STATISTICS - Branch of mathematics dealing with gathering
, analyzing, and making inferences from data. Originally associated with
government data (e.g., census data), the subject now has applications in all the sciences. Statistical tools not only summarize past data through such
indicators as the mean and the standard deviation
but can predict future events using frequency distribution functions.
Statistics provides ways to design efficient experiments that eliminate time
-consuming trial and error. Double-blind tests for polls, intelligence and
aptitude tests, and medical, biological, and industrial experiments all benefit
from statistical methods and theories. The results of all of them serve as predictors
of future performance, though reliability varies.
They declare that three hundred and forty-one generations separate the first king of Egypt
from the last mentioned (Hephaestus) . and that there was a king and a high priest
corresponding to each generation. Now reckon three generations as a hundred years, three
hundred generations make ten thousand years, and the remaining forty-one generations make
1,340 years more; thus one gets a total of 11,340 years.... Another example can be taken from Thucydides (460-400 BC). The following quotations are
from his History of the Peloponnesian War. Aristotle defines in his Nicomachean Ethics also a more philosophical form of the mean,
namely the mean relative to us. With this notion he explains what virtue is. About the
difference between the arithmetic mean and the mean relative to us he writes:
"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."
Later in the section he writes about virtue: Prequisite:College Algebra
Statistics is a subject,(unlike probability) that was mainly born outside mathematics
and was often not seen as part of mathematics. Its birth and growth was due to astronomy, demography, economics, medicine,
genetics, biometry, anthropology, the social sciences and many other areas (Porter 1986; Stigler
1986). This might explain why many histories of mathematics pay very little attention to
statistics. Perhaps the earliest use of statistics was that of estimation.
For example,
Herodotus on the Egyptians:
Here we see Herodotus makes an estimate that an average generation is approximately 33.333 years.
(The problem was for the Athenians) ... to force their way over the enemy's surrounding wall
... Their method was as follows: they constructed ladders to reach the top of the enemy's wall,
and they did this by calculating the height of the wall from the number of layers of bricks at a
point which was facing in their direction and had not been plastered. 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, especially as they counted the layers frequently and were not so far
away from the wall that they could not see it well enough for their purpose. Thus, guessing
what the thickness of a single brick was, they calculated how long their ladders would have to
be....
In Pythagoras' time (around 500 BC), three mean values were known in Greece: the arithmetic,
geometric, and harmonic mean (Heath 1921: 85; Iamblichus 1939). At least eleven different
mean values had been defined only some 200 years later (Heath 1921). The theory of the three
well-known mean values was developed in his school with reference to music theory and
arithmetic. Consider for example the musical proportions 6:8:9:12. 8 is the harmonic mean of 6
and 12, and 9 is the arithmetic mean. The proportions 6:8 = 9:12 (a fourth as a musical
interval), 6:9 = 8:12 (a fifth), 6:12 (an octave) all form consonant intervals; 8:9 is a second.
This example shows a connection between a phenomenon, musical intervals on strings, and the
related mathematical concepts, proportions that could be described with mean values. It also
shows that what comes early in history need not come early in education. It is not the intention
to teach geometric and harmonic means to students in grade 7 because their applications are far
less important than of the arithmetic mean. If we look at Greek propositions, for instance at a
proposition of Pappos on the mean values, it seems that the Greeks studied them for their
beauty and not for the applications.
"Virtue, therefore, is a mean state in the sense that it is able to hit the mean." (N.E. book II,
chapter vi).