The Magical Number Seven, Plus or Minus Two

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The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information is a 1956 paper by the cognitive psychologist George A. Miller. In it Miller showed a number of remarkable coincidences between the channel capacity of a number of human cognitive and perceptual tasks. In each case, the effective channel capacity is equivalent to between 5 and 9 equally-weighted error-less choices: on average, about 2.5 bits of information. Miller hypothesized that these may all be due to some common but unknown underlying mechanism.

Urban legends surrounding 7±2

A number of urban legends have grown up around the number 7±2 and human performance on various cognitive tasks. While Miller's paper is most often cited, by coincidence research into short term memory also threw up a 7±2 finding which seems to have added impetus to the claims made.

As outlined above, Miller's paper simply pointed out that channel capacity on various tasks was around 2.5 bits of information. Measurements of human short term memory capacity also found a 7±2 limit. However, this limit was eventually found to be a result of using subjects who were speakers of English to remember sequences of single digits. It turns out that one component of human working memory, the phonological loop, is capable of holding around 2 seconds of sound. Two seconds is the duration of the English spoken form of 7±2 digits (in Chinese it is around 9 and in Welsh around 6), the variation is highly correlated with the rate at which people speak.

The 7±2 urban legends are various rules specifying the maximum number of items that can occur in a given context (eg, in software engineering the maximum number of subroutines that should be called from the main program). Whether or not these 7±2 rules provide the benefits claimed of them can be verified only by experiments. However, neither Miller's paper or the early short term memory research is likely to provide the primary experimental evidence needed to back up such claims.

Working memory capacity

Working memory is generally considered to have limited capacity. The earliest quantification of the capacity limit associated with short-term memory was the magical number seven introduced by Miller (1956)^[1]. He noticed that the memory span of young adults was around seven elements, called chunks, regardless whether the elements were digits, letters, words, or other units. Later research revealed that span does depend on the category of chunks used (e.g., span is around seven for digits, around six for letters, and around 5 for words), and even on features of the chunks within a category. For instance, span is lower for long than 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, and on the lexical status of the contents (i.e., whether the contents are words known to the person or not)^[2]. 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)^[3] has proposed that working memory has a capacity of about four chunks in young adults (and less in children and old adults).

Software engineering related

Ed Yourdon in his Modern Structured Analysis (Prentice Hall, 1979) specified that the maximum number of subroutines that should be called from the main program should be between 5 and 9. This heuristic was not proposed as being due to any computer limit; rather, it was suggested that the programmer becomes confused when trying to understand the program.

Sanjiv Sidhu also claims to have arrived independently at the same 7±2 figure, his official biography states: Based on his observation that even the smartest people can juggle no more than nine variables when making decisions, he proposed a design for computer software based on artificial intelligence and advanced simulation techniques. This claim, however, may not be more than an other addition to the urban legend surrounding 7±2

Other cognitive numeric limits

The concept of a limit is illustrated by imagining the patterns on the faces of a die (see dice). It is easy for many people to visualise each of the six faces. Now imagine seven dots, eight dots, nine dots, ten dots, and so on. At some point it becomes impossible to visualise the dots as a single pattern (a process known as subitizing), and one thinks of, say, eight as two groups of four. The upper limit of your visualisation of a number represented as dots is your subsisting limit for that exercise.

The film Rain Man, starring Dustin Hoffman, portrayed an autistic savant, who was able to visualise the number represented by an entire box of toothpicks spilled on the floor. A similar feat was clinically observed by neuropsychologist Oliver Sacks and reported in his book The Man Who Mistook His Wife for a Hat. Therefore one might suppose that this limit is an arbitrary limit imposed by our cognition rather than necessarily being a physical limit.

Hrair from Watership Down and applications within programming

Hrair is a number too large to count. This term is from the fictional language Lapine used in Richard Adams's Watership Down. In this novel, a rabbit's hrair is greater than 4 whereas, for humans, hrair would be greater than 7 plus or minus 2. From a psychological perspective, hrair is the point where the person is overwhelmed by concepts or change. The interesting thing about a person reaching their hrair point is that we are not only unable to understand the new concept or stimulus when it is introduced, but it makes us unable to continue as effectively with what we were doing before. The term hrair limit as used by Ed Yourdon in his Modern Structured Analysis (Prentice Hall, 1979) is the maximum number of subroutines that should be called from the main program, again set at between 5 and 9. This heuristic was not proposed as being due to any computer limit; rather, it was suggested that the programmer becomes confused when trying to understand the program.

In organisation theory the limit has a similar meaning: the maximum number of projects that one can be involved in simultaneously before chaos starts to ensue.

References

• George A. Miller. The Magical Number Seven, Plus or Minus Two. The Psychological Review, 1956, vol. 63, pp. 81-97

See also

• Fitts' law
• Hick's law
• Subitizing
• Working memory
• Chunking (psychology)

External links

• George A. Miller. The Magical Number Seven, Plus or Minus Two. The Psychological Review, 1956, vol. 63, pp. 81-97
• Another online version of the same paper
• In-depth discussion on many myths around Miller's paper at Edward Tufte's site.
• The 7±2 Urban Legend (pdf file)