Download (Arithmetic) Mean

The Notion of “Average”
“Average” appears to be a commonsense
concept – simple and straighforward.
(A primary kid knows how to work out the
“average” for a set of numbers.)
Is it really that obvious to young kids?
Was it used to be that easy?
How did it occur to early people, including
mathematicians & scientists?
“Mean values” for the ancient Greek
In the semicircle
ADC with center O,
DB is perpendicular
to the diameter AC
and BF is
perpendicular to the
radius OD.
With magnitudes AB and BC,
OD is the arithmetic mean
BD is the geometric mean
FD is the harmonic mean
“(Arithmetic) Mean” for the Greek:
Aristotle (384-322 BC)
By the mean of a thing I denote a
point equally distant from
either extreme, which is one and the same for everybody; by the mean
relative to us, that amount which is neither too much nor too little,
and this is not one and the same for everybody. For example, let 10 be many
and 2 few; then one takes the mean with respect to the thing if one takes 6;
since 10-6 = 6-2, and this is the mean according to
arithmetical proportion [progression]. But we cannot arrive by this
method at the mean relative to us. Suppose that 10 lb. of food is a large ration for
anybody and 2 lb. a small one: it does not follow that a trainer will prescribe 6
lb., for perhaps even this will be a large portion, or a small one, for the particular
athlete who is to receive it; it is a small portion for Milo, but a large one for a
man just beginning to go in for athletics.
Nichomachean Ethics, Book II, Chapter 6 (italics added)
“(Arithmetic) Mean” for the Greek:
Aristotle (384-322 BC)
By the mean of a thing I denote a
point equally distant from
either extreme, which is one and the same for everybody; by the mean
relative to us, that amount which is neither too much nor too little,
and this is not one and the same for everybody. For example, let 10 be many
"Virtue,
therefore, is a mean state in the sense that
few; then one takes the mean with respect to the thing if one takes 6;
itandis2able
to hit the mean.“
since 10-6 = 6-2, and this is the mean according to
arithmetical proportion [progression]. But we cannot arrive by this
method
at the mean
to us. Suppose
lb. ethical
of food isideal.
a large ration for
For
Aristotle,
the relative
mean relative
to us that
was10an
anybody and
2 lb. a small
it does not
follow that a trainer
will prescribe
6
Aristotle
attempted
an one:
extension
of mathematical
means
(of
lb., for perhaps even this will be a large portion, or a small one, for the particular
geometric
of human
affairs
even
one
athlete who magnitudes)
is to receive it; ittoisdomain
a small portion
for Milo,
but a–large
one
for that
a
just beginning
go in for athletics.
isman
concerned
withtoethics.
Nichomachean Ethics, Book II, Chapter 6 (italics added)
“Arithmetic Mean” for the Greek:
The middle number b of a and c is called the arithmetic
mean if and only if
a - b = b - c.
Noteworthy is that:
 It is different from our modern definition: b
= (a+c)/2.
 It refers to the arithmetic mean of two numbers only.
And its definition makes the extension/generalisation to
more than two numbers not obvious.
 It quickly reveals the fact that the arithmetic mean lies
somewhere between the two given numbers.
“Arithmetic Mean”
in its modern sense
The arithmetic mean of n positive numbers:
a1  a2  a3  ...  an
n
Noteworthy is that:
 It involves a division (which might be cumbersome until
the invention of the decimal system in 1585).
 Measurement errors were well recognized among early
astronomers. Repeated observations were practised in
astronomy. However, the method of using the arithmetic
mean to reduce the measurement errors and/or produce a
single number from a set of discordant measurements
came into play no earlier than 16th century.
Demand for a mathematical
theory of measurements
The arithmetic mean of n positive numbers:
a1  a2  a3  ...  an
n
Background (around late 18th to early 19th century):
 Among astronomers, there was the need to reconcile
Newton’s Laws of motion with the astronomical
observations;
 There was also the rise of a new mode of rigorous
experimental physics (esp. in France), as distinguished
from the “mathematical scientists” and “experimental
philosophers”.
“Arithmetic Mean”
in its modern sense
a1  a2  a3  ...  an
n
The imprecision
of measurement became a major
issue in the mid-eighteenth century, when one of
the primary occupations of those working in celestial physics
and mathematics was the problem of reconciling
Newton’s Laws with the observed motions of the
moon and planets. One way to produce a single
number from a set of discordant measurements is to
take the average, or mean.
(Mlodinow, 2008, p.127)
Mlodinow, L. (2008). The Drunkard’s Walk: How Randomness Rules Our Lives.
New York: Pantheon Books.
a1  a2  a3  ...  an
n
“Arithmetic Mean”
in its modern sense (continued)
… It seems to have been young Issac Newton who, in his
optical investigations, first employed it for that purpose. But as
in many things, Newton was an anomaly. Most scientists in
Newton’s day, and in the following century, didn’t
take the mean. Instead, they chose the single “golden number”
from among their measurements – the number they deemed
mainly by hunch to be the most reliable result they had. That’s
because they regarded variation in measurement not as the
inevitable by-product of the measuring process but as evidence
of failure – with, at times, even moral consequences.
