VOL. 63, No. 1

MARCH, 1956

THE PSYCHOLOGICAL REVIEW
THE MAGICAL NUMBER SEVEN, PLUS OR MINUS TWO:SOME LIMITS ON OUR CAPACITY FOR
PROCESSING INFORMATION 1
GEORGE A. MILLER
Harvard University

My problem is that I have been persecuted by an integer. For seven years
this number has followed me around, has
intruded in my most private data, andhas assaulted me from the pages of ourmost public journals. This number assumes a variety of disguises, being sometimes a little larger and sometimes alittle smaller than usual, but neverchanging so much as to be unrecognizable. The persistence with which thisnumber plagues me is far more thana random accident. There is, to quotea famous senator, a design behind it,some pattern governing its appearances.
Either there really is something unusual
about the number or else I am sufferingfrom delusions of persecution.
I shall begin my case history by telling you about some experiments that
tested how accurately people can assign
numbers to the magnitudes of various
aspects of a stimulus. In the traditional language of psychology thesewould be called experiments in absolutejudgment. Historical accident, however, has decreed that they should haveanother name. We now call them experiments on the capacity of people totransmit information. Since these experiments would not have been done
without the appearance of information
theory on the psychological scene, andsince the results are analyzed in termsof the concepts of information theory,
I shall have to preface my discussion
with a few remarks about this theory.

INFORMATION MEASUREMENT
The "amount of information" is exactly the same concept that we havetalked about for years under the nameof "variance." The equations are different, but if we hold tight to the ideathat anything that increases the variance also increases the amount of information we cannot go far astray.
The advantages of this new wayof talking about variance are simple
enough. Variance is always stated interms of the unit of measurement—
1This paper was first read as an Invited inches, pounds, volts, etc.—whereas theAddress before the Eastern Psychological Association in Philadelphia on April IS, 19SS. amount of information is a dimensionPreparation of the paper was supported by less quantity. Since the information in
the Harvard Psycho-Acoustic Laboratory un- a discrete statistical distribution does
der Contract NSori-76 between Harvard Uni- not depend upon the unit of measureversity and the Office of Naval Research, U. S. ment, we can extend the concept toNavy (Project NR142-201, Report PNR-174).
Reproduction for any purpose of the U. S. situations where we have no metric andwe would not ordinarily think of using
Government is permitted.
81

GEORGE A. MILLER
the variance. And it also enables us tocompare results obtained in quite different experimental situations where it
would be meaningless to compare variances based on different metrics. So
there are some good reasons for adopting the newer concept.
The similarity of variance and amountof information might be explained thisway: When we have a large variance,
we are very ignorant about what is going to happen. If we are very ignorant,
then when we make the observation it
gives us a lot of information. On theother hand, if the variance is very small,
we know in advance how our observation must come out, so we get little information from making the observation.
If you will now imagine a communication system, you will realize that
there is a great deal of variability about
what goes into the system and also agreat deal of variability about whatcomes out. The input and the output
can therefore be described in terms of
their variance (or their information).
If it is a good communication system,
however, there must be some systematic relation between what goes in andwhat comes out. That is to say, theoutput will depend upon the input, or
will be correlated with the input. If wemeasure this correlation, then we cansay how much of the output variance isattributable to the input and how muchis due to random fluctuations or "noise"
introduced by the system during transmission. So we see that the measureof transmitted information is simply ameasure of the input-output correlation.
There are two simple rules to follow.
Whenever I refer to "amount of information," you will understand "variance." And whenever I refer to "amountof transmitted information," you will
understand "covariance" or "correlation."
The situation can be described graphically by two partially overlapping cir-cles. Then the left circle can be taken
to represent the variance of the input,
the right circle the variance of the output, and the overlap the covariance of
input and output. I shall speak of theleft circle as the amount of input information, the right circle as the amountof output information, and the overlap
as the amount of transmitted information.
In the experiments on absolute judgment, the observer is considered to be
a communication channel. Then theleft circle would represent the amountof information in the stimuli, the rightcircle the amount of information in his
responses, and the overlap the stimulusresponse correlation as measured by theamount of transmitted information. The
experimental problem is to increase theamount of input information and tomeasure the amount of transmitted information. If the observer's absolutejudgments are quite accurate, thennearly all of the input information willbe transmitted and will be recoverablefrom his responses. If he makes errors,then the transmitted information may
be considerably less than the input. We
expect that, as we increase the amountof input information, the observer will
begin to make more and more errors;
we can test the limits of accuracy of his
absolute judgments. If the human observer is a reasonable kind of communication system, then when we increase
the amount of input information thetransmitted information will increase at
first and will eventually level off at some
asymptotic value. This asymptotic value
we take to be the channel capacity ofthe observer: it represents the greatest
amount of information that he can give
us about the stimulus on the basis ofan absolute judgment. The channel capacity is the upper limit on the extent
to which the observer can match his responses to the stimuli we give him.
Now just a brief word about the bit

