Accuracy and precision

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Accuracy and precision are two measures of observational error .

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Accuracy is how close a given set of measurements (observations or readings) are to their true value .

Precision is how close the measurements are to each other.

In other words:

  1. More commonly, a description of systematic errors (a measure of statistical bias of a given measure of central tendency). Low accuracy causes a difference between a result and a true value. This secondary measure is referred to as trueness by ISO.
  2. 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 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

Accuracy is the proximity of measurement results to the accepted value; precision is the degree to which repeated (or reproducible) measurements under unchanged conditions show the same results. Accuracy and precision.svg
Accuracy is the proximity of measurement results to the accepted value; precision is the degree to which repeated (or reproducible) measurements under unchanged conditions show the same results.

In the fields of science and engineering, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's true value. [2] The precision of a measurement system, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditions show the same results. [2] [3] Although the two words precision and accuracy can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.

The field of statistics, where the interpretation of measurements plays a central role, prefers to use the terms bias and variability instead of accuracy and precision: bias is the amount of inaccuracy and variability is the amount of imprecision.

A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. The result would be a consistent yet inaccurate string of results from the flawed experiment. Eliminating the systematic error improves accuracy but does not change precision.

A measurement system is considered valid if it is both accurate and precise. Related terms include bias (non-random or directed effects caused by a factor or factors unrelated to the independent variable) and error (random variability).

The terminology is also applied to indirect measurements—that is, values obtained by a computational procedure from observed data.

In addition to accuracy and precision, measurements may also have a measurement resolution, which is the smallest change in the underlying physical quantity that produces a response in the measurement.

In numerical analysis, accuracy is also the nearness of a calculation to the true value; while precision is the resolution of the representation, typically defined by the number of decimal or binary digits.

In military terms, accuracy refers primarily to the accuracy of fire (justesse de tir), the precision of fire expressed by the closeness of a grouping of shots at and around the centre of the target. [4]

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. [5]

Ideally a measurement device is both accurate and precise, with measurements all close to and tightly clustered around the true value. The accuracy and precision of a measurement process is usually established by repeatedly measuring some traceable reference standard. Such standards are defined in the International System of Units (abbreviated SI from French: Système international d'unités) and maintained by national standards organizations such as the National Institute of Standards and Technology in the United States.

This also applies when measurements are repeated and averaged. In that case, the term standard error is properly applied: the precision of the average is equal to the known standard deviation of the process divided by the square root of the number of measurements averaged. Further, the central limit theorem shows that the probability distribution of the averaged measurements will be closer to a normal distribution than that of individual measurements.

With regard to accuracy we can distinguish:

A common convention in science and engineering is to express accuracy and/or precision implicitly by means of significant figures. 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 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 150,000.

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:

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. [6] 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. [7]

ISO definition (ISO 5725)

According to ISO 5725-1, accuracy consists of trueness (proximity of measurement results to the true value) and precision (repeatability or reproducibility of the measurement). Accuracy (trueness and precision).svg
According to ISO 5725-1, accuracy consists of trueness (proximity of measurement results to the true value) and precision (repeatability or reproducibility of the measurement).

A shift in the meaning of these terms appeared with the publication of the ISO 5725 series of standards in 1994, which is also reflected in the 2008 issue of the BIPM International Vocabulary of Metrology (VIM), items 2.13 and 2.14. [2]

According to ISO 5725-1, [1] the general term "accuracy" is used to describe the closeness of a measurement to the true value. When the term is applied to sets of measurements of the same measurand, it involves a component of random error and a component of systematic error. In this case trueness is the closeness of the mean of a set of measurement results to the actual (true) value and precision is the closeness of agreement among a set of results.

