Uncertainty

Last updated
Situations often arise wherein a decision must be made when the results of each possible choice are uncertain. Blank Fork.png
Situations often arise wherein a decision must be made when the results of each possible choice are uncertain.

Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable and/or stochastic environments, as well as due to ignorance, indolence, or both. [1] It arises in any number of fields, including insurance, philosophy, physics, statistics, economics, finance, psychology, sociology, engineering, metrology, meteorology, ecology and information science.

Information that which informs; the answer to a question of some kind; that from which data and knowledge can be derived

Information can be thought of as the resolution of uncertainty; it is that which answers the question of "what an entity is" and thus defines both its essence and nature of its characteristics. It is associated with data, as data represents values attributed to parameters, and information is data in context and with meaning attached. Information relates also to knowledge, as knowledge signifies understanding of an abstract or concrete concept.

Stochastic refers to a randomly determined process. The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable. The word, with its current definition meaning random, came from German, but it originally came from Greek στόχος (stókhos), meaning 'aim, guess'.

Ignorance is a lack of knowledge and information. The word "ignorant" is an adjective that describes a person in the state of being unaware, and can describe individuals who deliberately ignore or disregard important information or facts, or individuals who are unaware of important information or facts. Ignorance can appear in three different types: factual ignorance, objectual ignorance, and technical ignorance.

Contents

Concepts

Although the terms are used in various ways among the general public, many specialists in decision theory, statistics and other quantitative fields have defined uncertainty, risk, and their measurement as:

Decision theory is the study of an agent's choices. Decision theory can be broken into two branches: normative decision theory, which analyzes the outcomes of decisions or determines the optimal decisions given constraints and assumptions, and descriptive decision theory, which analyzes how agents actually make the decisions they do.

Statistics Study of the collection, analysis, interpretation, and presentation of data

Statistics is the discipline that concerns the collection, organization, displaying, analysis, interpretation and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.

Uncertainty
The lack of certainty, a state of limited knowledge where it is impossible to exactly describe the existing state, a future outcome, or more than one possible outcome.[ citation needed ]
Measurement of uncertainty
A set of possible states or outcomes where probabilities are assigned to each possible state or outcome – this also includes the application of a probability density function to continuous variables. [2]
Second order uncertainty
In statistics and economics, second-order uncertainty is represented in probability density functions over (first-order) probabilities. [3] [4] .
Opinions in subjective logic [5] carry this type of uncertainty.
Risk
A state of uncertainty where some possible outcomes have an undesired effect or significant loss.
Measurement of risk
A set of measured uncertainties where some possible outcomes are losses, and the magnitudes of those losses – this also includes loss functions over continuous variables. [6] [7] [8] [9]
Knightian uncertainty
In economics, in 1921 Frank Knight distinguished uncertainty from risk with uncertainty being lack of knowledge which is immeasurable and impossible to calculate; this is now referred to as Knightian uncertainty:

Uncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated.... The essential fact is that 'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating.... It will appear that a measurable uncertainty, or 'risk' proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all.

Frank Knight (1885–1972), Risk, Uncertainty, and Profit (1921), University of Chicago. [10]

Other taxonomies of uncertainties and decisions include a broader sense of uncertainty and how it should be approached from an ethics perspective: [11]

A taxonomy of uncertainty Uncertainty.svg
A taxonomy of uncertainty
There are some things that you know to be true, and others that you know to be false; yet, despite this extensive knowledge that you have, there remain many things whose truth or falsity is not known to you. We say that you are uncertain about them. You are uncertain, to varying degrees, about everything in the future; much of the past is hidden from you; and there is a lot of the present about which you do not have full information. Uncertainty is everywhere and you cannot escape from it.

Dennis Lindley, Understanding Uncertainty (2006)

For example, if it is unknown whether or not it will rain tomorrow, then there is a state of uncertainty. If probabilities are applied to the possible outcomes using weather forecasts or even just a calibrated probability assessment, the uncertainty has been quantified. Suppose it is quantified as a 90% chance of sunshine. If there is a major, costly, outdoor event planned for tomorrow then there is a risk since there is a 10% chance of rain, and rain would be undesirable. Furthermore, if this is a business event and $100,000 would be lost if it rains, then the risk has been quantified (a 10% chance of losing $100,000). These situations can be made even more realistic by quantifying light rain vs. heavy rain, the cost of delays vs. outright cancellation, etc.[ citation needed ]

Calibrated probability assessments are subjective probabilities assigned by individuals who have been trained to assess probabilities in a way that historically represents their uncertainty. For example, when a person has calibrated a situation and says they are "80% confident" in each of 100 predictions they made, they will get about 80% of them correct. Likewise, they will be right 90% of the time they say they are 90% certain, and so on.

