X. As an error of precision what this is saying is that the
"true" value of Xi is probably between Xi -
X and Xi +
X.In this section we discuss some theory about why errors of precision propagate as described in Section 9. Nothing in this section is required, and it may be skipped.
We will discuss the theory in two different ways. The discussion will not be rigorous, but will be correct.
Imagine a person leaves a bar in a highly intoxicated state and is staggering down the sidewalk. The person is so drunk that although each step is the same length, the direction of each step is completely random. Thus after N steps, the distance the drunk gets away from the bar is also random. But the most likely distance turns out to be the square root of N times the length of each step.
Now consider a set of repeated measurements Xi each
with error
X. As an error of precision what this is saying is that the
"true" value of Xi is probably between Xi -
X and Xi +
X.
But whether a measurement chosen at random from the set is too high or too low is completely random. Thus if we form the sum of all the measurements we must account for the fact that there is some possibility that the error in one particular measurement is cancelled by the error in another measurement. This is similar to the drunkard's walk, so we conclude that the error in the sum only goes up as the square root of the number of repeated measurements N.
This is exactly the behavior that we got by applying Rule 1 from Section 9 to the first part of the calculation of the error in the mean in Section 10.
This discussion assumes some knowledge of linear algebra.
We are combining two quantities X and Y, either by addition and subtraction.
We imagine some abstract space of errors. Since
X and
Y are the only errors, they span the space. Since the
error in the combination, Z, is zero only if both
X and
Y are zero, these two errors form a basis for the
space.
We all know that if we have a basis for a linear space, such as the two perpendicular sides of a right triangle, the sum, the length of the hypotenuse, is given by combining the lengths of the two sides in quadrature, i.e. using Pythagoras' theorem.
And that is why errors of precision are combined in quadrature too.
This document is Copyright © 2001, 2004 David M. Harrison
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