# Accuracy and precision

In measurement of a set,

**accuracy**is closeness of the measurements to a specific value, while

**precision**is the closeness of the measurements to each other.

*Accuracy*has two definitions:

- More commonly, it is a description of
*systematic errors*, a measure of statistical bias; low accuracy causes a difference between a result and a "true" value. ISO calls this*trueness*. - Alternatively, ISO defines accuracy as describing a combination of both types of observational error above, so high accuracy requires both high precision and high trueness.

*Precision*is a description of

*random errors*, a measure of statistical variability.

In simpler terms, given a set of data points from repeated measurements of the same quantity, the set can be said to be

*accurate*if their average is close to the

*true value*of the quantity being measured, while the set can be said to be

*precise*if the values are close to each other. In the first, more common definition of "accuracy" above, the two concepts are independent of each other, so a particular set of data can be said to be either accurate, or precise, or both, or neither.

## Common technical definition

In the fields of science and engineering, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's true value. The precision of a measurement system, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditions show the same results. Although the two words precision and accuracy can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.The field of statistics, where the interpretation of measurements plays a central role, prefers to use the terms

*bias*and

*variability*instead of accuracy and precision: bias is the amount of inaccuracy and variability is the amount of imprecision.

A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. The result would be a consistent yet inaccurate string of results from the flawed experiment. Eliminating the systematic error improves accuracy but does not change precision.

A measurement system is considered

*valid*if it is both

*accurate*and

*precise*. Related terms include

*bias*and

*error*.

The terminology is also applied to indirect measurements—that is, values obtained by a computational procedure from observed data.

In addition to accuracy and precision, measurements may also have a measurement resolution, which is the smallest change in the underlying physical quantity that produces a response in the measurement.

In numerical analysis, accuracy is also the nearness of a calculation to the true value; while precision is the resolution of the representation, typically defined by the number of decimal or binary digits.

In military terms, accuracy refers primarily to the accuracy of fire, the precision of fire expressed by the closeness of a grouping of shots at and around the centre of the target.

### Quantification

In industrial instrumentation, accuracy is the measurement tolerance, or transmission of the instrument and defines the limits of the errors made when the instrument is used in normal operating conditions.Ideally a measurement device is both accurate and precise, with measurements all close to and tightly clustered around the true 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 the United States.

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 reading of 8,000 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 × 10

^{3}m indicates that the first zero is significant while 8.000 × 10

^{3}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 × 10

^{3}m. In fact, it indicates a margin of 0.05 km. However, reliance on this convention can lead to false precision errors when accepting data from sources that do not obey it. For example, a source reporting a number like 153,753 with precision +/- 5,000 looks like it has precision +/- 0.5. Under the convention it would have been rounded to 154,000.

Precision includes:

*repeatability*— 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*— the variation arising using the same measurement process among different instruments and operators, and over longer time periods.

According to ISO 5725-1, the general term "accuracy" is used to describe the closeness of a measurement to the true value. When the term is applied to sets of measurements of the same measurand, it involves a component of random error and a component of systematic error. In this case trueness is the closeness of the mean of a set of measurement results to the actual value and precision is the closeness of agreement among a set of results.

ISO 5725-1 and VIM also avoid the use of the term "bias", previously specified in BS 5497-1, because it has different connotations outside the fields of science and engineering, as in medicine and law.

## 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 correct predictions among the total number of cases examined. To make the context clear by the semantics, it is often referred to as the "Rand accuracy" or "Rand index". It is a parameter of the test.

The formula for quantifying binary accuracy is:

where: TP = True positive; FP = False positive; TN = True negative; FN = False negative

Note that, in this context, the concepts of trueness and precision as defined by ISO 5725-1 are not applicable. One reason is that there is not a single "true value" of a quantity, but rather two possible true values for every case, while accuracy is an average across all cases and therefore takes into account both values. However, the term

*precision*is used in this context to mean a different metric originating from the field of information retrieval.

## In psychometrics and psychophysics

In psychometrics and psychophysics, the term*accuracy*is interchangeably used with validity and

*constant error*.

*Precision*is a synonym for reliability and

*variable error*. The validity of a measurement instrument or psychological test is established through experiment or correlation with behavior. Reliability is established with a variety of statistical techniques, classically through an internal consistency test like Cronbach's alpha to ensure sets of related questions have related responses, and then comparison of those related question between reference and target population.

## In logic simulation

In logic simulation, a common mistake in evaluation of accurate models is to compare a logic simulation model to a transistor circuit simulation model. This is a comparison of differences in precision, not accuracy. Precision is measured with respect to detail and accuracy is measured with respect to reality.## In information systems

Information retrieval systems, such as databases and web search engines, are evaluated by many different metrics, some of which are derived from the confusion matrix, which divides results into true positives, true negatives, false positives, and false negatives. Commonly used metrics include the notions of precision and recall. In this context, precision is defined as the fraction of retrieved documents which are relevant to the query, using a set of ground truth relevant results selected by humans. Recall is defined as the fraction of relevant documents retrieved compared to the total number of relevant documents. Less commonly, the metric of accuracy is used, is defined as the total number of correct classifications divided by the total number of documents.None of these metrics take into account the ranking of results. Ranking is very important for web search engines because readers seldom go past the first page of results, and there are too many documents on the web to manually classify all of them as to whether they should be included or excluded from a given search. Adding a cutoff at a particular number of results takes ranking into account to some degree. The measure precision at k, for example, is a measure of precision looking only at the top ten search results. More sophisticated metrics, such as discounted cumulative gain, take into account each individual ranking, and are more commonly used where this is important.