $or %hich

they are set than on attributes $or %hich they are not set. For e+ample, =hapman 4?5 used three di$$erent

attributes and compared the results obtained %hen the obser"ers %ere instructed be$ore the

tachistoschopic presentation %ith the results obtained %hen they %ere not told until a$ter thepresentation %hich one o$ the three attributes %as to be reported. hen the instruction %as #i"en in

ad"ance, the &ud#ments %ere more accurate. hen the instruction %as #i"en a$ter%ards, the sub&ects

presumably had to &ud#e all three attributes in order to report on any one o$ them and the accuracy %ascorrespondin#ly lo%er. 'his is in complete accord %ith the results %e ha"e &ust been considerin#, %here

the accuracy o$ &ud#ment on each attribute decreased as more dimensions %ere added. 'he point isprobably ob"ious, but ! shall make it anyho%, that the abstraction e+periments did not demonstrate thatpeople can &ud#e only one attribute at a time. 'hey merely sho%ed %hat seems )uite reasonable, that

people are less accurate i$ they must &ud#e more than one attribute simultaneously.

[p. 9;] Subitiing

! cannot lea"e this #eneral area %ithout mentionin#, ho%e"er brie$ly, the e+periments conducted at

Mount olyoke =olle#e on the discrimination o$ number 415. !n e+periments by Eau$man, ord,Deese, and 0olkmann random patterns o$ dots %ere $lashed on a screen $or 1 / o$ a second. 2ny%here

$rom 1 to more than ;; dots could appear in the pattern. 'he sub&ect6s task %as to report ho% many dots

there %ere.

'he $irst point to note is that on patterns containin# up to $i"e or si+ dots the sub&ects simply did not

make errors. 'he per$ormance on these small numbers o$ dots %as so di$$erent $rom the per$ormance

%ith more dots that is %as #i"en a special name. Celo% se"en the sub&ects %ere said to subiti#e7 abo"e

se"en they %ere said to esti$ate. 'his is, as you %ill reco#ni(e, %hat %e once optimistically called thespan o$ attention.

'his discontinuity at se"en is, o$ course, su##esti"e. !s this the same basic process that limits ourunidimensional &ud#ments to about se"en cate#ories@ 'he #enerali(ation is temptin#, but not sound in

my opinion. 'he data on number estimates ha"e not been analy(ed in in$ormational terms7 but on the

basis o$ the published data ! %ould #uess that the sub&ects transmitted somethin# more than $our bits o$in$ormation about the number o$ dots. Hsin# the same ar#uments as be$ore, %e %ould conclude that

there are about ; or ; distin#uishable cate#ories o$ numerousness. 'his is considerably more

in$ormation than %e %ould e+pect to #et $rom a unidimensional display. !t is, as a matter o$ $act, "erymuch like a t%o-dimensional display. 2lthou#h the dimensionality o$ the random dot patterns is not

entirely clear, these results are in the same ran#e as Elemmer and Frick6s $or their t%o-dimensional

display o$ dots in a s)uare. >erhaps the t%o dimensions o$ numerousness are area and density. hen the

sub&ect can subiti(e, area and density may not be the si#ni$icant "ariables, but %hen the sub&ect mustestimate perhaps they are si#ni$icant. !n any e"ent, the comparison is not so simple as it mi#ht seem at

$irst thou#ht.


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'his is one o$ the %ays in %hich the ma#ical number se"en has persecuted me. ere %e ha"e t%o

closely related kinds o$ e+periments, both o$ %hich point to the si#ni$icance o$ the number se"en as alimit on our capacities. 2nd yet %hen %e e+amine the matter more closely, there seems to be a

reasonable suspicion that it is nothin# more than a coincidence.

The Span of Imme*iate Memory

et me summari(e the situation in this %ay. 'here is a clear and de$inite limit to the accuracy %ith

%hich %e can identi$y absolutely the ma#nitude o$ a unidimensional stimulus "ariable. ! %ould proposeto call this limit thes"an of absolute %udg$ent, and ! maintain that $or unidimensional &ud#ments this

span is usually some%here in the nei#hborhood o$ se"en. e are not completely at the mercy o$ this

limited span, ho%e"er, because %e ha"e a "ariety o$ techni)ues $or #ettin# around it and increasin# theaccuracy o$ our &ud#ments. 'he three most important o$ these de"ices are 4a5 to make relati"e rather

than absolute &ud#ments.7 or, i$ that is not possible, 4b5 to increase the number o$ dimensions alon#

%hich the stimuli can di$$er7 or 4c5 to arran#e the task in such a %ay that %e make a se)uence o$ se"eralabsolute &ud#ments in a ro%.