THE MAGICAL NUMBER SEVEN
and we can begin to look at some data.
One bit of information is the amount of
information that we need to make adecision between two equally likely alternatives. If we must decide whether
a man is less than six feet tall or morethan six feet tall and if we know thatthe chances are SO-SO, then we need
one bit of information. Notice that
this unit of information does not referin any way to the unit of length that
we use—feet, inches, centimeters, etc.
However you measure the man's height,
we still need just one bit of information.
Two bits of information enable us todecide among four equally likely alternatives. Three bits of information enable us to decide among eight equally
likely alternatives. Four bits of information decide among 16 alternatives,
five among 32, and so on. That is tosay, if there are 32 equally likely alternatives, we must make five successive
binary decisions, worth one bit each, before we know which alternative is correct. So the general rule is simple:
every time the number of alternatives
is increased by a factor of two, one bitof information is added.
There are two ways we might increase the amount of input information.
We could increase the rate at which we
give information to the observer, so thatthe amount of information per unit time
would increase. Or we could ignore thetime variable completely and increase
the amount of input information byincreasing the number of alternative
stimuli. In the absolute judgment experiment we are interested in the second
alternative. We give the observer as
much time as he wants to make his response; we simply increase the numberof alternative stimuli among which hemust discriminate and look to see where
confusions begin to occur. Confusions
will appear near the point that we arecalling his "channel capacity."83

ABSOLUTE JUDGMENTS OF UNIDIMENSIONAL STIMULI
Now let us consider what happens
when we make absolute judgments oftones. Pollack (17) asked listeners to
identify tones by assigning numerals tothem. The tones were different with respect to frequency, and covered therange from 100 to 8000 cps in equal
logarithmic steps. A tone was sounded
and the listener responded by giving a
numeral. After the listener had madehis response he was told the correct
identification of the tone.
When only two or three tones wereused the listeners never confused them.
With four different tones confusions
were quite rare, but with five or moretones confusions were frequent. With
fourteen different tones the listeners
made many mistakes.
These data are plotted in Fig. 1.
Along the bottom is the amount of input information in bits per stimulus.
As the number of alternative tones wasincreased from 2 to 14, the input information increased from 1 to 3.8 bits. On
the ordinate is plotted the amount of

_2.5
BITS
a

UJ

2

en
z

PITCHES
100-8000 CPS
0

1

2

3

4

5

INPUT INFORMATION

FIG. 1. Data from Pollack (17, 18) on theamount of information that is transmitted bylisteners who make absolute judgments ofauditory pitch. As the amount of input information is increased by increasing from 2
to 14 the number of different pitches to be
judged, the amount of transmitted information approaches as its upper limit a channel
capacity of about 2.S bits per judgment.

84

GEORGE A. MILLER

transmitted information. The amountof transmitted information behaves inmuch the way we would expect a communication channel to behave; the transmitted information increases linearly up
to about 2 bits and then bends off toward an asymptote at about 2.5 bits.
This value, 2.5 bits, therefore, is what
we are calling the channel capacity of
the listener for absolute judgments ofpitch.
So now we have the number 2.5
bits. What does it mean? First, notethat 2.5 bits corresponds to about six
equally likely alternatives. The resultmeans that we cannot pick more thansix different pitches that the listener will
never confuse. Or, stated slightly differently, no matter how many alternative tones we ask him to judge, the best
we can expect him to do is to assign
them to about six different classes without error. Or, again, if we know that
there were N alternative stimuli, thenhis judgment enables us to narrow down
the particular stimulus to one out of
N/6.
Most people are surprised that the
number is as small as six. Of course,
there is evidence that a musically sophisticated person with absolute pitch
can identify accurately any one of 50
or 60 different pitches. Fortunately, I
do not have time to discuss these remarkable exceptions. I say it is fortunate because I do not know how to
explain their superior performance. So
I shall stick to the more pedestrian factthat most of us can identify about one
out of only five or six pitches before we
begin to get confused.
It is interesting to consider that psychologists have been using seven-point
rating scales for a long time, on theintuitive basis that trying to rate into
finer categories does not really add muchto the usefulness of the ratings. Pollack's results indicate that, at least for
pitches, this intuition is fairly sound.

1

2

3

4

5

INPUT INFORMATION

FIG. 2. Data from Garner (7) on the channel capacity for absolute judgments of auditory loudness.

Next you can ask how reproducible
this result is. Does it depend on thespacing of the tones or the various conditions of judgment? Pollack varied
these conditions in a number of ways.
The range of frequencies can be changed
by a factor of about 20 without changing the amount of information transmitted more than a small percentage.
Different groupings of the pitches decreased the transmission, but the loss
was small. For example, if you candiscriminate five high-pitched tones in
one series and five low-pitched tones inanother series, it is reasonable to expect that you could combine all ten intoa single series and still tell them allapart without error. When you try it,
however, it does not work. The channel capacity for pitch seems to be about
six and that is the best you can do.
While we are on tones, let us looknext at Garner's (7) work on loudness.
Garner's data for loudness are summarized in Fig. 2. Garner went to sometrouble to get the best possible spacingof his tones over the intensity range
from 15 to 110 db. He used 4, 5, 6, 7,
10, and 20 different stimulus intensities.
The results shown in Fig. 2 take into
account the differences among subjects
and the sequential influence of the immediately preceding judgment. Again
we find that there seems to be a limit.

THE MAGICAL NUMBER SEVENTASTJUDGMENTS OF SALINECONCENTRATION
1

2

3

4

INPUT INFORMATION

FIG. 3. Data from Beebe-Center, Rogers,
and O'Connell (1) on the channel capacity for
absolute judgments of saltiness.