(Mlodinow, 2008, p.127)
Mlodinow, L. (2008). The Drunkard’s Walk: How Randomness Rules Our Lives.
New York: Pantheon Books.
Arithmetic mean (as a statistical concept)
once was a problem of combining a set of
independent observations on the same quantity
traced from antiquity to the appearance in the
eighteenth century of the arithmetic mean as a
statistical concept
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.
Historical Example 1
Story of Nala in the great Indian epic Mahábarata
(well before A.D. 400)
• Nala took the job as charioteer to the foreign
potentate Rtuparna.
• Rtuparna was keen on mathematics.
• One day, he estimated the number of leaves and
fruit on two great branches of a spreading tree.
– examined a single twig
– estimated the number of twigs on the branches
– came to a total 2095 fruit
(Hacking, 1975, p.7)
Hacking, I. (1975/2006). The Emergence of Probability: A Philosophical Study of Early Ideas About Probability
Induction and Statistical Inference (2nd Edn.). Cambridge: Cambridge University Press.
Historical Example 1
(continued)
Story of Nala in the great Indian epic Mahábarata:
• Nala was in conflict with Kali, a demigod of dicing.
• Kali, in revenge, took possession of Nala’s body
and soul.
• As a result, Nala lost his kingdom “in a sudden
frenzy of gambling”.
• …
• Rtuparna taught the science of dicing and
mathematical skills to Nala in exchange for his
lessons in horsemanship.
(Hacking, 1975, p.7)
Hacking, I. (1975/2006). The Emergence of Probability: A Philosophical Study of Early Ideas About Probability
Induction and Statistical Inference (2nd Edn.). Cambridge: Cambridge University Press.
Historical Example 1
(continued)
Story of Nala in the great Indian epic Mahábarata:
Rtuparna says [in English translation in 1860],
“ I of dice possess the science
and in numbers thus am skilled.”
The storyline:
– Gambling (including dicing)
– Loss of fortune
– Mastering the science [of dice]
– Recover his loss (when playing with “less well-informed
people”)
(Hacking, 1975, pp.7-8)
Hacking, I. (1975/2006). The Emergence of Probability: A Philosophical Study of Early Ideas About Probability
Induction and Statistical Inference (2nd Edn.). Cambridge: Cambridge University Press.
Historical Example 1
(continued)
Story of Nala in the great Indian epic Mahábarata:
“More striking is the recognition that dicing has something
to do with estimating the number of leaves on a tree.
That indicates a very high level of sophistication. Even after the
European invention of probability around 1660 it took some
time before any substantial body of people could comprehend
that decisive connection. Indeed, although the Nala story was
almost the first piece of Sanskrit writing to be widely circulated
in modern Europe and was much admired by the German
romantics, no one paid any attention (so far as I know) to this
curious insight about the connection between dicing
and sampling.”
(Hacking, 1975, p.7)
Hacking, I. (1975/2006). The Emergence of Probability: A Philosophical Study of Early Ideas About Probability
Induction and Statistical Inference (2nd Edn.). Cambridge: Cambridge University Press.
Historical Example 1
Story of Nala in the great Indian epic Mahábarata:
Estimation of the number of leaves and fruit on two great
branches of a spreading tree.
– examined a single twig
– estimated the number of twigs on the branches
– came to a total 2095 fruit
• It can be seen “as an intuitive predecessor of arithmetic
mean”.
• “This use of an average, in our modern eyes, has to do with
compensation, balance, and representativenss.“
• The idea was recognized in the context of estimating a total.
(Bakker, 2003)
Bakker, A. (2003). The early history of average values and implications for education.
Journal of Statistics Education, 11(1). www.amstat.org/publications/jse/v11n1/bakker.html
Historical Example 2
Greek historian Herodotus (circa 485-420 BC):
• How many years had elapsed since the first king of Egypt to
the latest Hephaestus [in Herodotus’s historical account]
• It was said that there were 341 generations separating the
first king of Egypt from the last mentioned Hephaestus.
– Estimate: THREE generations as 100 years
– Thus, 300 generations make 10,000 years, and the
remaining 41 generations make 1,340 years more.
– Therefore, a total of 11,340 years.
Herodotus (circa 480-425 BC / 1996). Histories. Ware, Hertfordshire (UK): Wordsworth.
[ Book Two, Para 142 ]
Rubin, E. (1968). The Statistical World of Herodotus. The American Statistician, 22(1), 31-33.
Bakker, A. (2003). The early history of average values and implications for education.
Journal of Statistics Education, 11(1). www.amstat.org/publications/jse/v11n1/bakker.html
Historical Example 2 (continued)
Estimation of years elapsed from first Egyptian kings to the
last mentioned cited by Herodotus (circa 485-420 BC):
“The statistically important point in this quotation is the
assumption that three generations was reckoned a hundred
years. This assumption was made to estimate the total amount
of years between the first Egyptian King and Hephaestus. Of
course, three generations were not always exactly a hundred
years; sometimes a little less, sometimes a little more, but the
errors are roughly evened out. That is why this method may be
seen as a preliminary stage of the development of the average.
