# Identifying certain digits

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" ?

• Engineer's response: If the difference matters, you should improve your experiment to reduce the uncertainty rather than use a more sophisticated error analysis. Whoever you need to present your data to probably won't understand the more sophisticated error analysis anyway. (The only case this doesn't apply is if your audience is the readers of a metrology journal) May 8, 2020 at 16:14
• I have never heard of the "certain digits" rubric in measurement. You don't want your analysis to depend on the base of your integers (presumably base-10 has different 'certain' digits vs. base-64 or base-2).
– JEB
May 8, 2020 at 16:15
• @JEB you can see this definition here chem.libretexts.org/Bookshelves/Introductory_Chemistry/… May 8, 2020 at 16:29
• @ThePhoton I totally agree with you. My question is only a morbid curiosity about the definition of these "certain digits" found in some physics textbook. May 8, 2020 at 16:38

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.

• Thanks for the answer. Are you trying to tell me that the "certain digits" in one number is a different thing from the "certain digits" of an interval (like $57.64\pm0.38$) ? May 9, 2020 at 8:39
• idk. Counting tics and seeing the gauge come up short on the next one is a activity of extreme certainty. Averaging random measurements is not. Whence did you get $57.6\pm 0.4$? if it's $\bar x \pm \sigma_x$, it is not a non arbitrary interval.
– JEB
May 9, 2020 at 13:40
• $57.6 \pm 0.4$ is a $\bar x \pm \sigma_x$ May 9, 2020 at 14:26
• so then it is an arbitrary interval (the interval with 1 sigma probability, where "1" is an arbitrary custom), the endpoints don't mean anything definite w.r.t to the digits. Per a ruler, when you pass the 4th tick, and are short of the 5th, you are certain the answer is 4.x.
– JEB
May 9, 2020 at 17:12
• ok, it's more clearly now! May 9, 2020 at 22:19