# How do I denote a measurement with a margin of error using ±?

I am trying to express a quantity that is measured to be $$\pu{0.75 mL}$$, but could be anywhere between $$\pu{0.7 mL}$$ and $$\pu{0.8 mL}$$. Is this the correct way to do it?

$$\pu{0.75 mL} \pm \pu{0.05 mL}$$

• $\pu{0.75 \pm 0.05 mL}$ typically means a standard deviation of $\pu{0.05 mL}$, not a hard limit of $\pu{\pm 0.05 mL}$. – MaxW Mar 20 at 18:03
• @MaxW is right on the money: the notation most frequently means sample mean +/- one sample standard deviation, computed from N independent measurements. So degrees of freedom is N-1 and this information should get conveyed. I could add a lot to my answer, but the downvotes on the question mean it probably is not worth the effort. One more thing: the +/- term can reasonably convey hard limits in some cases, e.g., histogram bin center +/- ends of the histogram bin. Other possibilities are +/- half a confidence interval or a standard error. Context is everything. – Ed V Mar 20 at 20:20
• I'd say it should be written what the interval means, everything else is in practice too ambiguous: I regularly meet situations where it's not clear whether it is the standard deviation across measurements or the standar error of the mean. Not to mention the rather official recommendation in @EdV's answer to report extended uncertainty. – cbeleites unhappy with SX Mar 20 at 21:35
• @cbeleitesunhappywithSX I fully agree with you and I was going to reference this paper (D. Coleman, J. Auses, N. Grams, “Regulation – From an industry perspective or Relationships between detection limits, quantitation limits, and significant digits”, Chemom. Intell. Lab. Syst. 37 (1997) 71-80.), where Coleman et al. cogently criticize the inadequacy of the typical x ± y notation. Sadly, we are still having to deal with these needlessly ambiguous matters. – Ed V Mar 20 at 21:49

I strongly recommend looking at this: QUAM:2012.P1 (EURACHEM/CITAC Guide, “Quantifying Uncertainty in Analytical Measurement”, 3rd Ed., 2012., 141 pages.) You can easily find it online, for free, and it is all about quantifying uncertainties in measurements, though with an obvious focus on analytical chemistry-related measurements. As an example, the figure below shows the QUAM recommended way to report standard and expanded uncertainties:

9.3. Reporting standard uncertainty

9.3.1. When uncertainty is expressed as the combined standard uncertainty $$u_c$$ (that is, as a single standard deviation), the following form is recommended:

"(Result): $$x$$ (units) [with a] standard uncertainty of $$u_c$$ (units) [where standard uncertainty is as defined in the ISO/IEC Guide to the Expression of Uncertainty in Measurement and corresponds to one standard deviation.]"

NOTE: The use of the symbol $$\pm$$ is not recommended when using standard uncertainty as the symbol is commonly associated with intervals corresponding to high levels of confidence.

Terms in parentheses [] may be omitted or abbreviated as appropriate.

EXAMPLE:

• Total nitrogen: $$\pu{3.52 g}/\pu{100 g}$$

• Standard uncertainty: $$\pu{0.07 g}/\pu{100 g}$$*

• *Standard uncertainty corresponds to one standard deviation.

9.4. Reporting expanded uncertainty

9.4.1. Unless otherwise required, the result x should be stated together with the expanded uncertainty $$U$$ calculated using a coverage factor $$k=2$$ (or as described in section 8.3.3.). The following form is recommended:

"(Result): $$(x \pm U)$$ (units)

[where] the reported uncertainty is [an expanded uncertainty as defined in the International Vocabulary of Basic and General terms in metrology, 2nd ed., ISO 1993,] calculated using a coverage factor of 2, [which gives a level of confidence of approximately 95 %]"

Terms in parentheses [] may be omitted or abbreviated as appropriate. The coverage factor should, of course, be adjusted to show the value actually used.

EXAMPLE:

• Total nitrogen: $$(3.52 \pm 0.14)~\mathrm{g}/100~\mathrm{g}$$*
• *The reported uncertainty is an expanded uncertainty calculated using a coverage factor of 2 which gives a level of confidence of approximately 95 %.

This is from pages 30-31 of QUAM, referenced above. CITAC is the acronym for Co-Operation on International Traceability In Analytical Chemistry.

All that said, your notation is quite common, so no worries.

• @orthocresol Many thanks for the edit! – Ed V Apr 7 at 11:37
• Please consider giving the green checkmark to the most helpful of the posted answers. It encourages people to put some thought and time into crafting answers that are factually correct, relevant, understandable and likely to be of benefit to those, in future, who encounter the question and accepted answer. It is a small reward for those who volunteer their considerable time, effort and experience to aid others and they might well look favorably at future questions from the same person. Thanks for considering this! – Ed V May 26 at 1:36