As in the first example we see the aspect of compensation and
representativeness (typical number of years for generations).”
Bakker, A. (2003). The early history of average values and implications for education.
Journal of Statistics Education, 11(1). www.amstat.org/publications/jse/v11n1/bakker.html
Historical Example 3
Thucydides (circa 460-400 BC), one of the first scientific historians:
For the Athenians who wanted to force their ways
over their enemy's city wall, they had to construct
ladders long enough to reach the top of the city wall.
• the height of the wall = ?
• number of layers of bricks = ?
– the layers were counted by a lot of people at the same
time, and though some were likely to get the figure wrong,
the majority would get it right.
• thickness of a single brick = ?
“an implicit use of the
mode”
Rubin, E. (1971). Quantitative Commentary on Thucydides. The American Statistician, 25(4), 52-54.
Bakker, A. (2003). The early history of average values and implications for education.
Journal of Statistics Education, 11(1). www.amstat.org/publications/jse/v11n1/bakker.html
Historical Example 4
Thucydides (circa 460-400 BC), one of the first scientific historians:
“Homer gives the number of ships as 1,200 and says
that the crew of each Boetian ship numbered 120,
and the crews of Philoctetes were fifty men for each
ship. By this, [Thucydides, Rubin thinks] means to
express the maximum and minimum of the various
ships companies ... If, therefore, we reckon the
number by taking an average of the biggest and
smallest ships ...”
“midrange” in modern terms
Rubin, E. (1971). Quantitative Commentary on Thucydides. The American Statistician, 25(4), 52-54.
Bakker, A. (2003). The early history of average values and implications for education.
Journal of Statistics Education, 11(1). www.amstat.org/publications/jse/v11n1/bakker.html
History of mathematics
for mathematics education
The young learner recapitulates the learning process of
mankind, though in a modified way. He repeats
history not as it actually happened but as it would
have happened if people in the past would have
known something like what we do know now. It is a
revised and improved version of the historical learning
process that young learners recapitulate. “Ought to
recapitulate” - we should say. In fact we have not
understood the past well enough to give them this
chance to recapitulate it.
Freudenthal (1983) cited in Bakker (2003)
Freudenthal, H. (1983). The implicit philosophy of mathematics: History and education. Proceedings of the
International Congress of Mathematicians, pp.1695-1709. Warsaw and Amsterdam: Polish Scientific Publishers
and Elsevier Science Publishers.
Arithmetic mean (as a statistical concept)
• stemming from the practice in astronomical
observations (e.g. postions of stars, intervals
of time such as length of a “year”, etc)
• Babylonians (as early as 500B.C. to 300B.C.)
• Early Greeks, including Hipparchus (about
300B.C.)
• “The technique of taking the arithmetic mean
of a group of comparable observations had
not yet, however, made its appearance as a
general principle.” (p. 121)
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.
Arithmetic mean (as a statistical concept)
• stemming from the practice in astronomical
observations (e.g. postions of stars, intervals
of time such as length of a “year”, etc)
• Babylonians (as early as 500B.C. to 300B.C.)
• Early Greeks, including Hipparchus (since
300B.C.)
• “The technique of taking the arithmetic mean
of a group of comparable observations had
not yet, however, made its appearance as a
general principle.” (p. 121)
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.
Arithmetic mean (as a statistical concept)
• “The technique of repeating and combining
observations made on the same quantity
appears to have been introduced into
scientific method by Tycho Brahe towards the
end of the sixteenth century.” (p. 122)
• “We see that Tycho used the arithmetic mean
to eliminate systematic errors. The calculation
of the mean as a more precise value than a
single measurement is not far removed and
had certainly appeared about the end of the
seventeenth century, …” (p. 124)
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.
Arithmetic mean (as a statistical concept)
An example in 1736-37 on a French expedition
sent to Lapland:
“Each observer made his own observation of
the angles and wrote them down apart, they
then took the means of these observations for
each angle: the actual readings are not given,
but the mean is.”
(Clarke, 1880, cited by Plackett, p. 124)
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.
Arithmetic mean (as a statistical concept)
The DISTRIBUTION of the Arithmetic Mean …
the probability that the mean of t observations
is at most m/t for the following two
distributions:
(i) possible errors are –v, …, -2, -1, 0, 1, …, v and equal
probabilities are attached to them;
(ii) the same set of errors with probabilities
proportional to 1, 2, …, v+1, …, 2, 1 respectively.
(Simpson, 1755, cited by Plackett, p. 124)
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.
Arithmetic mean (as a statistical concept)
The DISTRIBUTION of the Arithmetic Mean …
Simpson “finds the probability that the mean is
nearer to zero than a single independent
observation.”
(Simpson, 1757, cited by Plackett, p. 124)
Lagrange (about the same time) also presented “a
detailed discussion of discrete error distributions, on
lines essentially the same as those folowed by
Simposon” (p. 125)
Plackett, R. L. (1958). The principle of the arithmetic mean. Biometrika, 45, 130-135. Reprinted in E. S. Pearson & M. G.
Kendall (Eds.) (1970), Studies in the history of statistics and probability (pp. 121-126). London: Charles Griffin & Co. Ltd.