# What does a number in brackets after another number mean? I.e. 21(1)cm-1

What does the number in brackets mean in these two examples?

$$21(1)\ \mathrm{cm^{-1}}$$

and

$$1.0(3)\times10^{-7}$$

The results of measurements and other numerical values of quantities are often given with an associated standard uncertainty. A numerical value and the associated uncertainty may be expressed as shown in the question:

\begin{align} y&=21(1)\ \mathrm{cm^{-1}}\\[6pt] &=a(b)\ \mathrm{cm^{-1}} \end{align}

where
$$y$$ is the estimate of the measurand (e.g. the result of a measurement) expressed in the unit $$\mathrm{cm^{-1}}$$,
$$a$$ is the numerical value, and
$$b$$ denotes a standard uncertainty expressed in terms of the least significant digit(s) in $$a$$.

It is important to note that the given uncertainty refers to the least significant digits of the given numerical value. For example, in the expression

$$l=23.4782(32)\ \mathrm m$$

the $$(32)$$ represents a standard uncertainty equal to

$$u(l)=0.0032\ \mathrm m$$

Many physical constants are also given in this form. For example, the previously recommended value for the molar gas constant $$R$$ as given by NIST from 25 June 2015 until the new value became available on 20 May 2019:

$$R=8.3144598(48)\ \mathrm{J\ mol^{-1}\ K^{-1}}$$

where the $$(48)$$ represents a standard uncertainty of

$$u(R)=0.0000048\ \mathrm{J\ mol^{-1}\ K^{-1}}$$

This form is in accordance with various current standards, in particular

The Guide to the Expression of Uncertainty in Measurement (GUM) also shows other permissible forms:

7.2.2 When the measure of uncertainty is $$u_\mathrm c(y)$$, it is preferable to state the numerical result of the measurement in one of the following four ways in order to prevent misunderstanding. (The quantity whose value is being reported is assumed to be a nominally 100 g standard of mass $$m_\mathrm S$$; the words in parentheses may be omitted for brevity if $$u_\mathrm c$$ is defined elsewhere in the document reporting the result.)

1) “$$m_\mathrm S=100{,}021\,47\ \mathrm g$$ with (a combined standard uncertainty) $$u_\mathrm c = 0{,}35\ \mathrm{mg}$$.”

2) “$$m_\mathrm S=100{,}021\,47(35)\ \mathrm g$$, where the number in parentheses is the numerical value of (the combined standard uncertainty) $$u_\mathrm c$$ referred to the corresponding last digits of the quoted result.”

3) “$$m_\mathrm S=100{,}021\,47(0{,}000\,35)\ \mathrm g$$, where the number in parentheses is the numerical value of (the combined standard uncertainty) $$u_\mathrm c$$ expressed in the unit of the quoted result.”

4) “$$m_\mathrm S=(100{,}021\,47\pm0{,}000\,35)\ \mathrm g$$, where the number following the symbol $$\pm$$ is the numerical value of (the combined standard uncertainty) $$u_\mathrm c$$ and not a confidence interval.”

Note that item 2) corresponds to the form given in the question.

Concerning item 4), however, the GUM notes

The ± format should be avoided whenever possible because it has traditionally been used to indicate an interval corresponding to a high level of confidence and thus may be confused with expanded uncertainty (…).

Furthermore, ISO 80000 notes

Uncertainties are often expressed in the following manner: $$(23{,}478\,2\pm0{,}003\,2)\ \mathrm m$$. This is, however, wrong from a mathematical point of view. $$23{,}478\,2\pm0{,}003\,2$$ means $$23{,}481\,4$$ or $$23{,}475\,0$$, but not all values between these two values. (…)

From Wikipedia:

In metrology, physics, and engineering, the uncertainty or margin of error of a measurement, when explicitly stated, is given by a range of values likely to enclose the true value. This may be denoted by error bars on a graph, or by the following notations:

• measured value ± uncertainty
• measured value $^{+uncertainty}_{−uncertainty}$
• measured value (uncertainty)