85

and Garner (8) asked observers to interpolate visually between two scale
markers. Their results are shown in
Fig. 4. They did the experiment in
two ways. In one version they let theobserver use any number between zeroand 100 to describe the position, although they presented stimuli at only
5, 10, 20, or SO different positions. The
results with this unlimited response
technique are shown by the filled circles
on the graph. In the other version theobservers were limited in their responses to reporting just those stimulus values that were possible. That is
to say, in the second version the number of different responses that the observer could make was exactly the sameas the number of different stimuli thatthe experimenter might present. The
results with this limited response technique are shown by the open circles on
the graph. The two functions are sosimilar that it seems fair to conclude
that the number of responses available
to the observer had nothing to do withthe channel capacity of 3.2S bits.
The Hake-Garner experiment has been
repeated by Coonan and Klemmer. Although they have not yet publishedtheir results, they have given me permission to say that they obtained channel capacities ranging from 3.2 bits for

The channel capacity for absolute judgments of loudness is 2.3 bits, or about
five perfectly discriminable alternatives.
Since these two studies were done indifferent laboratories with slightly different techniques and methods of analysis, we are not in a good position toargue whether five loudnesses is significantly different from six pitches. Probably the difference is in the right direction, and absolute judgments of pitch
are slightly more accurate than absolutejudgments of loudness. The important
point, however, is that the two answers
are of the same order of magnitude.
The experiment has also been done
for taste intensities. In Fig. 3 are theresults obtained by Beebe-Center, Rogers, and O'Connell (1) for absolutejudgments of the concentration of salt
-3.25
. BITS
solutions. The concentrations ranged
from 0.3 to 34.7 gm. NaCl per 100 8z
cc. tap water in equal subjective steps.
They used 3, 5, 9, and 17 different concentrations. The channel capacity is
1.9 bits, which is about four distinct
100
concentrations. Thus taste intensities
0
1
2
3
4
5
6
seem a little less distinctive than auditory stimuli, but again the order of
INPUT INFORMATION
magnitude is not far off.
FIG. 4. Data from Hake and Garner (8)On the other hand, the channel ca- on the channel capacity for absolute judgpacity for judgments of visual position ments of the position of a pointer in a linear
seems to be significantly larger. Hake interval.

86

GEORGE A. MILLER

very short exposures of the pointer position to 3.9 bits for longer exposures.
These values are slightly higher than
Hake and Garner's, so we must conclude that there'are between 10 and IS
distinct positions along a linear interval. This is the largest channel capacity that has been measured for any
unidimensional variable.
At the present time these four experiments on absolute judgments of simple,
unidimensional stimuli are all that haveappeared in the psychological journals.
However, a great deal of work on otherstimulus variables has not yet appearedin the journals. For example, Eriksen
and Hake (6) have found that thechannel capacity for judging the sizesof squares is 2.2 bits, or about fivecategories, under a wide range of experimental conditions. In a separate
experiment Eriksen (5) found 2.8 bits
for size, 3.1 bits for hue, and 2.3 bits
for brightness. Geldard has measured
the channel capacity for the skin byplacing vibrators on the chest region.
A good observer can identify about four
intensities, about five durations, andabout seven locations.
One of the most active groups in thisarea has been the Air Force Operational
Applications Laboratory. Pollack has
been kind enough to furnish me withthe results of their measurements forseveral aspects of visual displays. They
made measurements for area and forthe curvature, length, and direction oflines. In one set of experiments theyused a very short exposure of the stimulus—%0 second—and then they repeated the measurements with a 5second exposure. For area they got
2.6 bits with the short exposure and2.7 bits with the long exposure. For
the length of a line they got about 2.6
bits with the short exposure and about
3.0 bits with the long exposure. Direction, or angle of inclination, gave 2.8
bits for the short exposure and 3.3 bits

for the long exposure. Curvature wasapparently harder to judge. When thelength of the arc was constant, the result at the short exposure duration was
2.2 bits, but when the length of thechord was constant, the result was only
1.6 bits. This last value is the lowest
that anyone has measured to date. I
should add, however, that these values
are apt to be slightly too low because
the data from all subjects were pooled
before the transmitted information wascomputed.
Now let us see where we are. First,
the channel capacity does seem to be avalid notion for describing human observers. Second, the channel capacities
measured for these unidimensional variables range from 1.6 bits for curvature
to 3.9 bits for positions in an interval.
Although there is no question that thedifferences among the variables are real
and meaningful, the more impressive
fact to me is their considerable similarity. If I take the best estimates I
can get of the channel capacities for all
the stimulus variables I have mentioned,
the mean is 2.6 bits and the standard
deviation is only 0.6 bit. In terms of
distinguishable alternatives, this mean
corresponds to about 6.5 categories, one
standard deviation includes from 4 to
10 categories, and the total range isfrom 3 to IS categories. Considering
the wide variety of different variables
that have been studied, I find this tobe a remarkably narrow range.
There seems to be some limitation
built into us either by learning or by
the design of our nervous systems, alimit that keeps our channel capacities
in this general range. On the basis of
the present evidence it seems safe to
say that we possess a finite and rather
small capacity for making such unidimensional judgments and that this capacity does not vary a great deal fromone simple sensory attribute to another.

THE MAGICAL NUMBER SEVEN
ABSOLUTE JUDGMENTS OF MULTIDIMENSIONAL STIMULI
You may have noticed that I havebeen careful to say that this magicalnumber seven applies to one-dimensional
judgments. Everyday experience teaches
us that we can identify accurately any
one of several hundred faces, any oneof several thousand words, any one ofseveral thousand objects, etc. The story
certainly would not be complete if we
stopped at this point. We must havesome understanding of why the onedimensional variables we judge in thelaboratory give results so far out ofline with what we do constantly in our
behavior outside the laboratory. A possible explanation lies in the number of
independently variable attributes of thestimuli that are being judged. Objects,
faces, words, and the like differ fromone another in many ways, whereas thesimple stimuli we have considered thus
far differ from one another in only one
respect.
Fortunately, there are a few data on
what happens when we make absolutejudgments of stimuli that differ fromone another in several ways. Let us
look first at the results Klemmer andFrick (13) have reported for the absolute judgment of the position of a dot
in a square. In Fig. 5 we see their re-

POINTS IN A SQUARENO GRID.03 SEC. EXPOSURE
3

4

5

6

7

INPUT INFORMATION

FIG. S. Data from Klemmer and Frick (13)on the channel capacity for absolute judgments of the position of a dot in a square.

