🚀 go-pugleaf

In <1992Nov20.161226.27020@nosc.mil> marchett@antares.nosc.mil (Dave Marchette) writes:

>Let x1,...,xn be iid normal 0, sigma, and let x(1),...,x(n)
>be the order statistics.  Assume that only n and x(n)-x(1)
>are known.  That is, only the number of points and the total
>spread of the points is known.  What is the appropriate estimate 
>of sigma?

An unbiased estimator of sigma is the range divided by a constant (which is
a function of n). Unfortunately there is no nice closed expression for the
constant, since it is the solution to an integral equation. However the
constant is tabulated in lots of places, especially in quality control texts.
Some refs:

Pearson E.S. (1942) "The Probability Integral of the Range ...", 
	Biometrika, 32, p.301-310.
Dixon, W.J. & Massey, F.J. (1969) "Introduction to Statistical Analysis",
	McGraw-Hill, p.136, p.486.
Duncan, A.J. (1986) "Quality Control and Industrial Statistics", Irwin,  
	p.481, p.1005.
Montgomery, D.C. (1991) "Introduction to Statistical Quality Control", Wiley,
	p.204, p.A-15.

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