In the fields of
scienceScience is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
,
engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
,
industryIndustry refers to the production of an economic good or service within an economy.-Industrial sectors:There are four key industrial economic sectors: the primary sector, largely raw material extraction industries such as mining and farming; the secondary sector, involving refining, construction,...
and
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the
accuracy of a
measurementMeasurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
system is the degree of closeness of measurements of a
quantityQuantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...
to that quantity's actual (true)
valueIn mathematics, value commonly refers to the 'output' of a function. In the most basic case, that of unary, single-valued functions, there is one input and one output .The function f of the example is real-valued, since each and every possible function value is real...
. The
precision of a measurement system, also called
reproducibilityReproducibility is the ability of an experiment or study to be accurately reproduced, or replicated, by someone else working independently...
or
repeatabilityRepeatability or test-retest reliability is the variation in measurements if they would have been taken by a single person or instrument on the same item and under the same conditions. A less-than-perfect test-retest reliability causes test-retest variability. Such variability can be caused by, for...
, is the degree to which repeated measurements under unchanged conditions show the same
resultA result 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 of possible outcomes associated with an event depending on the point of...
s. Although the two words
reproducibilityReproducibility is the ability of an experiment or study to be accurately reproduced, or replicated, by someone else working independently...
and
repeatabilityRepeatability or test-retest reliability is the variation in measurements if they would have been taken by a single person or instrument on the same item and under the same conditions. A less-than-perfect test-retest reliability causes test-retest variability. Such variability can be caused by, for...
can be synonymous in colloquial use, they are deliberately contrasted in the context of the
scientific methodScientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
.
A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an
experimentAn experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...
contains a
systematic errorSystematic errors are biases in measurement which lead to the situation where the mean of many separate measurements differs significantly from the actual value of the measured attribute. All measurements are prone to systematic errors, often of several different types...
, then increasing the
sample sizeSample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...
generally increases precision but does not improve accuracy. The end 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 designated
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 variableThe terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
) 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 the case of full reproducibility, such as when rounding a number to a representable
floating pointIn computing, floating point describes a method of representing real numbers in a way that can support a wide range of values. Numbers are, in general, represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16...
number, the word
precision has a meaning not related to reproducibility. For example, in the IEEE 754-2008 standard it means the number of bits in the
significandThe significand is part of a floating-point number, consisting of its significant digits. Depending on the interpretation of the exponent, the significand may represent an integer or a fraction.-Examples:...
, so it is used as a measure for the relative accuracy with which an arbitrary number can be represented.
Accuracy versus precision: the target analogy
Accuracy is the degree of veracity while in some contexts precision may mean the degree of reproducibility.
The
analogyAnalogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
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 shot at a target. Accuracy describes the closeness of arrows to the
bullseyeThe bullseye, or bull's-eye, is the centre of a target , and by extension the name given to any shot that hits 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.
To continue the analogy, if a large number of arrows are shot, precision would be the size of the arrow cluster. (When only one arrow is shot, 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, even 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 probableIn the military science of ballistics, circular error probable is an intuitive measure of a weapon system's precision...
for application of precision to the science of
ballisticsBallistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...
.
Quantification
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
traceableTraceability refers to the completeness of the information about every step in a process chain.The formal definition: Traceability is the ability to chronologically interrelate uniquely identifiable entities in a way that is verifiable....
reference standard. Such standards are defined in the
International System of UnitsSi, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
and maintained by national
standards organizationA standards organization, standards body, standards developing organization , or standards setting organization is any organization whose primary activities are developing, coordinating, promulgating, revising, amending, reissuing, interpreting, or otherwise producing technical standards that are...
s such as the
National Institute of Standards and TechnologyThe National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...
.
This also applies when measurements are repeated and averaged. In that case, the term
standard errorThe standard error is the standard deviation of the sampling distribution of a statistic. The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate....
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 theoremIn probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
shows that the
probability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
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
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of the measurements and the reference value, the biasIn statistics, bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. Otherwise the estimator is said to be biased.In ordinary English, the term bias is...
. Establishing and correcting for bias is necessary for calibrationCalibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....
.
- 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 figuresThe significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
. 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 8,436 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 × 10
3 m indicates that the first zero is significant (hence a margin of 50 m) 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 (50 m). However, reliance on this convention can lead to
false precisionFalse precision occurs when numerical data are presented in a manner that implies better precision than is actually the case; since precision is a limit to accuracy, this often leads to overconfidence in the accuracy as well.In science and engineering, convention dictates that...
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 down to 1s 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 — 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 classificationBinary classification is the task of classifying the members of a given set of objects into two groups on the basis of whether they have some property or not. Some typical binary classification tasks are...
test correctly identifies or excludes a condition.
|
Condition as determined by Gold standard |
|
| True |
False |
Test
outcome |
Positive |
True positive |
False positive |
→ Positive predictive value In statistics and diagnostic testing, the positive predictive value, or precision rate is the proportion of subjects with positive test results who are correctly diagnosed. It is a critical measure of the performance of a diagnostic method, as it reflects the probability that a positive test...
or Precision |
| Negative |
False negative |
True negative |
→ Negative predictive value In statistics and diagnostic testing, the negative predictive value is a summary statistic used to describe the performance of a diagnostic testing procedure. It is defined as the proportion of subjects with a negative test result who are correctly diagnosed. A high NPV means that when the test...
|
|
↓
Sensitivity or recall |
↓
Specificity |
Accuracy |
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.
