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.

1 |

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

The original data were presented in a two-way, 13 x 13 frequency table,
`estimated`

(3:15) x `actual`

(3:15).

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

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/

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | ```
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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