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Accuracy and precision are two measures 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, an error is not necessarily a " mistake ...
''. ''Accuracy'' is how close a given set of measurements (
observation Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...
s or readings) are to their ''
true value In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population exa ...
'', while ''precision'' is how close the measurements are to each other. In other words, ''precision'' is a description of '' random errors'', a measure of statistical variability. ''Accuracy'' has two definitions: # More commonly, it is a description of only '' systematic errors'', a measure of statistical bias of a given measure of
central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications ...
; low accuracy causes a difference between a result and a
true value In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population exa ...
;
ISO ISO is the most common abbreviation for the International Organization for Standardization. ISO or Iso may also refer to: Business and finance * Iso (supermarket), a chain of Danish supermarkets incorporated into the SuperBest chain in 2007 * Iso ...
calls this ''trueness''. # Alternatively, ISO defines accuracy as describing a combination of both types of observational error (random and systematic), so high accuracy requires both high precision and high trueness. In the first, more common definition of "accuracy" above, the concept is independent of "precision", so a particular set of data can be said to be accurate, precise, both, or neither. In simpler terms, given a statistical sample or set of data points from repeated measurements of the same quantity, the sample or set can be said to be ''accurate'' if their
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
is close to the true value of the quantity being measured, while the set can be said to be ''precise'' if their standard deviation is relatively small.


Common technical definition

In the fields of
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
engineering Engineering is the use of scientific method, 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 rang ...
, the accuracy of a
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
system is the degree of closeness of measurements of a
quantity Quantity or amount is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate discontinuity (mathematics), discontinuity and continuum (theory), continuity. Quantities can be compared in terms o ...
to that quantity's true
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
.
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. For the findings of a study to be reproducible means that results obtained by an experiment or an observational study or in a ...
and repeatability, 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 of actions or events expressed qualitatively or quantitatively. Possible results include advantage, disadvantage, gain, injury, loss, value and victory. There may be a range ...
s. Although the two words precision and accuracy can be synonymous in
colloquial Colloquialism (), also called colloquial language, everyday language or general parlance, is the linguistic style used for casual (informal) communication. It is the most common functional style of speech, the idiom normally employed in conver ...
use, they are deliberately contrasted in the context of the
scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific m ...
. The field of
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, 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, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
'' 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 Sample size determination is the act of choosing the number of observations or Replication (statistics), replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make stat ...
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 usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
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 A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon. In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...
, 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). It is the study of ...
, 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: Symbols * Colours, standards and guidons, kinds of military signs * Standard (emblem), a type of a large symbol or emblem used for identification Norms, conventions or requirements * Standard (metrology), an object th ...
. 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 pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
(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 an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
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 or an estimate of that standard deviation. If the statistic is the sample mean, it is called the 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 In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
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 There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the '' ari ...
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, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
. 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 calibration standard of known accuracy. Such a standard could be another measurement device of know ...
. *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 digits in the number that are reliable and necessary to indicate the quantity of something. If a number expre ...
. 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 precision than is justified; since precision is a limit to accuracy ( ...
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 engineering, precision is often taken as three times Standard Deviation of measurements taken, representing the range that 99.73% of measurements can occur within. For example, an ergonomist measuring the human body can be confident that 99.73% of their extracted measurements fall within ± 0.7 cm - if using the GRYPHON processing system - or ± 13 cm - if using unprocessed data.


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,BS ISO 5725-1: "Accuracy (trueness and precision) of measurement methods and results - Part 1: General principles and definitions.", p.1 (1994) 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 Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
, 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 Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
", previously specified in BS 5497-1, because it has different connotations outside the fields of science and engineering, as in medicine and law.


In classification


In binary classification

''Accuracy'' is also used as a statistical measure of how well a
binary classification Binary classification is the task of classifying the elements of a set into two groups (each called ''class'') on the basis of a classification rule. Typical binary classification problems include: * Medical testing to determine if a patient has c ...
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 ...
s and
true negative 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 resul ...
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) before and after a diagnostic test, respectively. ''Post-test probabi ...
. To make the context clear by the semantics, it is often referred to as the "Rand accuracy" or "
Rand index The RAND Corporation (from the phrase "research and development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces. It is financed ...
". It is a parameter of the test. The formula for quantifying binary accuracy is: \text=\frac where ; ; ; 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 digit ...
'' is used in this context to mean a different metric originating from the field of information retrieval ( see below).


In multiclass classification

When computing accuracy in multiclass classification, accuracy is simply the fraction of correct classifications: \text=\frac This is usually expressed as a percentage. For example, if a classifier makes ten predictions and nine of them are correct, the accuracy is 90%. Accuracy is also called top-1 accuracy to distinguish it from top-5 accuracy, common in
convolutional neural network In deep learning, a convolutional neural network (CNN, or ConvNet) is a class of artificial neural network (ANN), most commonly applied to analyze visual imagery. CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Netwo ...
evaluation. To evaluate top-5 accuracy, the classifier must provide relative likelihoods for each class. When these are sorted, a classification is considered correct if the correct classification falls anywhere within the top 5 predictions made by the network. Top-5 accuracy was popularized by the ImageNet challenge. It is usually higher than top-1 accuracy, as any correct predictions in the 2nd through 5th positions will not improve the top-1 score, but do improve the top-5 score.