ISO 5725-1 and VIM also avoid the use of the term "bias", previously specified in BS 5497-1, [8] 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 test correctly identifies or excludes a condition. That is, the accuracy is the proportion of correct predictions (both true positives and true negatives) among the total number of cases examined. [9] As such, it compares estimates of pre- and post-test probability. To make the context clear by the semantics, it is often referred to as the "Rand accuracy" or "Rand index". [10] [11] [12] It is a parameter of the test. The formula for quantifying binary accuracy is:

where TP = True positive; FP = False positive; TN = True negative; FN = False negative

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 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: [13]

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 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 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. [14] [15]

In information systems

Information retrieval systems, such as databases and web search engines, are evaluated by many different metrics, some of which are derived from the confusion matrix, which divides results into true positives (documents correctly retrieved), true negatives (documents correctly not retrieved), false positives (documents incorrectly retrieved), and false negatives (documents incorrectly not retrieved). Commonly used metrics include the notions of precision and recall. In this context, precision is defined as the fraction of retrieved documents which are relevant to the query (true positives divided by true+false positives), using a set of ground truth relevant results selected by humans. Recall is defined as the fraction of relevant documents retrieved compared to the total number of relevant documents (true positives divided by true positives+false negatives). Less commonly, the metric of accuracy is used, is defined as the total number of correct classifications (true positives plus true negatives) divided by the total number of documents.

None of these metrics take into account the ranking of results. Ranking is very important for web search engines because readers seldom go past the first page of results, and there are too many documents on the web to manually classify all of them as to whether they should be included or excluded from a given search. Adding a cutoff at a particular number of results takes ranking into account to some degree. The measure precision at k, for example, is a measure of precision looking only at the top ten (k=10) search results. More sophisticated metrics, such as discounted cumulative gain, take into account each individual ranking, and are more commonly used where this is important.

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) 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. [16] [17] [18] To measure augmented cognition 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.

See also

Related Research Articles

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Observational error is the difference between a measured value of a quantity and its unknown true value. Such errors are inherent in the measurement process; for example lengths measured with a ruler calibrated in whole centimeters will have a measurement error of several millimeters. The error or uncertainty of a measurement can be estimated, and is specified with the measurement as, for example, 32.3 ± 0.5 cm.

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 known accuracy, a device generating the quantity to be measured such as a voltage, a sound tone, or a physical artifact, such as a meter ruler.

<span class="mw-page-title-main">Rounding</span> Replacing a number with a simpler value

Rounding or rounding off means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $23.4476 with $23.45, the fraction 312/937 with 1/3, or the expression √2 with 1.414.

Statistical bias, in the mathematical field of statistics, is a systematic tendency in which the methods used to gather data and generate statistics present an inaccurate, skewed or biased depiction of reality. Statistical bias exists in numerous stages of the data collection and analysis process, including: the source of the data, the methods used to collect the data, the estimator chosen, and the methods used to analyze the data. Data analysts can take various measures at each stage of the process to reduce the impact of statistical bias in their work. Understanding the source of statistical bias can help to assess whether the observed results are close to actuality. Issues of statistical bias has been argued to be closely linked to issues of statistical validity.

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Significant figures, also referred to as significant digits or sig figs, are specific digits within a number written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement, if the number of digits exceeds what the measurement instrument can resolve, only the number of digits within the resolution's capability are dependable and therefore considered significant.

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<span class="mw-page-title-main">Receiver operating characteristic</span> Diagnostic plot of binary classifier ability

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<span class="mw-page-title-main">Surface plate</span> Flat surface used as a reference plane

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<span class="mw-page-title-main">F-score</span> Statistical measure of a tests accuracy

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A test method is a method for a test in science or engineering, such as a physical test, chemical test, or statistical test. It is a definitive procedure that produces a test result. In order to ensure accurate and relevant test results, a test method should be "explicit, unambiguous, and experimentally feasible.", as well as effective and reproducible.

Precision bias also known as numeracy 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. More particularly, in assessing the merits of an argument, a measurement, or a report, an observer or assessor falls prey to precision bias when they believe that greater precision implies greater accuracy ; the observer or assessor are said to provide false precision.