87

suits. Now the channel capacity seems
to have increased to 4.6 bits, whichmeans that people can identify accurately any one of 24 positions in thesquare.
The position of a dot in a square isclearly a two-dimensional proposition.
Both its horizontal and its vertical position must be identified. Thus it seems
natural to compare the 4.6-bit capacity
for a square with the 3.25-bit capacity
for the position of a point in an interval. The point in the square requires
two judgments of the interval type. If
we have a capacity of 3.2S bits for estimating intervals and we do this twice,
we should get 6.5 bits as our capacity
for locating points in a square. Adding
the second independent dimension gives
us an increase from 3.2S to 4.6, but it
falls short of the perfect addition thatwould give 6.5 bits.
Another example is provided by BeebeCenter, Rogers, and O'Connell. Whenthey asked people to identify both thesaltiness and the sweetness of solutions
containing various concentrations of salt
and sucrose, they found that the channel capacity was 2.3 bits. Since the capacity for salt alone was 1.9, we might
expect about 3.8 bits if the two aspectsof the compound stimuli were judged
independently. As with spatial locations, the second dimension adds a little
to the capacity but not as much as it
conceivably might.
A third example is provided by Pollack (18), who asked listeners to judge
both the loudness and the pitch of puretones. Since pitch gives 2.S bits andloudness gives 2.3 bits, we might hopeto get as much as 4.8 bits for pitch andloudness together. Pollack obtained 3.1
bits, which again indicates that thesecond dimension augments the channel
capacity but not so much as it might.
A fourth example can be drawn fromthe work of Halsey and Chapanis (9)on confusions among colors of equal

88

GEORGE A. MILLER

luminance. Although they did not analyze their results in informational terms,
they estimate that there are about 11 to
IS identifiable colors, or, in our terms,
about 3.6 bits. Since these colors variedin both hue and saturation, it is probably correct to regard this as a twodimensional judgment. If we compare
this with Eriksen's 3.1 bits for hue(which is a questionable comparison todraw), we again have something lessthan perfect addition when a second
dimension is added.
It is still a long way, however, fromthese two-dimensional examples to themultidimensional stimuli provided by
faces, words, etc. To fill this gap wehave only one experiment, an auditory
study done by Pollack and Picks (19).
They managed to get six different acoustic variables that they could change:
frequency, intensity, rate of interruption, on-time fraction, total duration,and spatial location. Each one of thesesix variables could assume any one offive different values, so altogether there
were S8, or 15,625 different tones thatthey could present. The listeners madea separate rating for each one of thesesix dimensions. Under these conditions
the transmitted information was 7.2 bits,
which corresponds to about 150 different categories that could be absolutely
identified without error. Now we are
beginning to get up into the range that
ordinary experience would lead us to
expect.
Suppose that we plot these data,fragmentary as they are, and make aguess about how the channel capacitychanges with the dimensionality of thestimuli. The result is given in Fig. 6.
In a moment of considerable daring I
sketched the dotted line to indicate
roughly the trend that the data seemed
to be taking.
Clearly, the addition of independently
variable attributes to the stimulus increases the channel capacity, but at a

I

2

3

4

5

6

7

NUMBER OF VARIABLE ASPECTS

FIG. 6. The general form of the relation between channel capacity and the number of independently variable attributes of the stimuli.

decreasing rate. It is interesting tonote that the channel capacity is increased even when the several variables
are not independent. Eriksen (5) reports that, when size, brightness, andhue all vary together in perfect correlation, the transmitted information is 4.1
bits as compared with an average ofabout 2.7 bits when these attributes arevaried one at a time. By confounding
three attributes, Eriksen increased thedimensionality of the input without increasing the amount of input information ; the result was an increase in channel capacity of about the amount thatthe dotted function in Fig. 6 would lead
us to expect.
The point seems to be that, as we
add more variables to the display, we
increase the total capacity, but we decrease the accuracy for any particularvariable. In other words, we can makerelatively crude judgments of several
things simultaneously.
We might argue that in the course ofevolution those organisms were mostsuccessful that were responsive to thewidest range of stimulus energies intheir environment. In order to survive
in a constantly fluctuating world, it was
better to have a little information about
a lot of things than to have a lot of information about a small segment of theTHE MAGICAL NUMBER SEVEN

