# What's the number of significant digits for error bars? [duplicate]

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What's the number of significant digits for error bars? For example, the Particle Data Group lists the proton mass as,

$$938.2720813 \pm 0.000058$$

Should I take $$0.000058$$ as having 2 significant digits or 10, the same as that for $$938.2720813$$?

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## 1 Answer

$$0.0000058$$ (the number actually in that link) technically has two significant figures by itself, but its number of significant figures is irrelevant in this context.

Significant figures are a way to show how certain you are in your measurement. For example, if I just reported the measurement as $$938.2720813$$, what I am saying is that I know for certain that my measurement falls between $$938.272081$$ and $$938.272082$$, and I estimate it to be around $$938.2720813$$. This would be like if I used a ruler with tick marks only every centimeter to measure a distance. I would find between which two tick marks my distance lies between (for example, between $$3$$ and $$4$$), and then I would estimate the next decimal place (by eye it looks like $$3.3\ \rm{cm}$$, but since I have no finer tick marks, I am not entirely sure about the $$0.3\ \rm{cm}$$).

However, this is not as precise as we would like. This is when we include actual errors, or numerical values for our uncertainty in the measurement. So the number you give really says "I think the true value lies somewhere $$0.0000058$$ above or below $$938.2720813$$." In my previous example this would be like reporting the measurement as $$3.3\pm 0.1\ \rm{cm}$$ (I think it is about $$3.3$$, but I might be off by a tenth of a centimeter on either side based on my eyesight. Of course, this a crude example; there are much more sophisticated and precise ways of determining errors or uncertainty).

In other words, significant figures are a good way to show estimates on your measurements, but it is more precise to give uncertainties in terms of an actual number. This then removes the need to worry about significant figures. All you need to do is make sure your measurement and the corresponding uncertainly goes out to the same number of decimals, since it wouldn't make sense to have a more precise knowledge of one over the other.

• Doesn't a reported figure of $938.2720813$ mean that you're certain that actual value is between $938.27208129$ and $938.27208131$? Why do you interpret the LSD as so uncertain as in your answer? – Ruslan Dec 21 '18 at 11:27
• @Ruslan No, you are getting too precise then. The final number reported is the one you are uncertain about. If the reported number was perhaps $938.27208130$ then I think what you say would make more sense. Go to the ruler example. Saying $3.3\ \rm{cm}$ doesn't mean we might be off by a hundreth of a centimeter. Your are supposed to go one decimal place past what you are sure about based on the instrument. – Aaron Stevens Dec 21 '18 at 13:36
• @Ruslan I am not saying that in using significant figures the uncertainty is as big as between $938.272081$ and $938.272082$. I'm saying we are entirely sure it is between those values. Based on significant figures we might say the actual value is between $938.2720812$ and $938.2720814$. Or maybe the uncertainty is a little bit bigger, say between $938.2720811$ and $938.2720815$. We don't know for sure what the uncertainty is when using significant figures. That's why it's better to report actual uncertainty with a number. – Aaron Stevens Dec 21 '18 at 13:42