'he study o$ relati"e &ud#ments is one o$ the oldest topics in e+perimental psycholo#y, and ! %ill not

pause to re"ie% it no%. 'he second de"ice, increasin# the dimensionality, %e ha"e &ust considered. !t

seems that by addin# [p. 91] more dimensions and re)uirin# crude, binary, yes-no &ud#ments on eachattribute %e can e+tend the span o$ absolute &ud#ment $rom se"en to at least 1;. Jud#in# $rom our

e"eryday beha"ior, the limit is probably in the thousands, i$ indeed there is a limit. !n my opinion, %ecannot #o on compoundin# dimensions inde$initely. ! suspect that there is also a s"an of "erce"tual

di$ensionality and that this span is some%here in the nei#hborhood o$ ten, but ! must add at once that

there is no ob&ecti"e e"idence to support this suspicion. 'his is a )uestion sadly needin# e+perimentale+ploration.

=oncernin# the third de"ice, the use o$ successi"e &ud#ments, ! ha"e )uite a bit to say because this

de"ice introduces memory as the handmaiden o$ discrimination. 2nd, since mnemonic processes are atleast as comple+ as are perceptual processes, %e can anticipate that their interactions %ill not be easily

disentan#led.

uppose that %e start by simply e+tendin# sli#htly the e+perimental procedure that %e ha"e been usin#.

Hp to this point %e ha"e presented a sin#le stimulus and asked the obser"er to name it immediately

therea$ter. e can e+tend this procedure by re)uirin# the obser"er to %ithhold his response until %e

ha"e #i"en him se"eral stimuli in succession. 2t the end o$ the se)uence o$ stimuli he then makes hisresponse. e still ha"e the same sort o$ input-output situation that is re)uired $or the measurement o$

transmitted in$ormation. Cut no% %e ha"e passed $rom an e+periment on absolute &ud#ment to %hat is

traditionally called an e+periment on immediate memory.

Ce$ore %e look at any data on this topic ! $eel ! must #i"e you a %ord o$ %arnin# to help you a"oid some

ob"ious associations that can be con$usin#. *"erybody kno%s that there is a $inite span o$ immediatememory and that $or a lot o$ di$$erent kinds o$ test materials this span is about se"en items in len#th. !ha"e &ust sho%n you that there is a span o$ absolute &ud#ment that can distin#uish about se"en cate#ories

and that there is a span o$ attention that %ill encompass about si+ ob&ects at a #lance. hat is more

natural than to think that all three o$ these spans are di$$erent aspects o$ a sin#le underlyin# process@2nd that is a $undamental mistake, as ! shall be at some pains to demonstrate. 'his mistake is one o$ the

malicious persecutions that the ma#ical number se"en has sub&ected me to.

My mistake %ent somethin# like this. e ha"e seen that the in"ariant $eature in the span o$ absolute

&ud#ment is the amount o$ in$ormation that the obser"er can transmit. 'here is a real operational


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similarity bet%een the absolute &ud#ment e+periment and the immediate memory e+periment. !$

immediate memory is like absolute &ud#ment, then it should $ollo% that the in"ariant $eature in the spano$ immediate memory is also the amount o$ in$ormation that an obser"er can retain. !$ the amount o$

in$ormation in the span o$ immediate memory is a constant, then the span should be short %hen the

indi"idual items contain a lot o$ in$ormation and the span should be lon# %hen the items contain littlein$ormation. For e+ample, decimal di#its are %orth . bits apiece. e can recall about se"en o$ them,

$or a total o$ bits o$ in$ormation. !solated *n#lish %ords are %orth about 1; bits apiece. !$ the total

amount o$ in$ormation is to remain constant at bits, then %e should be able to remember only t%o or

three %ords chosen at random. !n this %ay ! #enerated a theory about ho% the span o$ immediatememory should "ary as a $unction o$ the amount o$ in$ormation per item in the test materials.

'he measurements o$ memory span in the literature are su##esti"e on this [p. 9] )uestion, but not

de$initi"e. 2nd so it %as necessary to do the e+periment to see. ayes 41;5 tried it out %ith $i"e di$$erent

kinds o$ test materials: binary di#its, decimal di#its, letters o$ the alphabet, letters plus decimal di#its,

and %ith 1,;;; monosyllabic %ords. 'he lists %ere read aloud at the rate o$ one item per second and thesub&ects had as much time as they needed to #i"e their responses. 2 procedure described by ood%orth

4;5 %as used to score the responses.

