WEIGHTED AVERAGING AND THE CONCEPT OF "KNOWLEDGE"
Weighted averaging is required when combining measurements of a quantity that have different
uncertainties, e.g., the result of combining two independent measurements of a quantity a, a = 10.0 ± 0.5 and
a = 10.3 ± 0.2, to determine a single best value and uncertainty. Here we will derive the results stated in
Taylor (Equations 7.10 - 7.12). Our approach will be to start with a single measurement process of a
physical variable a that yields a value and uncertainty. We apply this process the "i-th" time and get ai = ai ±
δa, where ai is the best value obtained in the i-th measurement, and we assume that each time we apply this
process we will get the same uncertainty δa. Now lets combine several of these measurements, say 12
of them. We will then get an overall average <a>12 (meaning the average from combining 12 of
these independent measurements of a) of <a>12= (a1 + a2 + a3 … + a12)/12, and a standard deviation
of the mean δa12 of δa12 = δa/√12, in the usual way for calculating the overall uncertainty from repeated
measurements. Thus, for 12 measurements of a:
a12 = <a>12 ± δa12 = (a1 + a2 + a3 … + a12)/12 ± δa/√12 .
Similarly, for example, for a set of 23 additional such measurements of a:
a23 = <a>23 ± δa23 = (a13 + a14 + a15 … + a35)/23 ± δa/√23 .
However, these two sets of data could have been considered one set of 35 measurements with the result:
a35 = <a>35 ± δa35 = (a1 + a2 + a3 … + a35)/35 ± δa/√35
.
Thus, the appropriate combination of a12 and a21 must yield a35. Specifically:
δa12 = δa/√12 combines with δa23 = δa/√23 to give δa35 = δa/√35 .
The only mathematics that achieves this is the following: [1/δa35]2 = [1/δa12]2 + [1/δa23]2, such that
[1/δa35]2 = [1/δa12]2 + [1/δa23]2 = 12/δa2 + 23/δa2 = 35/δa2, and therefore δa35 = δa/√35.
This suggests that it would be useful to define the notion of knowledge of a quantity a ≡ κνa, which
is related to the uncertainty of the quantity a ≡ δa, by
κνa ≡ 1/δa2 .
In the above measurements, with δaN = δa/√N, we have κνaN = N/δa2. Thus knowledge goes up as
uncertainty goes down, and, in fact, knowledge increases linearly with the number of
measurements. Because of this linear dependence, knowledge is the appropriate weighting factor
when combining measurements. To combine two measurements you simply add their knowledge to
get the combined knowledge,
κνa35 ≡ κνa12 + κνa23 ,
in the above example giving [1/δa35]2 = [1/δa12]2 + [1/δa23]2 or δa35 = {12/δa2 + 23/δa2}-1/2 = δa/√35, the
correct combined uncertainty. To get the combined best value use knowledge as the weighting factor,
a35 = {a12(κνa12) + a23(κνa23)]/[κνa12 + κνa23] = [a12(κνa12) + a23(κνa23)]/κνa35 ,
the weighting that gives the correct overall average in the above example:
a35 = {[(a1 + a2 .. + a12)/12](12/δa2) + [(a13 + a14 .. + a35)/23](23/δa2)}/(35/δa2) = (a1 + a2 .. + a35)/35.
Combine N measurements a1, a2 … aN in the same way:
κνa = κνa1 + κνa2 … + κνaN , a = [a1(κνa1) + a2(κνa2)… + aN(κνaN)]/κνa.