The Magical Number Seven, Plus or Minus Two
Encyclopedia
"The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information" is one of the most highly cited papers in psychology. It was published in 1956 by the cognitive psychologist
George A. Miller of Princeton University
's Department of Psychology
in Psychological Review
. It supposedly argues that the number of objects an average human can hold in working memory
is 7 ± 2. This is frequently referred to as Miller's Law (not to be confused with his theory of communication: Miller's Law
).
Recent research has demonstrated that not only is the "law" based on a misinterpretation of Miller's paper, but that the correct number is probably around three or four.
s of information
, which corresponds to the ability to distinguish between 4 and 8 alternatives.
The second cognitive limitation Miller discusses is memory span
. Memory span refers to the longest list of items (e.g., digits, letters, words) that a person can repeat back immediately after presentation in correct order on 50% of trials. Miller observed that memory span of young adults is approximately 7 items. He noticed that memory span is approximately the same for stimuli with vastly different amount of information - for instance, binary digits have 1 bit each; decimal digits have 3.32 bits each; words have about 10 bits each. Miller concluded that memory span is not limited in terms of bits but rather in terms of chunks
. A chunk is the largest meaningful unit in the presented material that the person recognizes - thus, it depends on the knowledge of the person what counts as a chunk. For instance, a word is a single chunk for a speaker of the language but breaks down into as many chunks as the word has letters for someone who is totally unfamiliar with the language.
Miller recognized that the correspondence between the limits of one-dimensional absolute judgment and of short-term memory span was only a coincidence, because only the first limit, not the second, can be characterized in information-theoretic terms (i.e., as a roughly constant number of bits). Therefore, there is nothing "magical" about the number 7, and Miller used the expression only rhetorically. Nevertheless, the idea of a "magical number 7" inspired much theorizing, rigorous and less rigorous, about the capacity limits of human cognition.
and working memory
revealed that memory span is not a constant even when measured in terms of a number of chunks. The number of chunks a human can recall immediately after presentation depends on the category of chunks used (e.g., span is around seven for digits, around six for letters, and around five for words), and even on features of the chunks
within a category. For instance, span is lower for long words than it is for short words. In general, memory span for verbal contents (digits, letters, words, etc.) strongly depends on the time it takes to speak the contents aloud. Some researchers have therefore proposed that the limited capacity of short-term memory for verbal material is not a "magic number" but rather a "magic spell". Baddeley used this finding to postulate that one component of his model of working memory
, the phonological loop, is capable of holding around 2 seconds of sound. However, the limit of short-term memory cannot easily be characterized as a constant "magic spell" either, because memory span depends also on other factors besides speaking duration. For instance, span depends on the lexical status of the contents (i.e., whether the contents are words known to the person or not). Several other factors also affect a person's measured span, and therefore it is difficult to pin down the capacity of short-term or working memory to a number of chunks. Nonetheless, Cowan (2001) has proposed that working memory has a capacity of about four chunks in young adults (and less in children and older adults).
Tarnow (2010) finds that in a classic experiment typically argued as supporting a 4 item buffer by Murdock (1962). There is in fact no evidence for such and thus the "magical number", at least in the Murdock experiment, is 1.
portrayed an autistic savant, who was able to rapidly determine the number of toothpicks from an entire box spilled on the floor, apparently subitizing a much larger number than four objects. A similar feat was informally observed by neuropsychologist
Oliver Sacks
and reported in his book The Man Who Mistook His Wife for a Hat
. (Autistic expert Daniel Tammet
has suggested that the children Sacks observed may have pre-counted the matches in the box.) Therefore one might suppose that this limit is an arbitrary limit imposed by our cognition
rather than necessarily being a physical limit.
Cognitive psychology
Cognitive psychology is a subdiscipline of psychology exploring internal mental processes.It is the study of how people perceive, remember, think, speak, and solve problems.Cognitive psychology differs from previous psychological approaches in two key ways....
George A. Miller of Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
's Department of Psychology
Princeton University Department of Psychology
The Princeton University Department of Psychology, located in Green Hall, is an academic department of Princeton University on the corner of Washington St. and William St. in Princeton, New Jersey. For over a century, the department has been one of the most notable psychology departments in the...
in Psychological Review
Psychological Review
Psychological Review is a scientific journal that publishes articles on psychological theory. It was founded by Princeton psychologist James Mark Baldwin and Columbia psychologist James McKeen Cattell in 1894 as a publication vehicle for psychologists not connected with the Clark laboratory of G....
