Rounding Uncertainty

Assorted numbers in metal on a white background. 3D render with HDRI lighting and raytraced textures.

Rounding estimates of uncertainty in measurement has recently become a topic of great debate. For several years, many Accreditation Bodies (AB) and assessors have asserted the implementation of the ’round up’ method, which forced Certified Accredited Bodies (CAB) to round up their estimated Calibration and Measurement Capability (CMC) values. Although this method of rounding is acceptable, it has been considered less favorable among CAB members.

With the revision of the ILAC P14:01/2103, ‘ILAC Policy for Uncertainty in Calibration,’ many ABs are changing their policies to adhere to the new requirements. This means that CABs will be required to adhere to the new policies as well. However, CABs should be relieved, because the policy changes favor ‘common sense’ when it comes to rounding.

According to section 6.3b of the ILAC P14:01/2013, the policy states,

For the process of rounding, the usual rules for rounding of numbers shall be used, subject to the guidance on rounding provided i.e. in Section 7 of the GUM.

Following the lead of the ILAC P14:01/13, lets refer to section 7.2.6 of the JCGM 100:2008, ‘Guide to the Expression of Uncertainty in Measurement’ (GUM), which states,

In reporting final results, it may sometimes be appropriate to round uncertainties up rather than to the nearest digit. For example, uc(y) = 10,47 mΩ might be rounded up to 11 mΩ. However, common sense should prevail and a value such as u(xi) = 28,05 kHz should be rounded down to 28 kHz. Output and input estimates should be rounded to be consistent with their uncertainties; for example, if y = 10,05762 Ω with uc (y) = 27 mΩ, y should be rounded to 10,058 Ω. Correlation coefficients should be given with three-digit accuracy if their absolute values are near unity.

01 | Conventional rounding
This method may also be referred to as simple rounding, round half-up, round half towards plus infinity, or asymmetric rounding. When selecting this method, the following conditions shall apply.

a | When the digit next beyond the one to be retained is less than five, keep the retained figure unchanged (e.g. 2.54 becomes 2.5)
b | When the digit next beyond the one to be retained is greater than five, increase the retained figure by one (e.g. 2.47 becomes 2.5)
c | When the digit next beyond the one to be retained is exactly five, increase the retained figure by one (e.g. 2.45 becomes 2.5)

02 | Round to the nearest even digit
This method may also be referred to as odd/even rounding, unbiased rounding, convergent rounding, statistician’s rounding, Dutch rounding, Gaussian rounding, banker’s rounding, or broken rounding. When selecting this method, the following conditions shall apply.

a | When the digit next beyond the one to be retained is less than five, keep the retained figure unchanged (e.g. 2.54 becomes 2.5)
b | When the digit next beyond the one to be retained is greater than five, increase the retained figure by one (e.g. 2.47 becomes 2.5)
c | When the digit next beyond the one to be retained is exactly five and the retained digit is even, leave it unchanged (e.g. 2.45 becomes 2.4). Conversely, if the digit is odd, increase the retained figure by one (e.g. 2.55 becomes 2.6).

03 | Round up
This method may also be referred to as round up, round towards plus infinity, round away from zero, and round towards infinity. When selecting this method, the following conditions shall apply.

a | When the digit next beyond the one to be retained is less than five, increase the retained figure by one (e.g. 2.43 becomes 2.5)
b | When the digit next beyond the one to be retained is greater than five, increase the retained figure by one (e.g. 2.47 becomes 2.5)
c | When the digit next beyond the one to be retained is exactly five, increase the retained figure by one (e.g. 2.45 becomes 2.5)

With the release and implementation of the ILAC P14:01/13 policy, the debate regarding rounding uncertainty should diminish. However, I recommend reading the identified documents and familiarizing yourself with the identified policies, guidance, and methods regarding rounding. Should a debate ever arise, hopefully, you will be prepared to support or defend the method you choose to practice.

References
ILAC P14:01/2013 – ILAC Policy for Uncertainty in Calibration
JCGM 100:2008 – Guide to the Expression of Uncertainty in Measurement
NIST GLP9 – Rounding Expanded Uncertainties and Calibration Values
NIST SP811 – Guide for the Use of the International System of Units (See Appendix B7)
Symmetric and Asymmetric Rounding by Schneeweiss, H.; Komlos, J.; Ahmad, A.
Rounding by B. Baas

About the Author

Richard Hogan

Richard Hogan is the CEO of ISO Budgets, L.L.C., a U.S.-based consulting and data analysis firm. Services include measurement consulting, data analysis, uncertainty budgets, and control charts. Richard is a systems engineer who has laboratory management and quality control experience in the Metrology industry. He specializes in uncertainty analysis, industrial statistics, and process optimization. Richard holds a Masters degree in Engineering from Old Dominion University in Norfolk, VA. Connect with Richard on Google+ and LinkedIn.

2 Comments

  1. Richard, Thank you very much for sharing your compiled research, knowledge, interpretations, opinions, and applications. Your articles have been very helpful to me and for that I am much obliged. I look forward to employing your services one day.

    • Hi Ken,

      Thank you for your comments and feedback. I am glad that you like my articles and find them helpful. I would be happy to work with you one day. When are ready, just contact me to get started.

Leave a Reply

Your email address will not be published. Required fields are marked *