# A doubt related to Significant Digits

Could someone please explain this statement to me

"Reporting the result of measurement that includes more digits than significant digits is superfluous and also misleading since it would give a wrong idea about the precision of measurement."

Also, shouldn't the word accuracy be used instead of precision because isn't precision the closeness of various measurements for the same quantity and only one measurement is being talked of here.

• Precision is the right word because the closeness of various measurements of the same quantity determines the significant digits. – Aziraphale Apr 12 '18 at 6:30
• – Farcher Apr 12 '18 at 7:49
• More on significant figures. – Qmechanic Apr 12 '18 at 8:41

Suppose I have a ruler marked in millimeters, and I can estimate lengths measured with my ruler to about half a millimetre by estimating by eye how far between marks my length is. So when I measure a length my precision is $\pm 0.5$ mm.

If I measure a length then report it as $3.14159$ mm this would be be misleading because it would imply that I have measured it to an accuracy of one part in the last digit i.e. $\pm 0.00001$ mm. The correct way for me to report it would be $3.0 \pm 0.5$ mm. That's what the statement means.

Note that I used the word precision in my first paragraph. That's because the precision refers to how accurately I can use my measuring equipment i.e. my ruler. But suppose it turns out my one metre rule was actually only $90$ cm long due to a manufacturing error. That means no matter how precisely I do my measurement it's still going to have an inaccuracy of $10$%. The accuracy refers to the comparison of my reported result with the real length while the precision refers only to how precisely I can use my equipment.

In this case my experiment has a systematic error due to my badly manufactured ruler. Systematic errors are the bane of an experimenter's life because they are not easily detected. The precision of my measurement can be estimated just by doing the measurement lots of times and seeing how much variation there is in my results, but a systematic error affects all the measurements in the same way so it isn't easily detected.

In general, it's best to give all potentially relevant information. So, in general, that quote provides poor advice.

However, it's important to write your number correctly. For example, a number understood to be approximately represented as "$1.2$" means "$1.2{\pm}0.05$" or "$\left[1.15,\,1.25\right)$".

If you want to add more information, you can't just tack the numbers on because that changes the implied precision of the value. But you can still do it. For example, Wikipedia lists a current value of the fine-structure constant, $\alpha$, as$${\alpha} {\quad}={\quad}\frac{e^2}{4π{\epsilon}_{0}{\hbar}c} {\quad}={\quad}0.007~297~352~566~4~(17),$$in which the last "$64$" is understood to not be significant as it's modified by "$\left(17\right)$". Translated, this means: $${\alpha} {\quad}{\approx}{\quad} \begin{array}{rl} & 0.007~~297~~352~~566~~4 \\ {\pm} & 0.000~~000~~000~~001~~7 \end{array},$$which if we had to write that using regular significant figures, would just be$${\alpha} {\quad}{\approx}{\quad} 0.007~~297~~352~~57.$$So, this notation can be used if you want to express additional information in a context in which the reader would assume that the number's form implies its uncertainty by codifying the difference in perceived uncertainty with an appropriate error qualification.

In general, though, the exact statistics of measurements can be pretty complicated; for example, error doesn't need to be normally distributed. So when it really matters, care must be taken to precisely specify what's meant.

In practice, most folks try to keep it simple to avoid a hassle. The basic significant figures system has been designed to be simple while working well-enough in many simple cases.