On the other hand, precision or
positive predictive valueIn statistics and diagnostic testing, the positive predictive value, or precision rate is the proportion of subjects with positive test results who are correctly diagnosed. It is a critical measure of the performance of a diagnostic method, as it reflects the probability that a positive test...
is defined as the proportion of the true positives against all the positive results (both true positives and false positives)
An accuracy of 100% means that the measured values are exactly the same as the given values.
Also see
Sensitivity and specificitySensitivity and specificity are statistical measures of the performance of a binary classification test, also known in statistics as classification function. Sensitivity measures the proportion of actual positives which are correctly identified as such Sensitivity and specificity are statistical...
.
Accuracy may be determined from Sensitivity and Specificity, provided
PrevalenceIn epidemiology, the prevalence of a health-related state in a statistical population is defined as the total number of cases of the risk factor in the population at a given time, or the total number of cases in the population, divided by the number of individuals in the population...
is known, using the equation:
The
accuracy paradoxThe 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...
for
predictive analyticsPredictive analytics encompasses a variety of statistical techniques from modeling, machine learning, data mining and game theory that analyze current and historical facts to make predictions about future events....
states that predictive models with a given level of accuracy may have greater
predictive powerThe predictive power of a scientific theory refers to its ability to generate testable predictions. Theories with strong predictive power are highly valued, because the predictions can often encourage the falsification of the theory...
than models with higher accuracy. It may be better to avoid the accuracy metric in favor of other metrics such as
precision and recallIn pattern recognition and information retrieval, precision is the fraction of retrieved instances that are relevant, while recall is the fraction of relevant instances that are retrieved. Both precision and recall are therefore based on an understanding and measure of relevance...
. In situations where the minority class is more important, F-measure may be more appropriate, especially in situations with very skewed class imbalance. An alternate performance measure that treats both classes with equal importance is "balanced accuracy":
In psychometrics and psychophysics
In
psychometricsPsychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, personality traits, and educational measurement...
and
psychophysicsPsychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they effect. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual...
, the term
accuracy is interchangeably used with
validityIn science and statistics, validity has no single agreed definition but generally refers to the extent to which a concept, conclusion or measurement is well-founded and corresponds accurately to the real world. The word "valid" is derived from the Latin validus, meaning strong...
and
constant error.
Precision is a synonym for
reliabilityIn statistics, reliability is the consistency of a set of measurements or of a measuring instrument, often used to describe a test. Reliability is inversely related to random error.-Types:There are several general classes of reliability estimates:...
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 alphaCronbach's \alpha is a coefficient of reliability. It is commonly used as a measure of the internal consistency or reliability of a psychometric test score for a sample of examinees. It was first named alpha by Lee Cronbach in 1951, as he had intended to continue with further coefficients...
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 simulationLogic simulation is the use of a computer program to simulate the operation of a digital circuit. Logic simulation is the primary tool used for verifying the logical correctness of a hardware design. In many cases logic simulation is the first activity performed in the process of taking a hardware...
, a common mistake in evaluation of accurate models is to compare a
logic simulation modelElectronic circuit simulation uses mathematical models to replicate the behavior of an actual electronic device or circuit.Simulation software allows for modeling of circuit operation and is an invaluable analysis tool...
to a
transistorA transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...
circuit simulation modelTransistors are simple devices with complicated behavior. In order to ensure the reliable operation of circuits employing transistors, it is necessary to scientifically model the physical phenomena observed in their operation using transistor models. There exists a variety of different models that...
. 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
The concepts of accuracy and precision have also been studied in the context of data bases, information systems and their sociotechnical context. The necessary extension of these two concepts on the basis of theory of science suggests that they (as well as
data qualityData are of high quality "if they are fit for their intended uses in operations, decision making and planning" . Alternatively, the data are deemed of high quality if they correctly represent the real-world construct to which they refer...
and
information qualityInformation quality is a term to describe the quality of the content of information systems. It is often pragmatically defined as: "The fitness for use of the information provided."- Conceptual problems :...
) should be centered on accuracy defined as the closeness to the true value seen as the degree of agreement of readings or of calculated values of one same conceived entity, measured or calculated by different methods, in the context of maximum possible disagreement.
See also
- ± or Plus-minus sign
The plus-minus sign is a mathematical symbol commonly used either*to indicate the precision of an approximation, or*to indicate a value that can be of either sign....
- Accuracy class
Accuracy classes are defined and used in IEC and ANSI standards. Classes are denoted by either a letter or percentage. For example, Class B is a temperature accuracy from IEC-751 that requires accuracy of +/- 0.15 degrees Celsius...
- ANOVA Gauge R&R
- ASTM E177 Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
- Engineering tolerance
- Experimental uncertainty analysis
The purpose of this introductory article is to discuss the experimental uncertainty analysis of a derived quantity, based on the uncertainties in the experimentally measured quantities that are used in some form of mathematical relationship to calculate that derived quantity...
- Failure assessment
Failure assessment, as used in the context of software systems design, is the science of determining the circumstances under which a particular algorithm fails and then correcting the cause. It is directly related to the notion of algorithm accuracy, robustness, and reliability....
- Gain (information retrieval)
The gain, also called improvement over random can be specified for a classifier and is an important measure to describe the performance of it.- Definition :...
- Precision bias
Precision bias is a form of cognitive bias in which an evaluator of information commits a logical fallacy as the result of confusing accuracy and precision...
- Precision engineering
Precision engineering is a subdiscipline of electrical engineering, electronics engineering, mechanical engineering, and optical engineering concerned with designing machines, fixtures, and other structures that have exceptionally low tolerances, are repeatable, and are stable over time...
- Precision (statistics)
In statistics, the term precision can mean a quantity defined in a specific way. This is in addition to its more general meaning in the contexts of accuracy and precision and of precision and recall....
- Accepted and experimental value
External links