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Calculation techniques in combining uncertainties
From the face expressions of participants attending my measurement uncertainty
courses, I could tell that some of them had yet to grasp the important point of calculating
the combined uncertainty from a series of uncertainty components. I hope the following
notes can bring more clarity to this issue.
When we are presented with few standard uncertainties u(xi), expressed in standard
deviations from an uncertainty contributing component, we need to find out what the
combined standard uncertainty of this component is. The best approach to begin with is
to remember a basic definition that โthe squared standard deviation is a varianceโ.
So, the combined standard uncertainty can first be evaluated by its total or combined
variance which is, of course, the sum of various variances from the uncertainty
component. This is referred to as the Law of Propagation of Uncertainty.
Mathematically, we can express it as in equation [1], assuming that each uncertainty
contribution is independent:
๐ข(๐ฆ)2=โ๐๐2๐ข(๐ฅ๐)2 [1]
where ๐๐=(๐๐ฆ
๐๐ฅ๐) is the sensitive coefficient of xi.
This equation leads us to two simple situations:
โข If a quantity xi is simply added to or subtracted from all the others to obtain the
result y, the contribution to the uncertainty in y is simply the uncertainty u(xi) in
xi. For example,
given y = x1 + x2 with uncertainties u(x1) and u(x2), then
u(y)2 = u(x1)2 + u(x2)2 , or, ๐ข(๐ฆ)=โ๐ข(๐ฅ1)2+๐ข(๐ฅ2)2 [2]
