# Uncertain figures

I have doubts about the definition of uncertain figure.

If I have the following measurement (2.7 ± 0.1) m it means that the true value of the quantity is between 2.6 and 2.8 and therefore the uncertain figure in 2.7 is 7 because it can vary from 6 to 8 while 2 is the certain figure.

If I have the measure (1.9 ± 0.1) m, in this case the true value of the quantity is between 1.8 and 2.0 therefore to vary are both the digits 1 and 9 which would both seem uncertain and we would have no certain figure. Is my consideration correct?

Also sometimes you only write 3.0 m without the uncertainty, is this writing a convention to indicate (3.0 ± 0.1) m? Also for example 23 m stands for (23 ± 1) m?

(a) "If I have the measure (1.9 ± 0.1) m, in this case the true value of the quantity is between 1.8 and 2.0 therefore to vary are both the digits 1 and 9 which would both seem uncertain and we would have no certain figure. Is my consideration correct?"

Yes, but it is merely an accidental consequence of using the decimal (base 10) system that both digits are 'uncertain'; it has no deeper significance. Note also John Darby's answer.

(b)"Also sometimes you only write 3.0 m without the uncertainty, is this writing a convention to indicate (3.0 ± 0.1) m? Also for example 23 m stands for (23 ± 1) m?"

In the context of measurements the convention that I use is that $$x=3.0$$ m means that $$2.95\ \text m \leq x<3.05\ \text m$$ and that $$x=23$$ m means that $$22.5\ \text m \leq x<23.5\ \text m$$.

• Ok, about (b), is there an universal convention?
– asv
Oct 16, 2022 at 14:56
• @asv I believe that the convention that I've stated is standard. Oct 17, 2022 at 18:38

Let X be a random variable representing a set of measurements of a particular quantity. Typically, the uncertainty in X is represented as $$\bar X \pm \sigma_{\bar X}$$ where $$\bar X$$ is the mean of X and $$\sigma_{\bar X}$$ is the standard deviation of the mean of X, both evaluated from a set of measurements for X. The uncertainty is a standard deviation, not an absolute limit for the possible values of X.

See my answer to Uncertainty in repetitive measurements on this exchange for more details. This reference provides details on how to evaluate the standard deviation of both the data and of the mean.

Regarding significant figures. To avoid confusion, use scientific notation. The rightmost digit reported for the measurement should be uncertain, and the digit just to the left should be certain. The uncertainty should express the uncertainty in the right most digit.

For example, the rightmost digit reported for the mean should be uncertain and the digit just to the left should be certain. The standard deviation should express the uncertainty in the rightmost digit. For example, $$\bar X \pm \sigma_{\bar X} = 1.050 \times 10^1 \pm 4 \times 10^{-2}$$.

• Ok, the uncertainty is the standard deviation but what is the exact definition of uncertain figures of \bar{X} ? Also the exact definiton of significant figures?
– asv
Oct 16, 2022 at 14:56
• The exact definition of the std dev of the mean is provided in the referenced earlier topic in the above answer. I added a definition of significant figures. Oct 16, 2022 at 20:31