The importance of the Arithmetic Mean was articulated in 1809 by C.F. Gauss, one of the pioneers of mathematical statistics. S.M. Stigler, a distinguished statistician of our time, considers Arithmetic Mean as the first of the "five ideas that changed statistics and continue to change the way we think about the world''.
Historians of statistics had hitherto identified the earliest unambiguous statistical use of the general Arithmetic Mean in a work of the English astronomer Henry Gellibrand (1635) and believed that examples of mean-taking would not be found in ancient science. But, as in the case of numerous other scientific concepts, their account overlooked the clear, precise and abundant use of the Arithmetic Mean in the treatises of ancient Indian mathematicians to represent the depth, width or length of an irregular-shaped pool of water and thereby estimate its volume. In fact, Brahmagupta (628 CE) defines and applies the more sophisticated concept of the Weighted Arithmetic Mean. The treatment of the excavation problems through the Arithmetic Mean was possibly one of the factors which shaped the gradual development of calculus in India. The weighted arithmetic mean also appears in ancient Indian arithmetic treatises as an exact mathematical formula in computing the proportion of pure gold in an alloy formed by blending of several pieces of gold of different weights and purities.
In this talk, we shall discuss the emergence of some of the avatars of the Arithmetic Mean at different time points, in different contexts, in different mathematics cultures.
