TheInfoList

In a set of measurements, 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 Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In statistics Statistics is the discipline that co ...
'', a measure of
statistical bias Statistical bias is a feature of a statistics, statistical technique or of its results whereby the expected value of the results differs from the true underlying quantitative parameter#Statistics and econometrics, parameter being estimation theory, ...
; low accuracy causes a difference between a result and a "true" value.
ISO The International Organization for Standardization (ISO ) is an international standard An international standard is a technical standard A technical standard is an established norm (social), norm or requirement for a repeatable technical task w ...
calls this ''trueness''. # Alternatively, ISO defines accuracy as describing a combination of both types of
observational error Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In statistics Statistics is the discipline that co ...
above (random and systematic), so high accuracy requires both high precision and high trueness. ''Precision'' is a description of ''
random errors Observational error (or measurement error) is the difference between a measurement, measured value of a quantity and its true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In statistics, an error is not a "mistake". ...
'', a measure of
statistical variability In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wi ...
. 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 Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ... and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ... , the accuracy of a
measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ... system is the degree of closeness of measurements of a
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measu ...
to that quantity's true
value Value or values may refer to: * Value (ethics) In ethics Ethics or moral philosophy is a branch of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, E ...
.
JCGM 200:2008 International vocabulary of metrology
— Basic and general concepts and associated terms (VIM)
The precision of a measurement system, related to
reproducibility Reproducibility, also known as replicability and repeatability, is a major principle underpinning the scientific method The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the develop ...
and
repeatability Repeatability or test–retest reliability is the closeness of the agreement between the results of successive measurements Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object ...
, is the degree to which repeated measurements under unchanged conditions show the same
result A result (also called upshot) is the final consequence of a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes a ... s. Although the two words precision and accuracy can be
synonymous A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are al ...
in
colloquial Colloquialism or colloquial language is the linguistic style used for casual (informal) communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-Eur ...
use, they are deliberately contrasted in the context of the
scientific method The scientific method is an empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence ... . The field of
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ... , where the interpretation of measurements plays a central role, prefers to use the terms ''
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded Open-mindedness is receptiveness to new ideas. Open-mindedness relates to the way in which people approach the views and kn ...
'' 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 Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In statistics Statistics is the discipline that co ...
, then increasing the
sample size Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample In statistics and quantitative research methodology, a sample is a set of individuals or objects collected or selected ...
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 In common parlance, randomness is the apparent or actual lack of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric p ... or directed effects caused by a factor or factors unrelated to the
independent variable Dependent and Independent variables are variables in mathematical modeling A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to f ...
) 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 In the broadest definition, a sensor is a device, module, machine, or subsystem whose purpose is to detect events or changes in its environment and send the information to other electronics, frequently a computer processor A central proc ...
, which is the smallest change in the underlying physical quantity that produces a response in the measurement. In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). 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 (''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 Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requirement * Standard (metrology), an object that bears a defined relationship to a unit of ...
. Such standards are defined in the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ... (abbreviated SI from French: ''Système international d'unités'') and maintained by national
standards organization A standards organization, standards body, standards developing organization (SDO), or standards setting organization (SSO) is an organization whose primary function is developing, coordinating, promulgating, revising, amending, reissuing, interpr ...
s such as the
National Institute of Standards and Technology The National Institute of Standards and Technology (NIST) is a physical sciences Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, e ...
in the United States. This also applies when measurements are repeated and averaged. In that case, the term
standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpret ...
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 In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by express ...
shows that the
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
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 There are several kinds of mean in mathematics, especially in statistics. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
of the measurements and the reference value, the
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded Open-mindedness is receptiveness to new ideas. Open-mindedness relates to the way in which people approach the views and kn ...
. Establishing and correcting for bias is necessary for
calibration In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a Standard (metrology), calibration standard of known accuracy. Such a standard could be another measuremen ... . *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 Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are Numerical digit, digits in the number that are reliable and absolutely necessary to indicate the quantity of som ...
. Where 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 843 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 zeros and no decimal point, is ambiguous; the trailing zeros 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 zeros are significant, giving a margin of 0.5 m. Similarly, one can use a multiple of the basic measurement unit: 8.0 km is equivalent to 8.0 × 103 m. It indicates a margin of 0.05 km (50 m). However, reliance on this convention can lead to
false precision False precision (also called overprecision, fake precision, misplaced precision and spurious precision) occurs when numerical data are presented in a manner that implies better Accuracy and precision, precision than is justified; since 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. Alternatively, in a scientific context, if it is desired to indicate the margin of error with more precision, one can use a notation such as 7.54398(23) × 10−10 m, meaning a range of between 7.54375 and 7.54421 × 10−10 m. 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.