<span class="mw-page-title-main">Precision and recall</span> Pattern-recognition performance metrics

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<span class="mw-page-title-main">Evaluation of binary classifiers</span>

The evaluation of binary classifiers compares two methods of assigning a binary attribute, one of which is usually a standard method and the other is being investigated. There are many metrics that can be used to measure the performance of a classifier or predictor; different fields have different preferences for specific metrics due to different goals. For example, in medicine sensitivity and specificity are often used, while in computer science precision and recall are preferred. An important distinction is between metrics that are independent on the prevalence, and metrics that depend on the prevalence – both types are useful, but they have very different properties.

References

  1. 1 2 BS ISO 5725-1: "Accuracy (trueness and precision) of measurement methods and results - Part 1: General principles and definitions.", p.1 (1994)
  2. 1 2 3 JCGM 200:2008 International vocabulary of metrology — Basic and general concepts and associated terms (VIM)
  3. ↑ Taylor, John Robert (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books. pp. 128–129. ISBN   0-935702-75-X.
  4. ↑ North Atlantic Treaty Organization, NATO Standardization Agency AAP-6 – Glossary of terms and definitions, p 43.
  5. ↑ Creus, Antonio. Instrumentación Industrial[ citation needed ]
  6. ↑ Black, J. Temple (21 July 2020). DeGarmo's materials and processes in manufacturing. John Wiley & Sons. ISBN   978-1-119-72329-5. OCLC   1246529321.
  7. ↑ Parker, Christopher J.; Gill, Simeon; Harwood, Adrian; Hayes, Steven G.; Ahmed, Maryam (2021-05-19). "A Method for Increasing 3D Body Scanning's Precision: Gryphon and Consecutive Scanning". Ergonomics. 65 (1): 39–59. doi: 10.1080/00140139.2021.1931473 . ISSN   0014-0139. PMID   34006206.
  8. ↑ BS 5497-1: "Precision of test methods. Guide for the determination of repeatability and reproducibility for a standard test method." (1979)
  9. ↑ Metz, CE (October 1978). "Basic principles of ROC analysis" (PDF). Semin Nucl Med. 8 (4): 283–98. doi:10.1016/s0001-2998(78)80014-2. PMID   112681. Archived (PDF) from the original on 2022-10-09.
  10. ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2015-03-11. Retrieved 2015-08-09.{{cite web}}: CS1 maint: archived copy as title (link)
  11. ↑ Powers, David M. W. (2015). "What the F-measure doesn't measure". arXiv: 1503.06410 [cs.IR].
  12. ↑ David M W Powers. "The Problem with Kappa" (PDF). Anthology.aclweb.org. Archived (PDF) from the original on 2022-10-09. Retrieved 11 December 2017.
  13. ↑ "3.3. Metrics and scoring: quantifying the quality of predictions". scikit-learn. Retrieved 17 May 2022.
  14. ↑ Acken, John M. (1997). "none". Encyclopedia of Computer Science and Technology. 36: 281–306.
  15. ↑ Glasser, Mark; Mathews, Rob; Acken, John M. (June 1990). "1990 Workshop on Logic-Level Modelling for ASICS". SIGDA Newsletter. 20 (1).
  16. ↑ Fulbright, Ron (2020). Democratization of Expertise: How Cognitive Systems Will Revolutionize Your Life (1st ed.). Boca Raton, FL: CRC Press. ISBN   978-0367859459.
  17. ↑ Fulbright, Ron (2019). "Calculating Cognitive Augmentation – A Case Study". Augmented Cognition. Lecture Notes in Computer Science. Vol. 11580. Springer Cham. pp. 533–545. arXiv: 2211.06479 . doi:10.1007/978-3-030-22419-6_38. ISBN   978-3-030-22418-9. S2CID   195891648.
  18. ↑ Fulbright, Ron (2018). "On Measuring Cognition and Cognitive Augmentation". Human Interface and the Management of Information. Information in Applications and Services. Lecture Notes in Computer Science. Vol. 10905. Springer Cham. pp. 494–507. arXiv: 2211.06477 . doi:10.1007/978-3-319-92046-7_41. ISBN   978-3-319-92045-0. S2CID   51603737.
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