Apr

3

After reading the paper the Chair found, my memory has been jogged. Introduction to Statistical Quality Control, 5th ed., Douglas Montgomery, pages 95-6 discusses the use of the range to estimate the standard deviation.

An unbiased estimator of the standard deviation s of a normal distribution is s(hat) = R/d2
R = range
d2=variable depending on n

So the factor 1.6926 is really d2 for n=3 (the # of GPS measurements Schwarz used).

For n=1 to 10, Appendix Table VI on page 725 of Montgomery gives:
n d2
2 1.128
3 1.693
4 2.059
5 2.326
6 2.534
7 2.704
8 2.847
9 2.970
10 3.078

Montgomery notes that the range method works very well (retains high efficiency) for small samples sizes (n <= 6).

Victor Niederhoffer writes:

An interesting article on
ranges shows that a good estimate of the standard deviation from a normal
distribution is range/1.7. Sequential estimates of the standard deviation from
the range, for example:

              date    range   stand dev

                4 02     10        6
                3 30     22      14
                3 29    14         8
                3 28    12         7
                3 27     8          5
                3 26    15         9 

For S&P futures this might provide a good template for thinking about short-term volatility. 

Bruno Ombreux adds:

Here is one of the early articles on the ratio of range to standard deviation, featuring tables for the ratio. Of course, today one can use resampling methods to get these kinds of ratios, even from non-normal populations.


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