89

environment. If a compromise was nec- find out. There is a limit, however, atessary, the one we seem to have made is about eight or nine distinctive features
clearly the more adaptive.
in every language that has been studied,
Pollack and Picks's results are very and so when we talk we must resort tostrongly suggestive of an argument that still another trick for increasing our
linguists and phoneticians have been channel capacity. Language uses semaking for some time (11). According quences of phonemes, so we make sevto the linguistic analysis of the sounds eral judgments successively when weof human speech, there are about eight listen to words and sentences. That isor ten dimensions—the linguists call to say, we use both simultaneous andthem distinctive features—that distin- successive discriminations in order toguish one phoneme from another. These expand the rather rigid limits imposed
distinctive features are usually binary, by the inaccuracy of our absolute judgor at most ternary, in nature. For ex- ments of simple magnitudes.
ample, a binary distinction is made beThese multidimensional judgments are
tween vowels and consonants, a binary strongly reminiscent of the abstraction
decision is made between oral and nasal experiment of Kulpe (14). As you may
consonants, a ternary decision is made remember, Kiilpe showed that observers
among front, middle, and back pho- report more accurately on an attributenemes, etc. This approach gives us for which they are set than on attributes
quite a different picture of speech per- for which they are not set. For examception than we might otherwise obtain ple, Chapman (4) used three different
from our studies of the speech spectrum attributes and compared the results oband of the ear's ability to discriminate
tained when the. observers were inrelative differences among pure tones.
structed before the tachistoscopic presI am personally much interested in thisentation with the results obtained when
new approach (15), and I regret thatthey were not told until after the presthere is not time to discuss it here.
entation which one of the three attriIt was probably with this linguistic
theory in mind that Pollack and Picks butes was to be reported. When the
conducted a test on a set of tonal instruction was given in advance, thestimuli that varied in eight dimensions, judgments were more accurate. Whenbut required only a binary decision on the instruction was given afterwards,
each dimension. With these tones they the subjects presumably had to judge allmeasured the transmitted information three attributes in order to report on
at 6.9 bits, or about 120 recognizable any one of them and the accuracy was
kinds of sounds. It is an intriguing correspondingly lower. This is in comquestion, as yet unexplored, whether plete accord with the results we have
one can go on adding dimensions in- just been considering, where the accuracy of judgment on each attributedefinitely in this way.
In human speech there is clearly a decreased as more dimensions werelimit to the number of dimensions that added. The point is probably obvious,
we use. In this instance, however, it is but I shall make it anyhow, that the
not known whether the limit is imposed abstraction experiments did not demonby the nature of the perceptual ma- strate that people can judge only one
chinery that must recognize the sounds attribute at a time. They merely showed
or by the nature of the speech ma- what seems quite reasonable, that peochinery that must produce them. Some- ple are less accurate if they must judge
body will have to do the experiment to more than one attribute simultaneously.

90

GEORGE A. MILLER
SUBITIZING

I cannot leave this general area without mentioning, however briefly, the experiments conducted at Mount Holyoke
College on the discrimination of number (12). In experiments by Kaufman,
Lord, Reese, and Volkmann random
patterns of dots were flashed on a screen
for y5 of a second. Anywhere from 1
to more than 200 dots could appear in
the pattern. The subject's task was toreport how many dots there were.
The first point to note is that on patterns containing up to five or six dots
the subjects simply did not make errors.
The performance on these small numbers of dots was so different from theperformance with more dots that it wasgiven a special name. Below seven thesubjects were said to subitize; above
seven they were said to estimate. Thisis, as you will recognize, what we once
optimistically called "the span of attention."
This discontinuity at seven is, ofcourse, suggestive. Is this the same
basic process that limits our unidimensional judgments to about seven categories? The generalization is tempting,but not sound in my opinion. The dataon number estimates have not been analyzed in informational terms; but on
the basis of the published data I would
guess that the subjects transmitted
something more than four bits of information about the number of dots.
Using the same arguments as before, wewould conclude that there are about 20
or 30 distinguishable categories of numerousness. This is considerably more
information than we would expect toget from a unidimensional display. It
is, as a matter of fact, very much like a
two-dimensional display. Although thedimensionality of the random dot patterns is not entirely clear, these results
are in the same range as Klemmer andFrick's for their two-dimensional display of dots in a square. Perhaps thetwo dimensions of numerousness arearea and density. When the subject
can subitize, area and density may not
be the significant variables, but when
the subject must estimate perhaps they
are significant. In any event, the comparison is not so simple as it mightseem at first thought.
This is one of the ways in which themagical number seven has persecutedme. Here we have two closely related
kinds of experiments, both of which
point to the significance of the number
seven as a limit on our capacities. And
yet when we examine the matter moreclosely, there seems to be a reasonablesuspicion that it is nothing more thana coincidence.
THE SPAN OF IMMEDIATE MEMOKY
Let me summarize the situation inthis way. There is a clear and definite
limit to the accuracy with which we canidentify absolutely the magnitude ofa unidimensional stimulus variable. I
would propose to call this limit the
span of absolute judgment, and I
maintain that for unidimensional judgments this span is usually somewherein the neighborhood of seven. We are
not completely at the mercy of thislimited span, however, because we havea variety of techniques for getting
around it and increasing the accuracyof our judgments. The three most important of these devices are (a) tomake relative rather than absolute judgments; or, if that is not possible, (b)to increase the number of dimensions
along which the stimuli can differ; or(c) to arrange the task in such a waythat we make a sequence of several absolute judgments in a row.
The study of relative judgments is
one of the oldest topics in experimental
psychology, and I will not pause to review it now. The second device, increasing the dimensionality, we have justconsidered. It seems that by adding

THE MAGICAL NUMBER SEVEN
more dimensions and requiring crude,
binary, yes-no judgments on each attribute we can extend the span of absolute judgment from seven to at least
ISO. Judging from our everyday behavior, the limit is probably in thethousands, if indeed there is a limit. In
my opinion, we cannot go on compounding dimensions indefinitely. I suspectthat there is also a span of perceptual
dimensionality and that this span issomewhere in the neighborhood of ten,but I must add at once that there is noobjective evidence to support this suspicion. This is a question sadly needing experimental exploration.
Concerning the third device, the useof successive judgments, I have quite abit to say because this device introduces
memory as the handmaiden of discrimination. And, since mnemonic processes
are at least as complex as are perceptual
processes, we can anticipate that theirinteractions will not be easily disentangled.
Suppose that we start by simply extending slightly the experimental procedure that we have been using. Up
to this point we have presented a single
stimulus and asked the observer to nameit immediately thereafter. We can extend this procedure by requiring the observer to withhold his response until wehave given him several stimuli in succession. At the end of the sequence ofstimuli he then makes his response. We
still have the same sort of input-output situation that is required for themeasurement of transmitted information. But now we have passed froman experiment on absolute judgment towhat is traditionally called an experiment on immediate memory.
Before we look at any data on this
topic I feel I must give you a word of
warning to help you avoid some obvious associations that can be confusing.
Everybody knows that there is a finite
span of immediate memory and that for91