# In binary classification

''Accuracy'' is also used as a statistical measure of how well a
binary classification Binary classification is the task of Statistical classification, classifying the elements of a Set (mathematics), set into two groups on the basis of a classification rule. Typical binary classification problems include: * Medical testing to determ ...
test correctly identifies or excludes a condition. That is, the accuracy is the proportion of correct predictions (both
true positive A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition such as a disease when the disease is not present, while a false negative is the opposite error where the test result inc ...
s and
true negative A false positive is an error in binary classification Binary classification is the task of Statistical classification, classifying the elements of a Set (mathematics), set into two groups on the basis of a classification rule. Typical binary classi ...
s) among the total number of cases examined. As such, it compares estimates of
pre- and post-test probability Pre-test probability and post-test probability (alternatively spelled pretest and posttest probability) are the probabilities of the presence of a condition (such as a disease A disease is a particular abnormal condition that negatively a ...
. To make the context clear by the semantics, it is often referred to as the "Rand accuracy" or "
Rand index The Rand index or Rand measure (named after William M. Rand) in statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, in ...
". 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 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 Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digits ...
'' is used in this context to mean a different metric originating from the field of information retrieval (
see below See or SEE may refer to: Arts, entertainment, and media * Music: ** See (album), ''See'' (album), studio album by rock band The Rascals *** "See", song by The Rascals, on the album ''See'' ** See (Tycho song), "See" (Tycho song), song by Tycho * T ...
).

# In psychometrics and psychophysics

In
psychometrics Psychometrics is a field of study within concerned with the theory and technique of . Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and related activities. P ...
and
psychophysics Psychophysics quantitatively investigates the relationship between physical stimulus (physiology), stimuli and the sensation (psychology), sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the re ...
, the term ''accuracy'' is interchangeably used with
validity Validity or Valid may refer to: Science/mathematics/statistics: * Validity (logic), a property of a logical argument * Scientific: ** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments ** ...
and ''constant error''. ''Precision'' is a synonym for
reliability Reliability, reliable, or unreliable may refer to: Science, technology, and mathematics Computing * Data reliability (disambiguation), Data reliability, a property of some disk arrays in computer storage * High availability * Reliability (computer ...
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 Tau-equivalent reliability (_), also known as Cronbach's alpha or coefficient alpha, is the most common test score reliability coefficient for single administration (i.e., the reliability of persons over items holding occasion fixed). Recent studie ...
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 Logic simulation is the use of simulation software to predict the behavior of digital circuits and hardware description language In computer engineering, a hardware description language (HDL) is a specialized computer language used to describe ...
, a common mistake in evaluation of accurate models is to compare a logic simulation model to a
transistor upright=1.4, gate Candi bentar, a typical Indonesian gate that is often found on the islands of Java">Indonesia.html" ;"title="Candi bentar, a typical Indonesia">Candi bentar, a typical Indonesian gate that is often found on the islands o ... 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
database In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and sof ... s and
web search engine A search engine is a software system A software system is a system A system is a group of interacting Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is ...
s, are evaluated by many different metrics, some of which are derived from the
confusion matrix In the field of machine learning and specifically the problem of statistical classification, a confusion matrix, also known as an error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a su ... , 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 , and , precision and recall are performance metrics that apply to data retrieved from a , or . Precision (also called ) is the fraction of relevant instances among the retrieved instances, while recall (also known as ) is the fraction of r ...
. 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 Ground truth is a term used in various fields to refer to information provided by direct observation (i.e. empirical evidence Empirical evidence is the information Information can be thought of as the resolution of uncertainty; it ans ...
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.

* Bias-variance tradeoff in statistics and machine learning * Accepted and experimental value *
Data quality Data quality refers to the state of or pieces of information. There are many definitions of data quality, but data is generally considered high quality if it is "fit for tsintended uses in , and ". Moreover, data is deemed of high quality if it c ...
*
Engineering tolerance Engineering tolerance is the permissible limit or limits of variation in: # a physical dimension; # a measured value or physical property of a material, manufacturing, manufactured object, system, or service; # other measured values (such as tem ...
* Exactness (disambiguation) *
Experimental uncertainty analysis Experimental uncertainty analysis is a technique that analyses a ''derived'' quantity, based on the uncertainties in the experimentally ''measured'' quantities that are used in some form of mathematical relationship (" model") to calculate that de ...
*
F-score In statistics, statistical analysis of binary classification, the F-score or F-measure is a measure of a test's Accuracy_and_precision#In_binary_classification, accuracy. It is calculated from the Precision (information retrieval), precision and ...
* Hypothesis tests for accuracy * Information quality *
Measurement uncertainty In metrology Metrology is the scientific study of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to c ...
*
Precision (statistics) In statistics, precision is the Multiplicative inverse, reciprocal of the variance, and the precision matrix (also known as concentration matrix) is the matrix inverse of the covariance matrix. Thus, if we are considering a single random variable ...
*
Probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ... * Random and systematic errors *
Sensitivity and specificity ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition, in comparison to a ‘Gold standard (test), Gold Standard’ or definition. *Sensitivity (True Positive Rate ... *
Significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are Numerical digit, digits in the number that are reliable and absolutely necessary to indicate the quantity of som ...
*
Statistical significance In statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to determine a possible conclusion from two different, and likely conflicting, hypotheses. In a statistical hypothesis test, a null hypothe ...

# References

BIPM - Guides in metrology
''Guide to the Expression of Uncertainty in Measurement (GUM) and International Vocabulary of Metrology (VIM)''
"Beyond NIST Traceability: What really creates accuracy"
''Controlled Environments'' magazine
Precision and Accuracy with Three Psychophysical Methods
''Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results''
Accuracy vs Precision
— a brief video by Matt Parker
What's the difference between accuracy and precision?
by Matt Anticole at TED-Ed
Precision and Accuracy exam study guide
{{DEFAULTSORT:Accuracy And Precision Biostatistics Metrology Psychometrics ISO standards Software quality