A Circle Theorem in the Samuelson Domain


Mihir B. Banerjee,J.R. Gupta,R.G. Shandil ,Jyotirmoy Mukhopadhyay ,




Samuelson Domain ,mean ,standard deviation ,maximum and the minimum deviations,critical value,


In the subject matter of mathematical statistics, let the domain of mathematical activity that draws its inspiration from and nurtures the lead provided by the seminal paper of the American Economist and Nobel Prize (1970) winner P.A. Samuelson entitled, “How Deviant can you be?” and published in the Journal of the American Statistical Association in 1968, on the maximum and the minimum deviations, from the mean (denoted presently by m and 'm respectively) in a set of n observations with given mean μ and standard deviation σ, be henceforth defined as the Samuelson Domain. The present communication is in the Samuelson Domain. A circle theorem in the −σmplane is rigorously established and exhibited step by step for the sheer delight of its simplicity and elegance. A crude first approximation yields a result that is inferior to Samuelson’s but a more precise investigation of the consequences of the circle theorem shows that Samuelson’s famous work on the existence of bounds, for a set of n real numbers, in terms of σμ, and n can be improved upon provided n exceeds a critical value.


I. Samuelson,P.A., How deviant can you be?, J. American Statistical Association, 63,1522-1525,1968.

II. Banerjee, M.B. and Shandil, R.G., A theorem on mean and standard deviation of a statistical variate, Ganita,46, 21-23,1995.

Author(s) :Mihir B. Banerjee, J.R. Gupta, R.G. Shandil and Jyotirmoy Mukhopadhyay View Download