Margins of Error and the Propagation of 10%

            A problem with the original derivation of the Toltec Module becomes apparent when we consider the criterion for deciding whether or not any particular measurement within a site 'fits' a multiple of the TM.  Sherrod and Rolingson write that the unit, "was assigned a value of 47.5 m, although it has a range from 46 to 49 m, with most variation within 1.5 m of the 47.5 m" (1987:36).  The standard actually applied to distances, however, is a wider range, "The actual distance of the module varies by up to 10% or +/- 4.75 m from a precise 47.5 m" (1987:134).  While a margin of error must of course be taken into account in any such study, the size of the margin and exactly how it is applied must be carefully considered.  In the case of Sherrod and Rolingson's study, the margin of error is relatively large to begin with, and the wording and use of "margin of error" is somewhat ambiguous.  Strictly speaking, the margin of error employed is 20% of a TM, as the margin is applied to both sides of the intended target – in this case, 47.5 m +/- 4.75 means anywhere between 42.75 and 52.25.  By this standard, one out of every five measurements across the site would appear to fit the TM by chance alone.

            The problem in the original study is even deeper, though.  The margin of error is not held at a constant +/- 4.75 m, but is applied as a percentage of the entire measured distance, and therefore propagated with each multiple.  For example, when considering distances at the Cahokia site, Sherrod and Rolingson conclude that 1,960 m between Monk's Mound and Mound #1 corresponds to 41.3 TM, which they consider to be an error of only 0.6% (1987:101).  The nearest TM increments to 1,960 m are 41 TM at 1,947.5 m, and 42 TM at 1,995 m.  So how can 1,960 m, almost equidistant between two TM increments, be considered to have an error of only 0.6%?  Figure 2 illustrates how the margin of error is propagated with distance to achieve this result.  At one TM any measurement between 42.75 to 52.25 m would be considered within 10% of the distance; at five TM the 10% includes all distances between 213.75 to 261.25 m – an entire TM distance.  In other words, at a distance of five or more TM, it is impossible to find a measurement which does not fall within 10% of the stated margin of error.  Given this propagation of the margin of error, the larger the distance the smaller the margin of error it can possibly contain, and any measurement over 213.75 m will automatically fall within the 10% standard. 

Figure 2.  Diagram of the propagation of error as employed by Sherrod and Rolingson.  Because the 10% margin of error is applied to the entire distance being measured, it becomes larger with each TM increment.  At a distance of 5 TM, the margin of error equals 1 TM, and it becomes impossible to find any measurement which is not within 10% of the 'error'. 

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