The Notion of “Average”

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

With magnitudes AB and BC,

OD is the arithmetic mean

BD is the geometric mean

FD is the harmonic mean

In the semicircle ADC with center O, DB is perpendicular to the diameter AC and BF is perpendicular to the radius OD.

“(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 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)

"Virtue, therefore, is a mean state in the sense that it is able to hit the mean.“

For Aristotle, the mean relative to us was an ethical ideal. Aristotle attempted an extension of mathematical means (of geometric magnitudes) to domain of human affairs – even one that is concerned with ethics.

“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 senseThe arithmetic mean of n positive numbers:

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.

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Demand for a mathematicaltheory of measurements The arithmetic mean of n positive numbers:

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”.

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“Arithmetic Mean” in its modern sense

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)

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Mlodinow, L. (2008). The Drunkard’s Walk: How Randomness Rules Our Lives. New York: Pantheon Books.

“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)

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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 1Story 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.





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.



“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.”



Historical Example 1Story 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 2Greek 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


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).”


Historical Example 3Thucydides (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 = ?

Rubin, E. (1971). Quantitative Commentary on Thucydides. The American Statistician, 25(4), 52-54.


“an implicit use of

the mode”

Historical Example 4Thucydides (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 ...”

Rubin, E. (1971). Quantitative Commentary on Thucydides. The American Statistician, 25(4), 52-54.


“midrange” in modern terms

History of mathematics for mathematics educationThe 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.


• 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)



• 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)



• “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)



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.


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.


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)


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The Notion of “Average”

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