'he results are sho%n by the $illed circles in Fi#. . ere

the dotted line indicates %hat the span should ha"e been i$the amount o$ in$ormation in the span %ere constant. 'hesolid cur"es represent the data. ayes repeated the

e+periment usin# test "ocabularies o$ di$$erent si(es but all

containin# only *n#lish monosyllables 4open circles in Fi#.

5. 'his more homo#eneous test material did not chan#e thepicture si#ni$icantly. ith binary items the span is about

nine and, althou#h it drops to about $i"e %ith monosyllabic

*n#lish %ords, the di$$erence is $ar less than the hypothesiso$ constant in$ormation %ould re)uire.

'here is nothin# %ron# %ith ayes6s e+periment, because>ollack 41ollack took pains to measure

the amount o$ in$ormation transmitted and did not rely on

the traditional procedure $or scorin# the responses. isresults are plotted in Fi#. 8. ere it is clear that the amount o$ in$ormation transmitted is not a constant,

but increases almost linearly as the amount o$ in$ormation per item in the input is increased.

2nd so the outcome is per$ectly clear. !n spite o$ the coincidence that the ma#ical number se"en appears

in both places, the span o$ absolute &ud#ment and the span o$ immediate memory are )uite di$$erent

kinds o$ limitations that are imposed on our ability to process in$ormation. 2bsolute &ud#ment is limited

by the amount o$ in$ormation. !mmediate memory is limited by the number o$ items. !n order to capturethis distinction in some%hat pictures)ue terms, ! ha"e $allen into the custom o$ distin#uishin# bet%een

bitso$ in$ormation and chunks o$ in$ormation. 'hen ! can say that the number o$ bits o$ in$ormation is

constant $or absolute &ud#ment and the number o$ chunks o$ in$orma- [p. 9] tion is constant $orimmediate memory. 'he span o$ immediate memory seems to be almost independent o$ the number o$

bits per chunk, at least o"er the ran#e that has been e+amined to date.


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'he contrast o$ the terms bitand chunkalso ser"es to

hi#hli#ht the $act that %e are not "ery de$inite about %hatconstitutes a chunk o$ in$ormation. For e+ample, the memory

span o$ $i"e %ords that ayes obtained %hen each %ord %as

dra%n at random $rom a set o$ 1;;; *n#lish monosyllablesmi#ht &ust as appropriately ha"e been called a memory span

o$ 1 phonemes, since each %ord had about three phonemes

in it. !ntuiti"ely, it is clear that the sub&ects %ere recallin#

$i"e %ords, not 1 phonemes, but the lo#ical distinction isnot immediately apparent. e are dealin# here %ith a

process o$ or#ani(in# or #roupin# the input into $amiliar

units or chunks, and a #reat deal o$ learnin# has #one into the$ormation o$ these $amiliar units.

-eco*ing

!n order to speak more precisely, there$ore, %e must

reco#ni(e the importance o$ #roupin# or or#ani(in# the input se)uence into units or chunks. ince the

memory span is a $i+ed number o$ chunks, %e can increase the number o$ bits o$ in$ormation that it

contains simply by buildin# lar#er and lar#er chunks, each chunk containin# more in$ormation thanbe$ore.

2 man &ust be#innin# to learn radiotele#raphic code hears each dit and dahas a separate chunk. oon he

is able to or#ani(e these sounds into letters and then he can deal %ith the letters as chunks. 'hen the

letters or#ani(e themsel"es as %ords, %hich are still lar#er chunks, and he be#ins to hear %hole phrases.! do not mean that each step is a discrete process, or that plateaus must appear in his learnin# cur"e, $or

surely the le"els o$ or#ani(ation are achie"ed at di$$erent rates and o"erlap each other durin# the

learnin# process. ! am simply pointin# to the ob"ious $act that the dits and dahs are or#ani(ed bylearnin# into patterns and that as these lar#er chunks emer#e the amount o$ messa#e that the operator

can remember increases correspondin#ly. !n the terms ! am proposin# to use, the operator learns to

increase the bits per chunk.

!n the &ar#on o$ communication theory, this process %ould be called recoding. 'he input is #i"en in a

code that contains many chunks %ith $e% bits per chunk. 'he operator recodes the input into another

code that contains $e%er chunks %ith more bits per chunk. 'here are many %ays to do this recodin#, but

probably the simplest is to #roup the input e"ents, apply a ne% name to the #roup, and then rememberthe ne% name rather than the ori#inal input e"ents.

ince ! am con"inced that this process is a "ery #eneral and important one $or psycholo#y, ! %ant to tellyou about a demonstration e+periment that should make per$ectly e+plicit %hat ! am talkin# about. 'his

e+periment %as conducted by idney mith and %as reported by him be$ore the *astern >sycholo#ical

2ssociation in 19?.

