W. Stanley Jevons' data on numerical discrimination

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Description

In a remarkable brief note in Nature, 1871, W. Stanley Jevons described the results of an experiment he had conducted on himself to determine the limits of the number of objects an observer could comprehend immediately without counting them. This was an important philosophical question: How many objects can the mind embrace at once?

He carried out 1027 trials in which he tossed an "uncertain number" of uniform black beans into a box and immediately attempted to estimate the number "without the least hesitation". His questions, procedure and analysis anticipated by 75 years one of the most influential papers in modern cognitive psychology by George Miller (1956), "The magical number 7 plus or minus 2: Some limits on ..." For Jevons, the magical number was 4.5, representing an empirical law of complete accuracy.

Usage

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Format

A frequency data frame with 50 observations on the following 4 variables.

actual

Actual number: a numeric vector

estimated

Estimated number: a numeric vector

frequency

Frequency of this combination of (actual, estimated): a numeric vector

error

actual-estimated: a numeric vector

Details

The original data were presented in a two-way, 13 x 13 frequency table, estimated (3:15) x actual (3:15).

Source

Jevons, W. S. (1871). The Power of Numerical Discrimination, Nature, 1871, III (281-282)

References

Miller, G. A. (1956). The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information, Psychological Review, 63, 81-97, http://www.musanim.com/miller1956/

Examples

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data(Jevons)
# show as tables
xtabs(frequency ~ estimated+actual, data=Jevons)
xtabs(frequency ~ error+actual, data=Jevons)

# show as sunflowerplot with regression line
with(Jevons, sunflowerplot(actual, estimated, frequency, 
  main="Jevons data on numerical estimation"))
Jmod <-lm(estimated ~ actual, data=Jevons, weights=frequency)
abline(Jmod)

# show as balloonplots
if (require(gplots)) {

with(Jevons, balloonplot(actual, estimated, frequency, xlab="actual", ylab="estimated", 
  main="Jevons data on numerical estimation\nBubble area proportional to frequency",
  text.size=0.8))

with(Jevons, balloonplot(actual, error, frequency, xlab="actual", ylab="error", 
  main="Jevons data on numerical estimation: Errors\nBubble area proportional to frequency", 
  text.size=0.8))
}

# plot average error
if(require(reshape)) {
unJevons <- untable(Jevons, Jevons$frequency)
str(unJevons)

require(plyr)
mean_error <- function(df) mean(df$error, na.rm=TRUE)
Jmean <- ddply(unJevons, .(actual), mean_error)
with(Jmean, plot(actual, V1, ylab='Mean error', xlab='Actual number', type='b', main='Jevons data'))
abline(h=0)
}

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