Reporting Uncertainties

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #357; 6 January 2009.

Table 1. Uncertainty Budget for Measurements of
Flow Rate of Water Through a Pipe.

Source of Uncertainty	Type	Magnitude (gpm)
Calibration curve at 21°C	B	0.113
Fluctuations in flow and meter	A	0.051
Scale resolution of meter	B	0.025
Variations in fluid temperature	B	0.006
Total (combined) uncertainty	---	0.13

When reporting uncertainties we should cite each source, tell whether it was treated as type A or type B, and give the magnitude determined. This is conveniently done in a table, called an Uncertainty Budget.^[1] A sample uncertainty budget from an analysis of water flow-rate measurements is given in Table 1. Note that an uncertainty budget can be a useful tool in experimental design as well as in experimental analysis.

Reporting Formats

The preferred ways for reporting final values of uncertainties, as adopted in the ANSI guide,^[2] are summarized in Table 2. Note the following about entries in Table 2:

Table 2. Recommended Formats for Reporting Values of
Total and Expanded Uncertainties.[8]

Uncertainty		Reporting Format
Total	1.	y = 5.7 gpm with total uncertainty of u = 0.1 gpm
	2.	y = 5.7(1) gpm
Relative Total	3.	y = 5.7 gpm with relative total uncertainty of 2%
Expanded	4.	y = 5.7 ± 0.3 gpm at the 95% level of confidence
Relative Expanded	5.	y = 5.7 ± 5% at the 95% level of confidence

1. For the total (combined) uncertainty you may use either (1) or (2) in Table 2.
2. In case (2) the integer in parenthesis gives the combined uncertainty for the last digit in the mean.
3. When reporting expanded uncertainties, be sure to include the level of confidence adopted.
4. The ± notation is reserved for reporting the expanded uncertainty.

Finally, note that on x-y plots the “error bar” shown at each point should, in fact, be the expanded uncertainty. In the figure caption you should state the level of confidence that applies. So the term “error bar” is a really misnomer; it should be called an uncertainty bar.

References

1. ↑ B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,” NIST Technical Note 1297, U. S. Government Printing Office, Washington, D. C., 1994.
2. ↑ U. S. Guide to the Expression of Uncertainty in Measurement, ANSI/NCSL Z540-2-1997, American National Standards Institute, NCSL International, Boulder, CO, 1997.