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 are two measures of observational error. 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.
According to ISO 5725-1, accuracy consists of trueness (proximity of the mean of measurement results to the true value) and precision (repeatability or reproducibility of the measurement).
More commonly, a description of systematic errors (a measure of statistical bias of a given measure of central tendency, such as the mean). In this definition of "accuracy", the concept is independent of "precision", so a particular set of data can be said to be accurate, precise, both, or neither. This concept corresponds to ISO's trueness.
A combination of both precision and trueness, accounting for the two types of observational error (random and systematic), so that high accuracy requires both high precision and high trueness. This usage corresponds to ISO's definition of accuracy (trueness and precision).[2]
Common technical definition
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.
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.[3] 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.[3][4] 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.[5]
ISO definition (ISO 5725)
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.[3]
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, that is the systematic error, and precision is the closeness of agreement among a set of results, that is the random error.
ISO 5725-1 and VIM also avoid the use of the term "bias", previously specified in BS 5497-1,[6] because it has different connotations outside the fields of science and engineering, as in medicine and law.
Accuracy of a target grouping according to BIPM and ISO 5725
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.[7]
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:
the difference between the mean of the measurements and the reference value, the bias. Establishing and correcting for bias is necessary for calibration.
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. 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 843.6m, or 843.0m, or 800.0m would imply a margin of 0.05m (the last significant place is the tenths place), while a recording of 843m would imply a margin of error of 0.5m (the last significant digits are the units).
A reading of 8,000m, 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: 8.0×103m indicates that the first zero is significant (hence a margin of 50m) while 8.000×103m indicates that all three zeros are significant, giving a margin of 0.5m. Similarly, one can use a multiple of the basic measurement unit: 8.0km is equivalent to 8.0×103m. It indicates a margin of 0.05km (50m). 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:
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.[8] For example, an ergonomist measuring the human body can be confident that 99.73% of their extracted measurements fall within ± 0.7cm - if using the GRYPHON processing system - or ± 13cm - if using unprocessed data.[9]
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.[10] 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".[11][12][13] 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:[14][15] 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 sometimes also viewed as a micro metric, to underline that it tends to be greatly affected by the particular class prevalence in a dataset and the classifier's biases.[14]
Furthermore, it 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
Comparative Waveforms for Logic values, circuit voltages, and measure voltages
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 documents correctly retrieved compared to the documents retrieved (true positives divided by true positives plus false positives), using a set of ground truth relevant results selected by humans. Recall is defined as the fraction of documents correctly retrieved compared to the relevant documents (true positives divided by true positives plus false negatives). Less commonly, the metric of accuracy is used, is defined as the fraction of documents correctly classified compared to the documents (true positives plus true negatives divided by true positives plus true negatives plus false positives plus false negatives).
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 the same output.[18][19][20] 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.
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.
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.
Binary classification is the task of classifying the elements of a set into one of two groups. Typical binary classification problems include:
A decision tree is a decision support recursive partitioning structure that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains conditional control statements.
In the field of machine learning and specifically the problem of statistical classification, a confusion matrix, also known as error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one; in unsupervised learning it is usually called a matching matrix.
False precision occurs when numerical data are presented in a manner that implies better precision than is justified; since precision is a limit to accuracy, this often leads to overconfidence in the accuracy, named precision bias.
A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the performance of a binary classifier model at varying threshold values.
The positive and negative predictive values are the proportions of positive and negative results in statistics and diagnostic tests that are true positive and true negative results, respectively. The PPV and NPV describe the performance of a diagnostic test or other statistical measure. A high result can be interpreted as indicating the accuracy of such a statistic. The PPV and NPV are not intrinsic to the test ; they depend also on the prevalence. Both PPV and NPV can be derived using Bayes' theorem.
Given a population whose members each belong to one of a number of different sets or classes, a classification rule or classifier is a procedure by which the elements of the population set are each predicted to belong to one of the classes. A perfect classification is one for which every element in the population is assigned to the class it really belongs to. The bayes classifier is the classifier which assigns classes optimally based on the known attributes of the elements to be classified.
Choice-supportive bias or post-purchase rationalization is the tendency to retroactively ascribe positive attributes to an option one has selected and/or to demote the forgone options. It is part of cognitive science, and is a distinct cognitive bias that occurs once a decision is made. For example, if a person chooses option A instead of option B, they are likely to ignore or downplay the faults of option A while amplifying or ascribing new negative faults to option B. Conversely, they are also likely to notice and amplify the advantages of option A and not notice or de-emphasize those of option B.
In statistical analysis of binary classification and information retrieval systems, the F-score or F-measure is a measure of predictive performance. It is calculated from the precision and recall of the test, where the precision is the number of true positive results divided by the number of all samples predicted to be positive, including those not identified correctly, and the recall is the number of true positive results divided by the number of all samples that should have been identified as positive. Precision is also known as positive predictive value, and recall is also known as sensitivity in diagnostic binary classification.
In medicine and statistics, sensitivity and specificity mathematically describe the accuracy of a test that reports the presence or absence of a medical condition. If individuals who have the condition are considered "positive" and those who do not are considered "negative", then sensitivity is a measure of how well a test can identify true positives and specificity is a measure of how well a test can identify true negatives:
In mathematics and statistics, deviation serves as a measure to quantify the disparity between an observed value of a variable and another designated value, frequently the mean of that variable. Deviations with respect to the sample mean and the population mean are called errors and residuals, respectively. The sign of the deviation reports the direction of that difference: the deviation is positive when the observed value exceeds the reference value. The absolute value of the deviation indicates the size or magnitude of the difference. In a given sample, there are as many deviations as sample points. Summary statistics can be derived from a set of deviations, such as the standard deviation and the mean absolute deviation, measures of dispersion, and the mean signed deviation, a measure of bias.
Youden's J statistic is a single statistic that captures the performance of a dichotomous diagnostic test. (Bookmaker) Informedness is its generalization to the multiclass case and estimates the probability of an informed decision.
In pattern recognition, information retrieval, object detection and classification, precision and recall are performance metrics that apply to data retrieved from a collection, corpus or sample space.
In statistics, the phi coefficient is a measure of association for two binary variables.
Evaluation of a binary classifier typically assigns a numerical value, or values, to a classifier that represent its accuracy. An example is error rate, which measures how frequently the classifier makes a mistake.
P4 metric (also known as FS or Symmetric F ) enables performance evaluation of the binary classifier. It is calculated from precision, recall, specificity and NPV (negative predictive value). P4 is designed in similar way to F1 metric, however addressing the criticisms leveled against F1. It may be perceived as its extension.
References
1 2 BS ISO 5725-1: "Accuracy (trueness and precision) of measurement methods and results - Part 1: General principles and definitions.", p.1 (1994)
↑ Menditto, Antonio; Patriarca, Marina; Magnusson, Bertil (2007-01-09). "Understanding the meaning of accuracy, trueness and precision". Accreditation and Quality Assurance. 12 (1): 45–47. doi:10.1007/s00769-006-0191-z. ISSN0949-1775.
↑ Acken, John M. (1997). "none". Encyclopedia of Computer Science and Technology. 36: 281–306.
↑ Glasser, Mark; Mathews, Rob; Acken, John M. (June 1990). "1990 Workshop on Logic-Level Modelling for ASICS". SIGDA Newsletter. 20 (1).
↑ Fulbright, Ron (2020). Democratization of Expertise: How Cognitive Systems Will Revolutionize Your Life (1sted.). Boca Raton, FL: CRC Press. ISBN978-0367859459.
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Low accuracy due to low precision
Low accuracy even with high precision
