Precision is a description of random errors, a measure of statistical variability.
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 (random and systematic), so high accuracy requires both high precision and high trueness.
Common technical definition
Accuracy is the proximity of measurement results to the true value; precision is the degree to which repeated (or reproducible) measurements under unchanged conditions show the same results.
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 (non-random or directed effects caused by a factor or factors unrelated to the independent variable) and error (random variability).
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 (or "justesse de tir"), 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 (abbreviated SI from French: Système international d'unités) 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 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 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 × 103 m indicates that the first zero is significant (hence a margin of 50 m) while 8.000 × 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 × 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. 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.
ISO definition (ISO 5725)
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According
to ISO 5725-1, Accuracy consists of trueness (proximity of measurement
results to the true value) and precision (repeatability or
reproducibility of the measurement)
A shift in the meaning of these terms appeared with the publication
of the ISO 5725 series of standards in 1994, which is also reflected in
the 2008 issue of the "BIPM International Vocabulary of Metrology"
(VIM), items 2.13 and 2.14.
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 (true) 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.
Accuracy of a target grouping according to BIPM and ISO 5725
Low accuracy due to poor precision
Low accuracy due to poor trueness
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) 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:
- Accuracy = (TP+TN)/(TP+TN+FP+FN)
where: TP = True positive; FP = False positive; TN = True negative; FN = False negative
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.[citation needed]
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 (documents correctly
retrieved), true negatives (documents correctly not retrieved), false
positives (documents incorrectly retrieved), and false negatives
(documents incorrectly not retrieved). 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 (true positives divided by
true+false positives), 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 (true positives divided by true positives+false
negatives). Less commonly, the metric of accuracy is used, is defined
as the total number of correct classifications (true positives plus true
negatives) 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 (k=10) 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.
