The Early History of Average Values and Implications for Education

Figures & data

Table

What is the average?		Possible statistical interpretations
Jennifer	The half. The whole, and in between the half, that is the mean.	Part of the algorithm: dividing by two.
Charissa	Everything together.	Part of the algorithm: adding all values
Bart	You look between the highest and lowest.	Midrange? Somewhere in between?
Centina	The most.	Mode, typical?
Claire	What you think it is roughly.	Estimation or representativeness
Lisa	The mean is about a bit in balance.	Balance point
Kerster	In between.	Midrange?
Frank:	The midpoint.	Midrange, median or center of gravity?
Others:	Add and divide by the number.	Algorithm

Figure 1. Four student strategies for estimating the total number of elephants in the picture. (Reprinted with permission from Mathematics in Context © 1998 Encyclopaedia Britannica, Inc.).

Figure 2 Theorem of Pappus on arithmetic, geometric, and harmonic mean.

Figure 3 B as the arithmetic mean of A and C, in the way that numbers are represented in Euclid's Elements around 300 BC (Euclid 1956).

Table

Interviewer	How would you estimate the average annual temperature in the Netherlands from this graph or table?
Jennifer	Add everything.
Interviewer	And then?
Lisa	Divide by 2.
Interviewer	Why divide by 2?
Lisa	Because that is the average.

Figure 4 Compensation strategy in the battery problem (Brand D on the left, Brand K on the right).

Figure 5 What would have been an “average box”?

Euclid (1956), The Thirteen Books of the Elements, tr. T. H. Heath, New York: Dover.

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