In psychometrics and psychophysics

In
psychometrics Psychometrics is a field of study within psychology concerned with the theory and technique of measurement. Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and ...
and
psychophysics Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, m ...
, 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), a property of some disk arrays in computer storage * High availability * Reliability (computer networking), a ...
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 Cronbach's alpha (Cronbach's \alpha), also known as tau-equivalent reliability (\rho_T) or coefficient alpha (coefficient \alpha), is a reliability coefficient that provides a method of measuring internal consistency of tests and measures. Nume ...
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 languages. Simulation can be performed at varying degrees of physical abstraction, such as at the transistor level, gate le ...
, a common mistake in evaluation of accurate models is to compare a logic simulation model to a
transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch e ...
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, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases sp ...
s and
web search engine A search engine is a software system designed to carry out web searches. They search the World Wide Web in a systematic way for particular information specified in a textual web search query. The search results are generally presented in a ...
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 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 information that is known to be real or true, provided by direct observation and measurement (i.e. empirical evidence) as opposed to information provided by inference. Etymology The ''Oxford English Dictionary'' (s.v. "ground t ...
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 Evaluation measures for an information retrieval (IR) system assess how well an index, search engine or database returns results from a collection of resources that satisfy a user's query. They are therefore fundamental to the success of informati ...
, 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.


In cognitive systems

In cognitive systems, accuracy and precision is used to characterize and measure results of a cognitive process performed by biological or artificial entities where a cognitive process is a transformation of data, information, knowledge, or wisdom to a higher-valued form. (
DIKW Pyramid The DIKW pyramid, also known variously as the DIKW hierarchy, wisdom hierarchy, knowledge hierarchy, information hierarchy, information pyramid, and the data pyramid, refers loosely to a class of models for representing purported structural and/o ...
) Sometimes, a cognitive process produces exactly the intended or desired output but sometimes produces output far from the intended or desired. Furthermore, repetitions of a cognitive process do not always produce the same output. ''Cognitive accuracy'' (CA) is the propensity of a cognitive process to produce the intended or desired output. ''Cognitive precision'' (CP) is the propensity of a cognitive process to produce only the intended or desired output. To measure
augmented cognition Augmented cognition is an interdisciplinary area of psychology and engineering, attracting researchers from the more traditional fields of human-computer interaction, psychology, ergonomics and neuroscience. Augmented cognition research generally ...
in human/cog ensembles, where one or more humans work collaboratively with one or more cognitive systems (cogs), increases in cognitive accuracy and cognitive precision assist in measuring the degree of
cognitive augmentation Intelligence amplification (IA) (also referred to as cognitive augmentation, machine augmented intelligence and enhanced intelligence) refers to the effective use of information technology in augmenting human intelligence. The idea was first pro ...
.


See also

* Bias-variance tradeoff in statistics and machine learning *
Accepted and experimental value In science, and most specifically chemistry, the accepted value denotes a value of a substance accepted by almost all scientists and the experimental value denotes the value of a substance's properties found in a localized lab. See also *Accurac ...
*
Data quality Data quality refers to the state of qualitative or quantitative 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 operations, decision making a ...
*
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, manufactured object, system, or service; # other measured values (such as temperature, hum ...
*
Exactness (disambiguation) In mathematics, exactness may refer to: * Exact category * Exact functor ** Landweber exact functor theorem * Exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, ...
*
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 d ...
*
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 Information quality (IQ) is the quality of the content of information systems. It is often pragmatically defined as: "The fitness for use of the information provided". IQ frameworks also provides a tangible approach to assess and measure DQ/IQ in a ...
* Measurement uncertainty *
Precision (statistics) In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, P = \Sigma^. For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the ...
*
Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
*
Random and 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, an error is not necessarily a "mistake ...
*
Sensitivity and specificity ''Sensitivity'' and ''specificity'' mathematically describe the accuracy of a test which reports the presence or absence of a condition. Individuals for which the condition is satisfied are considered "positive" and those for which it is not are ...
*
Significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expre ...
* Statistical significance


References


External links


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 TED may refer to: Economics and finance * TED spread between U.S. Treasuries and Eurodollar Education * ''Türk Eğitim Derneği'', the Turkish Education Association ** TED Ankara College Foundation Schools, Turkey ** Transvaal Education Depa ...
-Ed {{DEFAULTSORT:Accuracy And Precision Biostatistics Metrology Psychometrics ISO standards Software quality