a lot of different kinds of test materials
this span is about seven items in length.
I have just shown you that there is aspan of absolute judgment that can distinguish about seven categories and that
there is a span of attention that willencompass about six objects at a glance.
What is more natural than to think that
all three of these spans are different aspects of a single underlying process?
And that is a fundamental mistake, asI shall be at some pains to demonstrate.
This mistake is one of the malicious
persecutions that the magical number
seven has subjected me to.
My mistake went something like this.
We have seen that the invariant feature in the span of absolute judgment
is the amount of information that theobserver can transmit. There is a real
operational similarity between the absolute judgment experiment and theimmediate memory experiment. If immediate memory is like absolute judgment, then it should follow that the invariant feature in the span of immediate
memory is also the amount of information that an observer can retain. If theamount of information in the span of
immediate memory is a constant, thenthe span should be short when the individual items contain a lot of information and the span should be long when
the items contain little information. For
example, decimal digits are worth 3.3
bits apiece. We can recall about sevenof them, for a total of 23 bits of information. Isolated English words areworth about 10 bits apiece. If the total
amount of information is to remain
constant at 23 bits, then we should be
able to remember only two or threewords chosen at random. In this way
I generated a theory about how the spanof immediate memory should vary as afunction of the amount of information
per item in the test materials.
The measurements of memory span in
the literature are suggestive on this92

GEORGE A. MILLER

question, but not definitive. And so it
was necessary to do the experiment tosee. Hayes (10) tried it out with fivedifferent kinds of test materials: binary
digits, decimal digits, letters of the alphabet, letters plus decimal digits, andwith 1,000 monosyllabic words. The
lists were read aloud at the rate of one
item per second and the subjects had asmuch time as they needed to give their
responses. A procedure described by
Woodworth (20) was used to score theresponses.
The results are shown by the filled
circles in Fig. 7. Here the dotted line
indicates what the span should havebeen if the amount of information in the
span were constant. The solid curves
represent the data. Hayes repeated theexperiment using test vocabularies ofdifferent sizes but all containing only
English monosyllables (open circles inFig. 7). This more homogeneous test
material did not change the picture significantly. With binary items the span
is about nine and, although it drops toabout five with monosyllabic English
words, the difference is far less thanthe hypothesis of constant information
would require.

P-

DU

1
£ 40
O
2
2
z

30

tft
2
K
—

20

BINARYDIGITSJ[-BETTERS|000.-LETTERS a DIGITS WORDS
U
1>i* CONSTANT\INFORMATION

\
\

\
»
\
\
\

fc
o:
m
5
3
z

DECIMAL
DIGITSi

10 • -y,

s

>
t* ^<
^~
- J —nWOROS-^~~I'_1O
10INFORMATION PER ITEM IN BITS

FIG. 7. Data from Hayes (10) on the spanof immediate memory plotted as a functionof the amount of information per item in thetest materials.

"

0

1

2

3

4

5

6

INFORMATION PER ITEM IN BITS

FIG. 8. Data from Pollack (16) on theamount of information retained after one
presentation plotted as a function of theamount of information per item in the test
materials.

There is nothing wrong with Hayes's
experiment, because Pollack (16) repeated it much more elaborately andgot essentially the same result. Pollack took pains to measure the amountof information transmitted and did not
rely on the traditional procedure forscoring the responses. His results areplotted in Fig. 8. Here it is clear thatthe amount of information transmitted
is not a constant, but increases almost
linearly as the amount of information
per item in the input is increased.
And so the outcome is perfectly clear.
In spite of the coincidence that themagical number seven appears in both
places, the span of absolute judgment
and the span of immediate memory arequite different kinds of limitations thatare imposed on our ability to process
information. Absolute judgment is limited by the amount of information. Immediate memory is limited by the number of items. In order to capture this distinction in somewhat picturesque terms,
I have fallen into the custom of distinguishing between bits of informationand chunks of information. Then I cansay that the number of bits of information is constant for absolute judgment
and the number of chunks of informa-