Some may represent the risk in this example as the "expected opportunity loss" (EOL) or the chance of the loss multiplied by the amount of the loss (10% × $100,000 = $10,000). That is useful if the organizer of the event is "risk neutral", which most people are not. Most would be willing to pay a premium to avoid the loss. An insurance company, for example, would compute an EOL as a minimum for any insurance coverage, then add onto that other operating costs and profit. Since many people are willing to buy insurance for many reasons, then clearly the EOL alone is not the perceived value of avoiding the risk.

Quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as probability theory, actuarial science, and information theory. Some also create new terms without substantially changing the definitions of uncertainty or risk. For example, surprisal is a variation on uncertainty sometimes used in information theory. But outside of the more mathematical uses of the term, usage may vary widely. In cognitive psychology, uncertainty can be real, or just a matter of perception, such as expectations, threats, etc.

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

Actuarial science discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries

Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty.

Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper titled "A Mathematical Theory of Communication". Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones, the development of the Internet, the study of linguistics and of human perception, the understanding of black holes, and numerous other fields.

Vagueness is a form of uncertainty where the analyst is unable to clearly differentiate between two different classes, such as 'person of average height.' and 'tall person'. This form of vagueness can be modelled by some variation on Zadeh's fuzzy logic or subjective logic.

In philosophy, vagueness refers to an important problem in semantics, metaphysics and philosophical logic. Definitions of this problem vary. A predicate is sometimes said to be vague if the bound of its extension is indeterminate, or appears to be so. The predicate "is tall" is vague because there seems to be no particular height at which someone becomes tall. Alternately, a predicate is sometimes said to be vague if there are borderline cases of its application, such that in these cases competent speakers of the language may faultlessly disagree over whether the predicate applies. The disagreement over whether a hotdog is a sandwich suggests that “sandwich” is vague.

Lotfi A. Zadeh Electrical engineer and computer scientist

Lotfi Aliasker Zadeh was a mathematician, computer scientist, electrical engineer, artificial intelligence researcher and professor emeritus of computer science at the University of California, Berkeley.

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1 both inclusive. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.

Ambiguity is a form of uncertainty where even the possible outcomes have unclear meanings and interpretations. The statement "He returns from the bank" is ambiguous because its interpretation depends on whether the word 'bank' is meant as "the side of a river" or "a financial institution". Ambiguity typically arises in situations where multiple analysts or observers have different interpretations of the same statements.[ citation needed ]

Uncertainty may be a consequence of a lack of knowledge of obtainable facts. That is, there may be uncertainty about whether a new rocket design will work, but this uncertainty can be removed with further analysis and experimentation.

At the subatomic level, uncertainty may be a fundamental and unavoidable property of the universe. In quantum mechanics, the Heisenberg uncertainty principle puts limits on how much an observer can ever know about the position and velocity of a particle. This may not just be ignorance of potentially obtainable facts but that there is no fact to be found. There is some controversy in physics as to whether such uncertainty is an irreducible property of nature or if there are "hidden variables" that would describe the state of a particle even more exactly than Heisenberg's uncertainty principle allows.[ citation needed ]

Measurements

The most commonly used procedure for calculating measurement uncertainty is described in the "Guide to the Expression of Uncertainty in Measurement" (GUM) published by ISO. A derived work is for example the National Institute for Standards and Technology (NIST) Technical Note 1297, "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results", and the Eurachem/Citac publication "Quantifying Uncertainty in Analytical Measurement". The uncertainty of the result of a measurement generally consists of several components. The components are regarded as random variables, and may be grouped into two categories according to the method used to estimate their numerical values:

By propagating the variances of the components through a function relating the components to the measurement result, the combined measurement uncertainty is given as the square root of the resulting variance. The simplest form is the standard deviation of a repeated observation.

In metereology, physics, and engineering, the uncertainty or margin of error of a measurement, when explicitly stated, is given by a range of values likely to enclose the true value. This may be denoted by error bars on a graph, or by the following notations:[ citation needed ]

In the last notation, parentheses are the concise notation for the ± notation. For example, applying 10 12 meters in a scientific or engineering application, it could be written 10.5 m or 10.50 m, by convention meaning accurate to within one tenth of a meter, or one hundredth. The precision is symmetric around the last digit. In this case it's half a tenth up and half a tenth down, so 10.5 means between 10.45 and 10.55. Thus it is understood that 10.5 means 10.5±0.05, and 10.50 means 10.50±0.005, also written 10.50(5) and 10.500(5) respectively. But if the accuracy is within two tenths, the uncertainty is ± one tenth, and it is required to be explicit: 10.5±0.1 and 10.50±0.01 or 10.5(1) and 10.50(1). The numbers in parentheses apply to the numeral left of themselves, and are not part of that number, but part of a notation of uncertainty. They apply to the least significant digits. For instance, 1.00794(7) stands for 1.00794±0.00007, while 1.00794(72) stands for 1.00794±0.00072. [12] This concise notation is used for example by IUPAC in stating the atomic mass of elements.

