# What does the plus-minus sign with a value in parentheses mean?

I always see the plus-minus sign with a value in parentheses after a physical property is described such as melting and boiling point, like this:

$$\text{Melting point }(\pm 1~^\circ\mathrm{C})$$

What does it mean?

• – Loong Mar 25 '17 at 9:33
• Where is that snippet from? Context helps. – Jack Aidley Mar 25 '17 at 10:30
• If you search up "CHEMISTRY INVESTIGATION FACTORS EFFECTING THE BOILING AND MELTING POINTS IN ORGANIC HOMOLOGOUS SERIES." It's the pdf. – idk wat else Mar 25 '17 at 14:51

## 2 Answers

It's simply describing the absolute error of the relevant quantities.

So if this precedes a table of melting points, you can expect the true value to vary by 1 degree, e.g.

$\ce{H2O}: 0.0 \pm 1~^{\circ}\mathrm{C}$

means the melting point is between $-1$ and $+1~^{\circ}\mathrm{C}$.

• I suggest to adhere to the less ambiguous, proper notation, that is either i) value and incertainty enclosed in parentheses, followed by the unit like (1.5 ± 0.3) kg. Or ii), without a parentheses like 1.5(3) kg. To state a measurement like 0.0 ± 1 ºC however rises questions. As one useful reference here, consult BPIM guide JCGM 100:2008, section 7.2.2. -- A document in consent with IUPAC, IUPAP, ISO, and others; equally worth to read for a general picture, too. (Source: bipm.org/en/publications/guides/gum.html) – Buttonwood Mar 24 '17 at 21:46

The errors you quote is the absolute error as described in the answer by @khaverim, however, what this means is not clear. This is because what the error measures is often not defined in tables or papers. Normally it is assumed to be one standard deviation but could be two standard deviations, or if there are only a few measurements might be just the range (largest-smallest value).

If the error is one standard deviation it means that by chance only $68$ % of the time will a measurement fall within the range, (and so by chance $32$ % outside the range), if two std. dev. then this increases to $95$ % so it matters quite a lot.

Recall also that the error can only be know if the true value of the quantity being measured is also known. Thus, although called an 'absolute' error it is only an estimate of the the error just as the mean of quantity is an estimate of its true value. Clearly as the number of measurements increases these estimates get better but the estimate of the error only increases as $\sqrt N$ for N measurements so it is easy to reach a point of diminishing returns.

[ Sometimes the Relative Error is quoted and this is (error)/(true value). The Relative Standard Deviation is the ratio (calculated std. dev. )/ (calculated mean value). When multiplied by $10^6$ these relative values become part per million, p.p.m. ]