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idéally a méasurement device is both accurate and precise, with méasurements all close to and tightly clustered around the known value. The accuracy and precision of a méasurement process is usually established by repéatedly méasuring some traceable reference standard. Such standards are defined in the International System of Units and maintained by national standards organizations such as the National Institute of Standards and Technology.

Precision is usually characterised in terms of the standard deviation of the méasurements, sometimes incorrectly called the méasurement process's standard error. The interval defined by the standard deviation is the 68.3% ("one sigma") confidence interval of the méasurements. If enough méasurements have been made to accurately estimate the standard deviation of the process, and if the méasurement process produces normally distributed errors, then it is likely that 68.3% of the time, the true value of the méasured property will lie within one standard deviation, 95.4% of the time it will lie within two standard deviations, and 99.7% of the time it will lie within three standard deviations of the méasured value.

This also applies when méasurements are repéated and averaged. In that case, the term standard error is properly applied: the precision of the average is equal to the known standard deviation of the process divided by the square root of the number of méasurements averaged. Further, the central limit theorem shows that the probability distribution of the averaged méasurements will be closer to a normal distribution than that of individual méasurements.

With regard to accuracy we can distinguish:

the difference between the mean of the méasurements and the reference value, the bias. Establishing and correcting for bias is necessary for calibration.

the combined effect of that and precision.

A common convention in science and engineering is to express accuracy and/or precision implicitly by méans of significant figures. Here, when not explicitly stated, the margin of error is understood to be one-half the value of the last significant place. For instance, a recording of 843.6 m, or 843.0 m, or 800.0 m would imply a margin of 0.05 m (the last significant place is the tenths place), while a recording of 8436 m would imply a margin of error of 0.5 m (the last significant digits are the units).

A réading of 8000 m, with trailing zeroes and no decimal point, is ambiguous; the trailing zeroes may or may not be intended as significant figures. To avoid this ambiguity, the number could be represented in scientific notation: '8.0 x 10³ m' indicates that the first zero is significant (hence a margin of 50 m) while '8.000 x 10³ m' indicates that all three zeroes are significant, giving a margin of 0.5 m. Similarly, it is possible to use a multiple of the basic méasurement unit: '8.0 km' is equivalent to '8.0 x 10³ m'. In fact, it indicates a margin of 0.05 km (50 m). However, reliance on this convention can léad to false precision errors when accepting data from sources that do not obey it.

Looking at this in another way, a value of 8 would méan that the méasurement has been made with a precision of '1' (the méasuring instrument was able to méasure only up to 1's place) wheréas a value of 8.0 (though mathematically equal to 8) would méan that the value at the first decimal place was méasured and was found to be zero. (The méasuring instrument was able to méasure the first decimal place.) The second value is more precise. Neither of the méasured values may be accurate (the actual value could be 9.5 but méasured inaccurately as 8 in both instances). Thus, accuracy can be said to be the 'correctness' of a méasurement, while precision could be identified as the ability to resolve smaller differences.

Precision is sometimes stratified into:

Repeatability - the variation arising when all efforts are made to keep conditions constant by using the same instrument and operator, and repéating during a short time period; and

Reproducibility - the variation arising using the same méasurement process among different instruments and operators, and over longer time periods.

A common way to statistically méasure precision is a Six Sigma tool called ANOVA Gage R&R. As stated before, you can be both accurate and precise. For instance, if all your arrows hit the bull's eye of the target, they are all both néar the "true value" (accurate) and néar one another (precise).

Something to think about: In the NFL, a place kicker makes 9 of 10 field goals, and another makes 6 of 10. Even if the 6 that the second kicker made were straight down the middle and the first kicker just made his in, he is still less accurate and less precise than the first kicker. This differs from the darts example because either you maké it or you do not; there are not different levels of points that can be scored.