THE MAGICAL NUMBER SEVEN

93

tlon is constant for immediate memory. achieved at different rates and overlap
The span of immediate memory seems each other during the learning process.
to be almost independent of the number I am simply pointing to the obviousof bits per chunk, at least over the fact that the dits and dahs are organrange that has been examined to date. ized by learning into patterns and that
The contrast of the terms bit and as these larger chunks emerge the
chunk also serves to highlight the fact amount of message that the operator
that we are not very definite about what can remember increases correspondingly.
constitutes a chunk of information. For In the terms I am proposing to use, theexample, the memory span of five words operator learns to increase the bits perthat Hayes obtained when each word chunk.
In the jargon of communication thewas drawn at random from a set of 1000
English monosyllables might just as ap- ory, this process would be called recedpropriately have been called a memory ing. The input is given in a code thatspan of IS phonemes, since each word contains many chunks with few bits per
had about three phonemes in it. Intui- chunk. The operator recedes the input
tively, it is clear that the subjects were into another code that contains fewer
recalling five words, not IS phonemes, chunks with more bits per chunk. Therebut the logical distinction is not im- are many ways to do this receding, but
mediately apparent. We are dealing probably the simplest is to group thehere with a process of organizing or input events, apply a new name to thegrouping the input into familiar units group, and then remember the new nameor chunks, and a great deal of learning rather than the original input events.
Since I am convinced that this prochas gone into the formation of theseess is a very general and important one
familiar units.
for psychology, I want to tell you about
RECODINGa demonstration experiment that should
In order to speak more precisely, make perfectly explicit what I am talktherefore, we must recognize the impor- ing about. This experiment was contance of grouping or organizing the in- ducted by Sidney Smith and was reput sequence into units or chunks. ported by him before the Eastern PsySince the memory span is a fixed num- chological Association in 1954.
ber of chunks, we can increase the numBegin with the observed fact that peober of bits of information that it con- ple can repeat back eight decimal digits,
tains simply by building larger and but only nine binary digits. Since there
larger chunks, each chunk containing is a large discrepancy in the amount of
information recalled in these two cases,
more information than before.
A man just beginning to learn radio- we suspect at once that a receding protelegraphic code hears each dit and dah cedure could be used to increase theas a separate chunk. Soon he is able span of immediate memory for binary
to organize these sounds into letters and digits. In Table 1 a method for groupthen he can deal with the letters as ing and renaming is illustrated. Alongchunks. Then the letters organize the top is a sequence of 18 binary digits,
themselves as words, which are still far more than any subject was able tolarger chunks, and he begins to hear recall after a single presentation. In
whole phrases. I do not mean that each the next line these same binary digits
step is a discrete process, or that pla- are grouped by pairs. Four possibleteaus must appear in his learning curve, pairs can occur: 00 is renamed 0, 01 isfor surely the levels of organization are renamed 1, 10 is renamed 2, and 11 isGEORGE A. MILLERTABLE 1
WAYS OP RECODING SEQUENCES OF BINARY DIGITS
Binary Digits (Bits)

1 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0

2:1 Chunks
Receding

10
2

3:1

ChunksReceding

101
5

4:1 Chunks
Receding

1010
10

5:1 Chunks
Receding

10100
20

10
2

renamed 3. That is to say, we recodefrom a base-two arithmetic to a basefour arithmetic. In the receded sequence there are now just nine digits toremember, and this is almost within the
span of immediate memory. In the nextline the same sequence of binary digits
is regrouped into chunks of three. Thereare eight possible sequences of three, so
we give each sequence a new name between 0 and 7. Now we have recededfrom a sequence of 18 binary digits
into a sequence of 6 octal digits, andthis is well within the span of immediate memory. In the last two lines thebinary digits are grouped by fours andby fives and are given decimal-digit
names from 0 to IS and from 0 to 31.
It is reasonably obvious that this kindof receding increases the bits per chunk,and packages the binary sequence into
a form that can be retained within the
span of immediate memory. So Smithassembled 20 subjects and measured
their spans for binary and octal digits.
The spans were 9 for binaries and 7 foroctals. Then he gave each receding
scheme to five of the subjects. They
studied the receding until they said
they understood it—for about S or 10
minutes. Then he tested their span forbinary digits again while they tried touse the receding schemes they hadstudied.

00
0

10
2
100
4

000
0

0010
2

01001
9

11
3

00
0

111
7

001
1

0111
7

0011
3

01
1

11001
25

11
3

10
2
110
6

10

110

The receding schemes increased theirspan for binary digits in every case.
But the increase was not as large as we
had expected on the basis of their spanfor octal digits. Since the discrepancy
increased as the receding ratio increased,
we reasoned that the few minutes the
subjects had spent learning the receding schemes had not been sufficient.
Apparently the translation from one
code to the other must be almost automatic or the subject will lose part of thenext group while he is trying to remember the translation of the last group.
Since the 4:1 and 5:1 ratios requireconsiderable study, Smith decided toimitate Ebbinghaus and do the experiment on himself. With Germanic patience he drilled himself on each receding successively, and obtained the results shown in Fig. 9. Here the data
follow along rather nicely with the results you would predict on the basis of
his span for octal digits. He could remember 12 octal digits. With the 2:1
receding, these 12 chunks were worth
24 binary digits. With the 3:1 receding they were worth 36 binary digits.
With the 4:1 and 5:1 recodings, they
were worth about 40 binary digits.
It is a little dramatic to watch a person get 40 binary digits in a row andthen repeat them back without error.
However, if you think of this merely asTHE MAGICAL NUMBER SEVEN
50
1
>

40

S

30

e

z 20

PREDICTEDFROM SPAN FOROCTAL DIGITS,*/NOBSERVED10
ONE HIGHLY PRACTICED SUBJECT

2:1

3:1

4:1

SM

RECOOINO RATIO

FIG. 9. The span of immediate memory forbinary digits is plotted as a function of thereceding procedure used. The predicted function is obtained by multiplying the span for
octals by 2, 3 and 3.3 for receding into base4, base 8, and base 10, respectively.