Ce#in %ith the obser"ed $act that people can repeat back ei#ht decimal di#its, but only nine binary

di#its. ince there is a lar#e discrepancy in the amount o$ in$ormation recalled in these t%o cases, %e

suspect at once that a recodin# procedure could be used to increase the span o$ immediate memory $orbinary di#its. !n 'able 1 a method $or #roupin# and renamin# is illustrated. 2lon# the top is a se)uence

o$ 18 binary di#its, $ar more than any sub&ect %as able to recall a$ter a sin#le presentation. !n the ne+t

line these same binary di#its are #rouped by pairs. Four possible pairs can occur: ;; is renamed ;, ;1 isrenamed 1, 1; is renamed , and 11 is [p. 9?] renamed . 'hat is to say, %e recode $rom a base-t%o

arithmetic to a base-$our arithmetic. !n the recoded se)uence there are no% &ust nine di#its to remember,

and this is almost %ithin the span o$ immediate memory. !n the ne+t line the same se)uence o$ binary


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di#its is re#rouped into chunks o$ three. 'here are ei#ht possible se)uences o$ three, so %e #i"e each

se)uence a ne% name bet%een ; and . o% %e ha"e recoded $rom a se)uence o$ 18 binary di#its into ase)uence o$ < octal di#its, and this is %ell %ithin the span o$ immediate memory. !n the last t%o lines the

binary di#its are #rouped by $ours and by $i"es and are #i"en decimal-di#it names $rom ; to 1 and $rom

; to 1.

!t is reasonably ob"ious that this kind o$ recodin# increases the bits per chunk, and packa#es the binary

se)uence into a $orm that can be retained %ithin the span o$ immediate memory. o mith assembled ;sub&ects and measured their spans $or binary and octal di#its. 'he spans %ere 9 $or binaries and $or

octals. 'hen he #a"e each recodin# scheme to $i"e o$ the sub&ects. 'hey studied the recodin# until they

said they understood it - $or about or 1; minutes. 'hen he tested their span $or binary di#its a#ain%hile they tried to use the recodin# schemes they had studied.

'he recodin# schemes increased their span $or binary di#its in e"ery case. Cut the increase %as not as

lar#e as %e had e+pected on the basis o$ their span $or octal di#its. ince the discrepancy increased as

the recodin# ratio increased, %e reasoned that the $e% minutes the sub&ects had spent learnin# therecodin# schemes had not been su$$icient. 2pparently the translation $rom one code to the other must be

almost automatic or the sub&ect %ill lose part o$ the ne+t #roup %hile he is tryin# to remember thetranslation o$ the last #roup.

ince the ?:1 and :1 ratios re)uire considerable study, mith decided to imitate *bbin#haus and do the

e+periment on himsel$. ith Bermanic patience he drilled himsel$ on each recodin# successi"ely, andobtained the results sho%n in Fi#. 9. ere the data $ollo% alon# rather nicely %ith the results you %ould

predict on the basis o$ his span $or octal di#its. e could remember 1 octal di#its. ith the :1

recodin#, these 1 chunks %ere %orth ? binary di#its. ith the :1 recodin# they %ere %orth < binarydi#its. ith the ?:1 and :1 recodin#s, they %ere %orth about ?; binary di#its.

!t is a little dramatic to %atch a person #et ?; binary di#its in a ro% and then repeat them back %ithout

error. o%e"er, i$ you think o$ this merely as [p. 9] a mnemonic trick $or e+tendin# the memory span,you %ill miss the more important point that is implicit in nearly all such mnemonic de"ices. 'he point is

that recodin# is an e+tremely po%er$ul %eapon $or increasin# the amount o$ in$ormation that %e can

deal %ith. !n one $orm or another %e use recodin# constantly in our daily beha"ior.

!n my opinion the most customary kind o$ recodin# that %e do all the time is to translate into a "erbal

code. hen there is a story or an ar#ument or an idea that %e %ant to remember, %e usually try torephrase it in our o%n %ords. hen %e %itness some e"ent %e %ant to remember, %e make a "erbal


13/15

description o$ the e"ent and then remember our

"erbali(ation. Hpon recall %e recreate by secondaryelaboration the details that seem consistent %ith the

particular "erbal recodin# %e happen to ha"e made. 'he

%ell-kno%n e+periment by =armichael, o#an, and alter45 on the in$luence that names ha"e on the recall o$ "isual

$i#ures is one demonstration o$ the process.

