Identifying certain digits

Question

The result of a measurement is $57.64$ $\pm$ $0.38$ (for the sake of simplicity, I omit the measurement unit). I would know which are the certain digits in $57.64$ $\pm$ $0.38$. I know that $57.64$ has $4$ significant digits. According to the rule

the significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit

I should say that $57.64$ $\pm$ $0.38$ has $2$ certain digits, in particular they are $5$ and $7$. But, if I rewrite $57.64$ $\pm$ $0.38$ as the interval $[57.26; 58.02]$, I see that the only digit that remains the same is $5$. Therefore, is it still possible to say that $7$ is a certain digit ? What is the definition of "certain digit" ?

Thank you in advance.

Answer 1

Per:

https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Map%3A_Introductory_Chemistry_(Tro)/02%3A_Measurement_and_Problem_Solving/2.02%3A_Significant_Figures_-_Writing_Numbers_to_Reflect_Precision

it appears "certain digits" refers to a single measurement on a graduated analog device such as a gauge (such as a thermometer or manometer), a clock, or ruler.

In such a case you can count tic marks with certainty, and then can estimate the fraction of the gap until the next tic for an uncertain "digit" (e.g., $\frac 1 2 \rightarrow 0.5$).

This idea does not carry over the traditional measurement:

$$ \bar x \pm \sigma_x $$

in which $\bar x$ is the average of many measurements and $\sigma_x$ is their standard deviation (or unbiased estimator thereof).

Per the example:

$$ \bar x = 57.64 $$

and $$ \sigma_x = 0.38 $$

the true value could be $x=60.0$. It a 6.2 sigma fluctuation, which is very unlikely, but it is a possibility, meaning no digit is certain.