a mnemonic trick for extending thememory span, you will miss the more
important point that is implicit innearly all such mnemonic devices. The
point is that receding is an extremely
powerful weapon for increasing theamount of information that we candeal with. In one form or another we
use receding constantly in our daily
behavior.
In my opinion the most customarykind of receding that we do all the timeis to translate into a verbal code. When
there is a story or an argument or an
idea that we want to remember, we usually try to rephrase it "in our own
words." When we witness some event
we want to remember, we make a verbal
description of the event and then remember our verbalization. Upon recall
we recreate by secondary elaboration
the details that seem consistent withthe particular verbal recoding we happen to have made. The well-known experiment by Carmichael, Hogan, andWalter (3) on the influence that names
have on the recall of visual figures is
one demonstration of the process.
The inaccuracy of the testimony ofeyewitnesses is well known in legal psychology, but the distortions of testimony are not random—they follow
naturally from the particular recodingthat the witness used, and the particular recoding he used depends upon hiswhole life history. Our language is tremendously useful for repackaging material into a few chunks rich in information. I suspect that imagery is aform of recoding, too, but images seemmuch harder to get at operationally andto study experimentally than the more
symbolic kinds of recoding.
It seems probable that even memorization can be studied in these terms.
The process of memorizing may be simply the formation of chunks, or groups
of items that go together, until there
are few enough chunks so that we canrecall all the items. The work by Bousfield and Cohen (2) on the occurrenceof clustering in the recall of words is
especially interesting in this respect.
SUMMARY
I have come to the end of the datathat I wanted to present, so I would
like now to make some summarizing remarks.
First, the span of absolute judgment
and the span of immediate memory impose severe limitations on the amountof information that we are able to receive, process, and remember. By organizing the stimulus input simultaneously into several dimensions and successively into a sequence of chunks, wemanage to break (or at least stretch)
this informational bottleneck.
Second, the process of recoding is avery important one in human psychology and deserves much more explicit attention than it has received. In particular, the kind of linguistic recodingthat people do seems to me to be thevery lifeblood of the thought processes.
Recoding procedures are a constantconcern to clinicians, social psycholo-

96

GEORGE A. MILLER

gists, linguists, and anthropologists andyet, probably because receding is lessaccessible to experimental manipulation
than nonsense syllables or T mazes, thetraditional experimental psychologist hascontributed little or nothing to theiranalysis.
Nevertheless, experimental
techniques can be used, methods of recoding can be specified, behavioral indicants can be found. And I anticipate
that we will find a very orderly set of
relations describing what now seems an
uncharted wilderness of individual differences.
Third, the concepts and measuresprovided by the theory of information
provide a quantitative way of getting at
some of these questions. The theory
provides us with a yardstick for calibrating our stimulus materials and formeasuring the performance of our subjects. In the interests of communication I have suppressed the technical details of information measurement andhave tried to express the ideas in morefamiliar terms; I hope this paraphrase
will not lead you to think they are notuseful in research. Informational concepts have already proved valuable inthe study of discrimination and of language; they promise a great deal in the
study of learning and memory; and it
has even been proposed that they canbe useful in the study of concept formation. A lot of questions that seemed
fruitless twenty or thirty years ago may
now be worth another look. In fact, I
feel that my story here must stop justas it begins to get really interesting.
And finally, what about the magicalnumber seven? What about the sevenwonders of the world, the seven seas,
the seven deadly sins, the seven daughters of Atlas in the Pleiades, the sevenages of man, the seven levels of hell,
the seven primary colors, the seven notesof the musical scale, and the seven daysof the week? What about the sevenpoint rating scale, the seven categories

for absolute judgment, the seven objects in the span of attention, and theseven digits in the span of immediate
memory? For the present I propose towithhold judgment. Perhaps there issomething deep and profound behind allthese sevens, something just calling out
for us to discover it. But I suspectthat it is only a pernicious, Pythagorean
coincidence.
REFERENCES
1. BEEBE-CENTER, J. G., ROGERS, M. S., &O'CoNNELL, B. N. Transmission of information about sucrose and saline solutions through the sense of taste. J.
Psychol., 19SS, 39, 157-160.
2. BOUSFIELD, W. A., & COHEN, B. H. The
occurrence of clustering in the recall ofrandomly arranged words of different
frequencies-of-usage. /. gen. Psychol.,
1955, 52, 83-9S.
3. CARMICHAEL, L., HOGAN, H. P., & WALTER,
A. A. An experimental study of theeffect of language on the reproductionof visually perceived form. /. exp.
Psychol., 1932, IS, 73-86.
4. CHAPMAN, D. W. Relative effects of determinate and indeterminate Aufgaben.
Amer. J. Psychol, 1932, 44, 163-174.
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6. ERIKSEN, C. W., & HAKE, H. W. Absolute judgments as a function of thestimulus range and the number ofstimulus and response categories. /.
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7. GARNER, W. R. An informational analysis of absolute judgments of loudness.
/. exp. Psychol., 1953, 46, 373-380.
8. HAKE, H. W., & GARNER, W. R. The effect of presenting various numbers of
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9. HALSEY, R. M., & CHAPANIS, A. Chromaticity-confusion contours in a complex viewing situation. J. Opt. Soc.
Amer., 1954, 44, 442-4S4.
10. HAYES, J. R. M. Memory span for several vocabularies as a function of vocabulary size. In Quarterly Progress
Report, Cambridge, Mass.: Acoustics
Laboratory, Massachusetts Institute ofTechnology, Jan.-June, 1952.

THE MAGICAL NUMBER SEVEN
11. JAKOBSON, R,, FANT, C. G. M., & HALLE,
M, Preliminaries to speech analysis.
Cambridge, Mass.: Acoustics Laboratory, Massachusetts Institute of Technology, 19S2. (Tech. Rep. No. 13.)
12. KAUFMAN, E. L., LORD, M. W., REESE,
T. W., & VOLKMANN, J. The discrimination of visual number. Amer. J.
Psychol, 1949, 62, 498-525.
13. KLEMMEE, E. T., & FRICK, F. C. Assimilation of information from dot andmatrix patterns. /. exp. Psychol., 1953,
45, 15-19.
14. KTJLPE, O. Versuche tiber Abstraktion.
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English consonants. J. Acoust. Soc.
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(Received May 4, 1955)