. It supposedly argues that the number of objects an average human can hold in working memory
Working memory
Working memory has been defined as the system which actively holds information in the mind to do verbal and nonverbal tasks such as reasoning and comprehension, and to make it available for further information processing...
is 7 ± 2. This is frequently referred to as Miller's Law (not to be confused with his theory of communication: Miller's Law
Miller's law
-In communication:Miller's law, part of his theory of communication, was formulated by George Miller, Princeton Professor and psychologist.It instructs us to suspend judgment about what someone is saying so we can first understand them without imbuing their message with our own personal...
).
Recent research has demonstrated that not only is the "law" based on a misinterpretation of Miller's paper, but that the correct number is probably around three or four.
Miller's article
In his article, Miller discussed a coincidence between the limits of one-dimensional absolute judgment and the limits of short-term memory. In a one-dimensional absolute-judgment task, a person is presented with a number of stimuli that vary on one dimension (e.g., 10 different tones varying only in pitch) and responds to each stimulus with a corresponding response (learned before). Performance is nearly perfect up to 5 or 6 different stimuli but declines as the number of different stimuli is increased. The task can be described as one of information transmission: The input consists of one out of n possible stimuli, and the output consists of one out of n responses. The information contained in the input can be determined by the number of binary decisions that need to be made to arrive at the selected stimulus, and the same holds for the response. Therefore, people's maximum performance on one-dimensional absolute judgement can be characterized as an information channel capacity with approximately 2 to 3 bitBit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
s of information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
, which corresponds to the ability to distinguish between 4 and 8 alternatives.
The second cognitive limitation Miller discusses is memory span
Memory span
In psychology and neuroscience, memory span is the longest list of items that a person can repeat back in correct order immediately after presentation on 50% of all trials. Items may include words, numbers, or letters. The task is known as digit span when numbers are used. Memory span is a common...
. Memory span refers to the longest list of items (e.g., digits, letters, words) that a person can repeat back immediately after presentation in correct order on 50% of trials. Miller observed that memory span of young adults is approximately 7 items. He noticed that memory span is approximately the same for stimuli with vastly different amount of information - for instance, binary digits have 1 bit each; decimal digits have 3.32 bits each; words have about 10 bits each. Miller concluded that memory span is not limited in terms of bits but rather in terms of chunks
Chunking (psychology)
Chunking, in psychology, is a phenomenon whereby individuals group responses when performing a memory task. Tests where individuals can illustrate "chunking" commonly include serial and free recall, as these both require the individual to reproduce items that he or she had previously been...
. A chunk is the largest meaningful unit in the presented material that the person recognizes - thus, it depends on the knowledge of the person what counts as a chunk. For instance, a word is a single chunk for a speaker of the language but breaks down into as many chunks as the word has letters for someone who is totally unfamiliar with the language.
Miller recognized that the correspondence between the limits of one-dimensional absolute judgment and of short-term memory span was only a coincidence, because only the first limit, not the second, can be characterized in information-theoretic terms (i.e., as a roughly constant number of bits). Therefore, there is nothing "magical" about the number 7, and Miller used the expression only rhetorically. Nevertheless, the idea of a "magical number 7" inspired much theorizing, rigorous and less rigorous, about the capacity limits of human cognition.
The "magical number 7" and working memory capacity
Later research on short-term memoryShort-term memory
Short-term memory is the capacity for holding a small amount of information in mind in an active, readily available state for a short period of time. The duration of short-term memory is believed to be in the order of seconds. A commonly cited capacity is 7 ± 2 elements...
and working memory
Working memory
Working memory has been defined as the system which actively holds information in the mind to do verbal and nonverbal tasks such as reasoning and comprehension, and to make it available for further information processing...
revealed that memory span is not a constant even when measured in terms of a number of chunks. The number of chunks a human can recall immediately after presentation depends on the category of chunks used (e.g., span is around seven for digits, around six for letters, and around five for words), and even on features of the chunks
Chunking (psychology)
Chunking, in psychology, is a phenomenon whereby individuals group responses when performing a memory task. Tests where individuals can illustrate "chunking" commonly include serial and free recall, as these both require the individual to reproduce items that he or she had previously been...
within a category. For instance, span is lower for long words than it is for short words. In general, memory span for verbal contents (digits, letters, words, etc.) strongly depends on the time it takes to speak the contents aloud. Some researchers have therefore proposed that the limited capacity of short-term memory for verbal material is not a "magic number" but rather a "magic spell". Baddeley used this finding to postulate that one component of his model of working memory
Working memory
Working memory has been defined as the system which actively holds information in the mind to do verbal and nonverbal tasks such as reasoning and comprehension, and to make it available for further information processing...
