Expanded Uncertainty

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #421; 6 January 2009.

Community Portal → Data Analysis → Expanded Uncertainty


After we have estimated the total (combined) uncertainty, we often want to be able to state that some interval captures a large fraction of the values that could be measured for the quantity; that is, we need to report an expanded uncertainty U to which we can attach a level of confidence. This expanded uncertainty is computed from our total uncertainty u by multiplying by a coverage factor k [8],


(1)
U = k\,u


We commonly take the coverage factor from a normal distribution, unless some other distribution is known to apply. Therefore, we use

k = 1 to claim a level of confidence of 68.3%,
k = 2 to claim 95.5%,
k = 2.58 to claim 99%, and
k = 3 to claim 99.7%.

For type A uncertainties, coverage factors from the normal distribution can usually be used when the number of measurements N is large. But when the number is small (say, N < 12), the sampling distribution is ill-defined and the coverage factor from the normal distribution is too small. In such cases, we separate the type A coverage factor kA from the type B factor kB. Then instead of (1), we use


(2)
U \ = \ \sqrt{(k_A u_A)^2 + (k_B u_B)^2 }


Table 1. Values of type A coverage factor kA for N measurements at three levels of confidence.[1]
N 68% 95% 99%
∞ 1 1.96 2.6
20 1.06 2.1 2.8
10 1.09 2.2 3.2
8 1.11 2.3 3.4
6 1.13 2.4 3.7
4 1.2 2.8 4.6
3 1.25 3.2 5.8
2 1.39 4.3 9.9

Values for kB are still taken from a normal distribution. But values for kA are taken from Table 1 at either the 68, 95, and 99% levels of confidence. Note that as N increases, the values of kA approach those for a normal distribution; that is, k_A \rarr \, k_B and (2) reverts to (1).

To have a numerical example, we consider flow rate of water through a pipe, measured ten times, giving a mean of 5.68 gpm with a type A uncertainty of 0.051 gpm, type B uncertainty of 0.12 gpm, and combined uncertainty of 0.13 gpm. Then at the 95% confidence level, the value for the expanded uncertainty of our ten flow rates is given by (2) as


(3)
U \ = \ \sqrt{(2.2 \times 0.051)^2 + (1.96 \times 0.12)^2 } \ = \ 0.3


and we would report the measured flow rate as


(4)

y = 5.7 ± 0.3 gpm


at the 95% confidence level. Note that the position of the one significant figure in the uncertainty determines the position of the last significant figure in the reported mean.[2] In contrast, a reported mean of 5.70 would be misleading because the final “0” is insignificant compared to the uncertainty of ±0.3 gpm.


References

  1. ↑ B. W. Lindgren and G. W. McElrath, Introduction to Probability and Statistics, Macmillan, New York, 1959.
  2. ↑ J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.
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