'he inaccuracy o$ the testimony o$ eye%itnesses is %ellkno%n in le#al psycholo#y, but the distortions o$ testimony

are not random - they $ollo% naturally $rom the particularrecodin# that the %itness used, and the particular recodin#

he used depends upon his %hole li$e history. 3ur lan#ua#e is

tremendously use$ul $or repacka#in# material into a $e%

chunks rich in in$ormation. ! suspect that ima#ery is a $ormo$ recodin#, too, but ima#es seem much harder to #et at

operationally and to study e+perimentally than the more

symbolic kinds o$ recodin#.

!t seems probable that e"en memori(ation can be studied in these terms. 'he process o$ memori(in# maybe simply the $ormation o$ chunks, or #roups o$ items that #o to#ether, until there are $e% enou#h

chunks so that %e can recall all the items. 'he %ork by Cous$ield and =ohen 45 on the occurrence o$clusterin# in the recall o$ %ords is especially interestin# in this respect.

Summary

! ha"e come to the end o$ the data that ! %anted to present, so ! %ould like no% to make some

summari(in# remarks.

First, the span o$ absolute &ud#ment and the span o$ immediate memory impose se"ere limitations on the

amount o$ in$ormation that %e are able to recei"e, process, and remember. Cy or#ani(in# the stimulusinput simultaneously into se"eral dimensions and successi"ely into a se)uence o$ chunks, %e mana#e tobreak 4or at least stretch5 this in$ormational bottleneck.

econd, the process o$ recodin# is a "ery important one in human psycholo#y and deser"es much more

e+plicit attention than it has recei"ed. !n particular, the kind o$ lin#uistic recodin# that people do seemsto me to be the "ery li$eblood o$ the thou#ht processes. Decodin# procedures are a constant concern to

clinicians, social psycholo- [p. 9


14/15

can be use$ul in the study o$ concept $ormation. 2 lot o$ )uestions that seemed $ruitless t%enty or thirty

years a#o may no% be %orth another look. !n $act, ! $eel that my story here must stop &ust as it be#ins to#et really interestin#.

2nd $inally, %hat about the ma#ical number se"en@ hat about the se"en %onders o$ the %orld, these"en seas, the se"en deadly sins, the se"en dau#hters o$ 2tlas in the >leiades, the se"en a#es o$ man,

the se"en le"els o$ hell, the se"en primary colors, the se"en notes o$ the musical scale, and the se"en

days o$ the %eek@ hat about the se"en-point ratin# scale, the se"en cate#ories $or absolute &ud#ment,

the se"en ob&ects in the span o$ attention, and the se"en di#its in the span o$ immediate memory@ For thepresent ! propose to %ithhold &ud#ment. >erhaps there is somethin# deep and pro$ound behind all these

se"ens, somethin# &ust callin# out $or us to disco"er it. Cut ! suspect that it is only a pernicious,>ytha#orean coincidence.

.ootnotes

[1] 'his paper %as $irst read as an !n"ited 2ddress be$ore the *astern >sycholo#ical 2ssociation in

>hiladelphia on 2pril 1, 19. >reparation o$ the paper %as supported by the ar"ard >sycho-2coustic

aboratory under =ontract ori-< bet%een ar"ard Hni"ersity and the 3$$ice o$ a"al Desearch,H.. a"y 4>ro&ect D1?-;1, Deport >D-1?5. Deproduction $or any purpose o$ the H..

Bo"ernment is permitted.

-eferences

415 Ceebe-=enter, J. B., Do#ers, M. ., K 36=onnell, A. . 'ransmission o$ in$ormation about sucrose

and saline solutions throu#h the sense o$ taste.& Psychol., 19, 9, 1-1


15/15

41;5 ayes, J. D. M. Memory span $or se"eral "ocabularies as a $unction o$ "ocabulary si(e. !n

1uarterly Progress Re"ort, =ambrid#e, Mass.: 2coustics aboratory, Massachusetts !nstitute o$'echnolo#y. Jan - June 19.

4115 Jakobson, D., Fant, =. B. M., K alle, M.Preli$inaries to s"eech analysis. =ambrid#e, Mass.:2coustics aboratory, Massachusetts !nstitute o$ 'echnolo#y, 19. 4'ech. Dep. o. 1.5

415 Eau$man, *. ., ord, M. ., Deese, '. ., K 0olkmann, J. 'he discrimination o$ "isual number.

*$er & Psychol., 19?9,

The Magical Numbers Seven, Plus or Minus Two

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