Accuracy and precision

Accuracy indicates proximity of measurement results to the true value, precision to the repeatability, or reproducibility of the measurement

In the fields of science, engineering, industry, and statistics, the accuracy^[1] of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual (true) value. The precision^[1] of a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.^[2] Although the two words reproducibility and repeatability can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.

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 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 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 the case of full reproducibility, such as when rounding a number to a representable floating point 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 significand, so it is used as a measure for the relative accuracy with which an arbitrary number can be represented.

1 Accuracy versus precision: the target analogy
2 Quantification
3 Terminology of ISO 5725
4 In binary classification
5 In psychometrics and psychophysics
6 In logic simulation
7 In information systems
8 See also
9 References
10 External links

Accuracy versus precision: the target analogy

High accuracy, but low precision.

High precision, but low accuracy

Accuracy is the degree of veracity while in some contexts precision may mean the degree of reproducibility. Accuracy is dependent on how data is collected, and is usually judged by comparing several measurements from the same or different sources.^{[citation needed]}

The analogy 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 bullseye at the target center. Arrows that strike closer to the bullseye are considered more accurate. The closer a system's measurements are 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 probable for application of precision to the science of ballistics.

Quantification

Terminology of ISO 5725

A shift in the meaning of these terms appeared with the publication of the ISO 5725 series of standards. According to ISO 5725-1, the terms trueness and precision are used to described the accuracy of a measurement. Trueness refers to the closeness of the mean of the measurement results to the "correct" value and precision refers to the closeness of agreement within individual results. Therefore, according to the ISO standard, the term "accuracy" refers to both trueness and precision. The standard also avoids the use of the term bias, because it has different connotations outside the fields of science and engineering, as in medicine and law.^[3] The terms "accuracy" and "trueness" were again redefined in 2008 with a slight shift in their exact meanings in the "BIMP International Vocabulary of Metrology", items 2.13 and 2.14 ^[1]

In binary classification

Accuracy is also used as a statistical measure of how well a binary classification 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 or Precision
Test outcome	Negative	False negative	True negative	→ Negative predictive value
		↓ Sensitivity or recall	↓ Specificity (or its complement, Fall-Out)	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.

$\text{accuracy}=\frac{\text{number of true positives}+\text{number of true negatives}}{\text{number of true positives}+\text{false positives} + \text{false negatives} + \text{true negatives}}$

On the other hand, precision or positive predictive value is defined as the proportion of the true positives against all the positive results (both true positives and false positives)

$\text{precision}=\frac{\text{number of true positives}}{\text{number of true positives}+\text{false positives}}$

An accuracy of 100% means that the measured values are exactly the same as the given values.

Also see Sensitivity and specificity.

Accuracy may be determined from Sensitivity and Specificity, provided Prevalence is known, using the equation:

$\text{accuracy}=(\text{sensitivity})(\text{prevalence}) + (\text{specificity})(1-\text{prevalence})$

The 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. It may be better to avoid the accuracy metric in favor of other metrics such as precision and recall.^{[citation needed]} In situations where the minority class is more important, F-measure may be more appropriate, especially in situations with very skewed class imbalance.

Another useful performance measure is the balanced accuracy which avoids inflated performance estimates on imbalanced datasets. It is defined as the arithmetic mean of sensitivity and specificity, or the average accuracy obtained on either class:

$\text{balanced accuracy}=\frac{\text{sensitivity} + \text{specificity}}{2}$

$= \frac{0.5*\text{true positives}}{\text{true positives}+\text{false negatives}} + \frac{0.5*\text{true negatives}}{\text{true negatives}+\text{false positives}}$

If the classifier performs equally well on either class, this term reduces to the conventional accuracy (i.e., the number of correct predictions divided by the total number of predictions). In contrast, if the conventional accuracy is above chance only because the classifier takes advantage of an imbalanced test set, then the balanced accuracy, as appropriate, will drop to chance.^[4] A closely related chance corrected measure is:

$\text{Informedness} = \text{sensitivity} + \text{specificity} - 1=2*\text{balanced accuracy}-1$ ^[5]

while a direct approach to debiasing and renormalizing Accuracy is Cohen's kappa whilst Informedness has been shown to be a Kappa family debiased renormalization of Recall.^[6] Informedness and Kappa have the advantage that chance level is defined to be 0, and they have the form of a probability. Informedness has the stronger property that it is the probability that an informed decision is made (rather than a guess), when positive. When negative this is still true for the absolutely value of Informedness, but the information has been used to force an incorrect response.^[5]

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.^[7]^[8]

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 quality and information quality) 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.^[9]

Further information: Precision and recall

References

^ ^a ^b ^c JCGM 200:2008 International vocabulary of metrology — Basic and general concepts and associated terms (VIM)
^ John Robert Taylor (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books. pp. 128–129. ISBN 0-935702-75-X.
^ BS ISO 5725-1: "Accuracy (trueness and precision) of measurement methods and reults - Part 1: General principles and definitions", pp.1 (1994)
^ K.H. Brodersen, C.S. Ong, K.E. Stephan, J.M. Buhmann (2010). The balanced accuracy and its posterior distribution. Proceedings of the 20th International Conference on Pattern Recognition, 3121-3124.
^ ^a ^b Powers, David M W (2007/2011). "Evaluation: From Precision, Recall and F-Factor to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies 2 (1): 37–63.
^ Powers, David M. W. (2012). "The Problem with Kappa". Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUS-UNSUP Workshop.
^ John M. Acken, Encyclopedia of Computer Science and Technology, Vol 36, 1997, page 281-306
^ 1990 Workshop on Logic-Level Modelling for ASICS, Mark Glasser, Rob Mathews, and John M. Acken, SIGDA Newsletter, Vol 20. Number 1, June 1990
^ Ivanov, K. (1972). "Quality-control of information: On the concept of accuracy of information in data banks and in management information systems".

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, Appendix D.1: Terminology
Accuracy and Precision
Accuracy vs Precision — a brief, clear video by Matt Parker

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