Accuracy and precision

In the fields of science, engineering, industry and statistics, accuracy is the degree of closeness of a measured or calculated quantity to its actual (true) value. Accuracy is closely related to precision, also called reproducibility or repeatability, the degree to which further measurements or calculations show the same or similar results. The results of calculations or a measurement can be accurate but not precise; precise but not accurate; neither; or both. A measurement system or computational method is called valid if it is both accurate and precise. The related terms are bias (non-random or directed effects caused by a factor or factors unrelated by the independent variable) and error (random variability), respectively.

Accuracy versus precision; the target analogy
Accuracy is the degree of veracity while precision is the degree of reproducibility. The analogy used here to explain the difference between accuracy and precision is the target comparison. In this analogy, repeated measurements are compared to arrows that are fired at a target. Accuracy describes the closeness of arrows to the bullseye at the target center. Arrows that strike closer to the bullseye are considered more accurate. The closer a system's measurements to the accepted value, the more accurate the system is considered to be so to be it.

To continue the analogy, if a large number of arrows are fired, precision would be the size of the arrow cluster. (When only one arrow is fired, precision is the size of the cluster one would expect if this were repeated many times under the same conditions.) When all arrows are grouped tightly together, the cluster is considered precise since they all struck close to the same spot, if not necessarily near the bullseye. The measurements are precise, though not necessarily accurate.

However, it is not possible to reliably achieve accuracy in individual measurements without precision — if the arrows are not grouped close to one another, they cannot all be close to the bullseye. (Their average position might be an accurate estimation of the bullseye, but the individual arrows are inaccurate.) See also Circular error probable for application of precision to the science of ballistics.

Quantifying accuracy and precision
Ideally a measurement device is both accurate and precise, with measurements all close to and tightly clustered around the known value. The accuracy and precision of a measurement process is usually established by repeatedly measuring 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.



In some literature, precision is defined as the reciprocal of variance, while many others still confuse precision with the confidence interval. The interval defined by the standard deviation is the 68.3% ("one sigma") confidence interval of the measurements. If enough measurements have been made to accurately estimate the standard deviation of the process, and if the measurement process produces normally distributed errors, then it is likely that 68.3% of the time, the true value of the measured 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 measured value.

This also applies when measurements are repeated 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 measurements averaged. Further, the central limit theorem shows that the probability distribution of the averaged measurements will be closer to a normal distribution than that of individual measurements.

With regard to accuracy we can distinguish:
 * the difference between the mean of the measurements 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 means 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 reading 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 &times; 103 m' indicates that the first zero is significant (hence a margin of 50 m) while '8.000 &times; 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 measurement unit: '8.0 km' is equivalent to '8.0 &times; 103 m'. In fact, it indicates a margin of 0.05 km (50 m). However, reliance on this convention can lead 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 mean that the measurement has been made with a precision of '1' (the measuring instrument was able to measure only up to 1's place) whereas a value of 8.0 (though mathematically equal to 8) would mean that the value at the first decimal place was measured and was found to be zero. (The measuring instrument was able to measure the first decimal place.) The second value is more precise. Neither of the measured values may be accurate (the actual value could be 9.5 but measured inaccurately as 8 in both instances). Thus, accuracy can be said to be the 'correctness' of a measurement, while precision could be identified as the ability to resolve smaller differences.

Precision is sometimes stratified into:
 * Repeatability &mdash; the variation arising when all efforts are made to keep conditions constant by using the same instrument and operator, and repeating during a short time period; and
 * Reproducibility &mdash; the variation arising using the same measurement process among different instruments and operators, and over longer time periods.

A common way to statistically measure 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 near the "true value" (accurate) and near one another (precise).

Accuracy in binary classification
"Accuracy" is also used as a statistical measure of how well a binary classification test correctly identifies or excludes a condition.

That is, the accuracy is the proportion of true results (both true positives and true negatives) in the population. It is a parameter of the test.


 * $$\text{accuracy}=\frac{\text{number of true positives}+\text{number of true negatives}}{\text{numbers of true positives}+\text{false positives} + \text{false negatives} + \text{true negatives}}$$

An accuracy of 100% means that the test identifies all sick and well people correctly.

Also see Sensitivity and specificity.

Accuracy may be determined from Sensitivity and Specificity, provided Prevalence is known, using the equation:


 * $$\text{accuracy}=(\text{sensitivity})(\text{prevalence}) + (\text{specificity})(1-\text{prevalence}) $$

The accuracy paradox for predictive analytics states that predictive models with a given level of accuracy may have greater predictive power than models with higher accuracy. It may be better to avoid the accuracy metric in favor of other metrics such as precision and recall.

Accuracy and precision in psychometrics and psychophysics
In psychometrics and psychophysics, the term accuracy is interchangeably used with validity and 'constant error', whereas 'precision' is a synonym for reliability and 'variable error' respectively. Validity of a measurement instrument or psychological test is established through experiment or correlation with behavior. Reliability is established with a variety of statistical technique (classically Cronbach's alpha).