19.4 Expanded Uncertainty

In many cases we want to use the standard uncertainty to create an interval in which we have a specified level of confidence that it contains the value of the measurand. This is called the expanded uncertainty, as is acheived by multiplying the standard uncertainty by a coverage factor (\(k\)). For measurand \(X\) with best approximation \(x\) with a standard uncertainty \(u(x)\), we can construct the interval \([x - k \times u(x), x + k \times u(x)]\), which can also be written as:

\[ X = x \pm k u(x) \]

To determine \(k\) for the required level of confidence \(p\), we need to solve the intergral equation:

\[ p = \int_{x - k u(x)}^{x + k u(x)} f(x') dx' \]

where \(f\) is the PDF.

The equations for the coverage factors for a given confidence level, as well as some example confidence levels and coverage factors, are shown for a normal, rectangular and triangular distribution in the table below. Note the \(erf^{-1}\) in the normal distribution’s equation is the inverse error-function, which cannot be solved analytically.

Confidence level	Normal Distribution \(k\)	Rectangular Distribution \(k\)	Triangular Distribution \(k\)
p	\(\sqrt{2}erf^{-1}(p)\)	\(\sqrt{3} p\)	\(\sqrt{6}\left(1 - \sqrt{1 - p}\right)\)
68.27%	1	1.18	1.07
95.45%	2	1.65	1.93
99.73%	3	1.73	2.32

Note

For combined standard uncertianties we may assume a normal distribution by evoking the central limit theorom.