, the phonological loop, is capable of holding around 2 seconds of sound. However, the limit of short-term memory cannot easily be characterized as a constant "magic spell" either, because memory span depends also on other factors besides speaking duration. For instance, span depends on the lexical status of the contents (i.e., whether the contents are words known to the person or not). Several other factors also affect a person's measured span, and therefore it is difficult to pin down the capacity of short-term or working memory to a number of chunks. Nonetheless, Cowan (2001) has proposed that working memory has a capacity of about four chunks in young adults (and less in children and older adults).
Tarnow (2010) finds that in a classic experiment typically argued as supporting a 4 item buffer by Murdock (1962). There is in fact no evidence for such and thus the "magical number", at least in the Murdock experiment, is 1.
Other cognitive numeric limits
Cowan also noted a number of other limits of cognition that point to a "magical number four", and different from Miller, he argued that this correspondence is no coincidence. One other process that seems to be limited at about four elements is subitizing, the rapid enumeration of small numbers of objects. When a number of objects is flashed briefly, their number can be determined very quickly, at a glance, when the number does not exceed the subitizing limit, which is about four objects. Larger numbers of objects must be counted, which is a slower process. The film Rain ManRain Man
Rain Man is a 1988 drama film written by Barry Morrow and Ronald Bass and directed by Barry Levinson. It tells the story of an abrasive and selfish yuppie, Charlie Babbitt, who discovers that his estranged father has died and bequeathed all of his multimillion-dollar estate to his other son,...
portrayed an autistic savant, who was able to rapidly determine the number of toothpicks from an entire box spilled on the floor, apparently subitizing a much larger number than four objects. A similar feat was informally observed by neuropsychologist
Neuropsychology
Neuropsychology studies the structure and function of the brain related to specific psychological processes and behaviors. The term neuropsychology has been applied to lesion studies in humans and animals. It has also been applied to efforts to record electrical activity from individual cells in...
Oliver Sacks
Oliver Sacks
Oliver Wolf Sacks, CBE , is a British neurologist and psychologist residing in New York City. He is a professor of neurology and psychiatry at Columbia University, where he also holds the position of Columbia Artist...
and reported in his book The Man Who Mistook His Wife for a Hat
The Man Who Mistook His Wife for a Hat
The Man Who Mistook His Wife for a Hat and Other Clinical Tales is a 1985 book by neurologist Oliver Sacks describing the case histories of some of his patients. The title of the book comes from the case study of a man with visual agnosia...
. (Autistic expert Daniel Tammet
Daniel Tammet
Daniel Tammet is a British writer. His best selling 2006 memoir, Born On A Blue Day, about his life with high-functioning autism and savant syndrome, was named a "Best Book for Young Adults" in 2008 by the American Library Association.Tammet's second book, Embracing the Wide Sky, was named one of...
has suggested that the children Sacks observed may have pre-counted the matches in the box.) Therefore one might suppose that this limit is an arbitrary limit imposed by our cognition
Cognition
In science, cognition refers to mental processes. These processes include attention, remembering, producing and understanding language, solving problems, and making decisions. Cognition is studied in various disciplines such as psychology, philosophy, linguistics, and computer science...
rather than necessarily being a physical limit.
See also
- 7 (number)
- Baddeley's model of working memoryBaddeley's Model of Working MemoryAlan Baddeley and Graham Hitch proposed a model of working memory in 1974, in an attempt to describe a more accurate model of short-term memory....
- Chunking (psychology)Chunking (psychology)Chunking, in psychology, is a phenomenon whereby individuals group responses when performing a memory task. Tests where individuals can illustrate "chunking" commonly include serial and free recall, as these both require the individual to reproduce items that he or she had previously been...
- Free recallFree recallFree recall is a basic paradigm in the psychological study of memory. In this paradigm, participants study a list of items on each trial, and then are prompted to recall the items in any order...
- Fitts's law
- Hick's lawHick's lawHick's Law, named after British psychologist William Edmund Hick, or the Hick–Hyman Law , describes the time it takes for a person to make a decision as a result of the possible choices he or she has. The Hick-Hyman Law assesses cognitive information capacity in choice reaction experiments...
- Subitizing
- Working memoryWorking memoryWorking memory has been defined as the system which actively holds information in the mind to do verbal and nonverbal tasks such as reasoning and comprehension, and to make it available for further information processing...
External links
- Version of the paper with figures adapted for HTML and proofread and approved by Miller in 1997
- In-depth discussion on many myths around Miller's paper at Edward Tufte'sEdward TufteEdward Rolf Tufte is an American statistician and professor emeritus of political science, statistics, and computer science at Yale University. He is noted for his writings on information design and as a pioneer in the field of data visualization....
site. - The 7±2 Urban Legend (pdf file)