# What is precision?

I’ve learnt that accuracy is how close to the true value a measurement is (i.e. low discrepancy between the experimental and theoretical values). Precision, on the other hand, is the reproducibility of experimental results—high precision means that the data points are closely located. However, precision can also refer to how many significant figures a measurement has.

Are those two definitions of precision the same thing? What exactly is precision?

• Although not a precise duplicate I would suggest that @LubošMotl 's answer explains the difference between accuracy and precision very well? It is also worth reading some of the other answers. physics.stackexchange.com/q/126895 Sep 3, 2016 at 8:10
• @Farcher I think my original question was a little ambiguous and I’ve made a slight change. Hopefully it now reflects what I’m asking—not the difference between accuracy and precision, but the definition of precision. Sep 3, 2016 at 8:17
• In the @LubošMotl answer he writes "Before 2008, people would agree that "precision" refers to the typical difference between individual measurements, and the precision is good if a "statistical error" is low or if the measurements are producing "many significant figures" for the result." and that probably is still a good way of explaining what precision is? Sep 3, 2016 at 8:23
• @Farcher So many significant figures = low difference between individual measurements? Why is that? Sep 3, 2016 at 9:23
• in radar measurement $precision = \frac {1}{standard \: deviation}$ Sep 3, 2016 at 15:24

I would say that precision is the resolution of a measurement (eg how many decimal places a measurement has) and accuracy is how true that number is traceable to an official standard.

So you can buy a cheap digital thermometer that will read out to 0.01deg but will be 1deg wrong in the absolute value if you put it on eg melting ice.

• So how does that relate to precision’s definition as the closeness of measurement values? Sep 3, 2016 at 13:26
• this kind of usage, that is equating accuracy with resolution, has created immense amount of confusion for the word $resolution$ could be given another very good and useful meaning, namely the ability to distinguish two equal strength "sources" (of information) in noise. See, Rayleigh's use of the notion of "diffraction limited resolution" of two point sources. Sep 3, 2016 at 15:29
• @hyportnex - I meant the exact opposite. Precision = resolution not accuracy. If I say c=3E8 m/s that is accurate (ie correct) but not precise. If I say it is 300000000 m/s that is precise but inacurate ie wrong Sep 3, 2016 at 18:36

Think of data points in just 1 dimension: you can plot their distribution with values on an axis and frequencies on the other axis. If the points are closely located, their dispersion spans a small region on the value axis. Therefore you need more significant figures to express the region in which the data are located. You can think of an higher precision, or a smaller dispersion, as a deeper insight in determining the location parameter (the mean value) and the dispersion parameter (standard deviation) of the distribution.

Those two definitions relate to the same concept but in two different frames: structure of data (they are closely located) and their rapresentation (how many significant figures you need to express the parameters of the distribution).

in my opinion precision tells us to what resolution or limit the quantity is measured, since the number of significant figures implies resolution in measurement, we say that more number of significant figures implies higher is the precision.consider an example, i m taking measurement of a rod and i have got two sets of results 1. 5.5cm and 5.6cm , 2. 5.0005cm and 5.0006cm, since second set of measurement contain mesurements with 5 number of significant figures hence measurements in second set are highly precise. hope you got what i m trying to say

When measuring the amount of a quantity, any measuring instrument can (only) ascertain that the actual value of the amount of the quantity lies somewhere between two amounts.

The smallest difference between two amounts between which the measuring instrument can always ascertain that the actual amount of any measured quantity will lie is defined as the precision of the measuring instrument. Eg. The precision of a meter scale with 1mm divisions is 1mm. Supposing the length of a certain (mythical/hypothetical) rod is preordained to be exactly equal to 1.499900000.... meters. A meter stick with 1mm division can only ascertain when measuring the rod, that the actual value lies between 1.499 m (or 1499 mm) and 1.500 m (or 1500 mm).

"Significant figures" is actually just a set of rules for writing the numerical value of the amount of a measured quantity such that by just looking at the number, a person (who knows these rules, and knows that the number was written following these rules) can easily figure out the precision of measuring instrument with which the measurement was made.