The middle notation is used when the error is not symmetrical about the value – for example 3.4+0.3
−0.2
. This can occur when using a logarithmic scale, for example.

Uncertainty of a measurement can be determined by repeating a measurement to arrive at an estimate of the standard deviation of the values. Then, any single value has an uncertainty equal to the standard deviation. However, if the values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of the number of measurements. This procedure neglects systematic errors, however.[ citation needed ]

When the uncertainty represents the standard error of the measurement, then about 68.3% of the time, the true value of the measured quantity falls within the stated uncertainty range. For example, it is likely that for 31.7% of the atomic mass values given on the list of elements by atomic mass, the true value lies outside of the stated range. If the width of the interval is doubled, then probably only 4.6% of the true values lie outside the doubled interval, and if the width is tripled, probably only 0.3% lie outside. These values follow from the properties of the normal distribution, and they apply only if the measurement process produces normally distributed errors. In that case, the quoted standard errors are easily converted to 68.3% ("one sigma"), 95.4% ("two sigma"), or 99.7% ("three sigma") confidence intervals.[ citation needed ]

In this context, uncertainty depends on both the accuracy and precision of the measurement instrument. The lower the accuracy and precision of an instrument, the larger the measurement uncertainty is. Precision is often determined as the standard deviation of the repeated measures of a given value, namely using the same method described above to assess measurement uncertainty. However, this method is correct only when the instrument is accurate. When it is inaccurate, the uncertainty is larger than the standard deviation of the repeated measures, and it appears evident that the uncertainty does not depend only on instrumental precision.

In the media

Uncertainty in science, and science in general, may be interpreted differently in the public sphere than in the scientific community. [13] This is due in part to the diversity of the public audience, and the tendency for scientists to misunderstand lay audiences and therefore not communicate ideas clearly and effectively. [13] One example is explained by the information deficit model. Also, in the public realm, there are often many scientific voices giving input on a single topic. [13] For example, depending on how an issue is reported in the public sphere, discrepancies between outcomes of multiple scientific studies due to methodological differences could be interpreted by the public as a lack of consensus in a situation where a consensus does in fact exist. [13] This interpretation may have even been intentionally promoted, as scientific uncertainty may be managed to reach certain goals. For example, climate change deniers took the advice of Frank Luntz to frame global warming as an issue of scientific uncertainty, which was a precursor to the conflict frame used by journalists when reporting the issue. [14]

"Indeterminacy can be loosely said to apply to situations in which not all the parameters of the system and their interactions are fully known, whereas ignorance refers to situations in which it is not known what is not known." [15] These unknowns, indeterminacy and ignorance, that exist in science are often "transformed" into uncertainty when reported to the public in order to make issues more manageable, since scientific indeterminacy and ignorance are difficult concepts for scientists to convey without losing credibility. [13] Conversely, uncertainty is often interpreted by the public as ignorance. [16] The transformation of indeterminacy and ignorance into uncertainty may be related to the public's misinterpretation of uncertainty as ignorance.

Journalists may inflate uncertainty (making the science seem more uncertain than it really is) or downplay uncertainty (making the science seem more certain than it really is). [17] One way that journalists inflate uncertainty is by describing new research that contradicts past research without providing context for the change. [17] Journalists may give scientists with minority views equal weight as scientists with majority views, without adequately describing or explaining the state of scientific consensus on the issue. [17] In the same vein, journalists may give non-scientists the same amount of attention and importance as scientists. [17]

Journalists may downplay uncertainty by eliminating "scientists' carefully chosen tentative wording, and by losing these caveats the information is skewed and presented as more certain and conclusive than it really is". [17] Also, stories with a single source or without any context of previous research mean that the subject at hand is presented as more definitive and certain than it is in reality. [17] There is often a "product over process" approach to science journalism that aids, too, in the downplaying of uncertainty. [17] Finally, and most notably for this investigation, when science is framed by journalists as a triumphant quest, uncertainty is erroneously framed as "reducible and resolvable". [17]

Some media routines and organizational factors affect the overstatement of uncertainty; other media routines and organizational factors help inflate the certainty of an issue. Because the general public (in the United States) generally trusts scientists, when science stories are covered without alarm-raising cues from special interest organizations (religious groups, environmental organizations, political factions, etc.) they are often covered in a business related sense, in an economic-development frame or a social progress frame. [18] The nature of these frames is to downplay or eliminate uncertainty, so when economic and scientific promise are focused on early in the issue cycle, as has happened with coverage of plant biotechnology and nanotechnology in the United States, the matter in question seems more definitive and certain. [18]

Sometimes, stockholders, owners, or advertising will pressure a media organization to promote the business aspects of a scientific issue, and therefore any uncertainty claims which may compromise the business interests are downplayed or eliminated. [17]

Applications

See also

Related Research Articles

Standard deviation dispersion of the values of a random variable around its expected value

In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

In measurement of a set, accuracy refers to closeness of the measurements to a specific value, while precision refers to the closeness of the measurements to each other.

Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that

In mathematical optimization and decision theory, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its negative, in which case it is to be maximized.

Standard error statistical property

The standard error (SE) of a statistic is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the parameter or the statistic is the mean, it is called the standard error of the mean (SEM).

Propagation of uncertainty effect of variables uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them

In statistics, propagation of uncertainty is the effect of variables' uncertainties on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations which propagate due to the combination of variables in the function.

In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (gambles)⁠—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes. The introduction of St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis. This hypothesis has proven useful to explain some popular choices that seem to contradict the expected value criterion, such as occur in the contexts of gambling and insurance.

Value of information is the amount a decision maker would be willing to pay for information prior to making a decision.

In economics, Knightian uncertainty is a lack of any quantifiable knowledge about some possible occurrence, as opposed to the presence of quantifiable risk. The concept acknowledges some fundamental degree of ignorance, a limit to knowledge, and an essential unpredictability of future events.

The Ellsberg paradox is a paradox in decision theory in which people's choices violate the postulates of subjective expected utility. It is generally taken to be evidence for ambiguity aversion. The paradox was popularized by Daniel Ellsberg, although a version of it was noted considerably earlier by John Maynard Keynes.

A percentage point or percent point is the unit for the arithmetic difference of two percentages. For example, moving up from 40% to 44% is a 4 percentage point increase, but is a 10 percent increase in what is being measured. In the literature, the percentage point unit is usually either written out, or abbreviated as pp or p.p. to avoid ambiguity. After the first occurrence, some writers abbreviate by using just "point" or "points".

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

Detection limit for a given analytical procedure, concentration or quantity derived from the smallest measure that can be detected with reasonable certainty

In analytical chemistry, the detection limit, lower limit of detection, or LOD, is the lowest quantity of a substance that can be distinguished from the absence of that substance with a stated confidence level. The detection limit is estimated from the mean of the blank, the standard deviation of the blank and some confidence factor. Another consideration that affects the detection limit is the accuracy of the model used to predict concentration from the raw analytical signal.

In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by a statement of the associated uncertainty, such as the standard deviation. By international agreement, this uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity value. It is a non-negative parameter.

In decision theory and economics, ambiguity aversion is a preference for known risks over unknown risks. An ambiguity-averse individual would rather choose an alternative where the probability distribution of the outcomes is known over one where the probabilities are unknown. This behavior was first introduced through the Ellsberg paradox.

Statistical dispersion general term for the value describing how spread out the data are

In statistics, dispersion is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

Risk is the potential for uncontrolled loss of something of value. Values can be gained or lost when taking risk resulting from a given action or inaction, foreseen or unforeseen. Risk can also be defined as the intentional interaction with uncertainty. Uncertainty is a potential, unpredictable, and uncontrollable outcome; risk is an aspect of action taken in spite of uncertainty.

References

  1. Peter Norvig; Sebastian Thrun. "Introduction to Artificial Intelligence". Udacity .
  2. Kabir, H. D., Khosravi, A., Hosen, M. A., & Nahavandi, S. (2018). Neural Network-based Uncertainty Quantification: A Survey of Methodologies and Applications. IEEE Access. Vol. 6, Pages 36218 - 36234, doi : 10.1109/ACCESS.2018.2836917
  3. Gärdenfors, Peter; Sahlin, Nils-Eric (1982). "Unreliable probabilities, risk taking, and decision making". Synthese. 53 (3): 361–386. doi:10.1007/BF00486156.
  4. David Sundgren and Alexander Karlsson. Uncertainty levels of second-order probability. Polibits, 48:5–11, 2013.
  5. Audun Jøsang. Subjective Logic: A Formalism for Reasoning Under Uncertainty. Springer, Heidelberg, 2016.
  6. Douglas Hubbard (2010). How to Measure Anything: Finding the Value of Intangibles in Business, 2nd ed. John Wiley & Sons. Description Archived 2011-11-22 at the Wayback Machine , contents Archived 2013-04-27 at the Wayback Machine , and preview.
  7. Jean-Jacques Laffont (1989). The Economics of Uncertainty and Information, MIT Press. Description Archived 2012-01-25 at the Wayback Machine and chapter-preview links.
  8. Jean-Jacques Laffont (1980). Essays in the Economics of Uncertainty, Harvard University Press. Chapter-preview links.
  9. Robert G. Chambers and John Quiggin (2000). Uncertainty, Production, Choice, and Agency: The State-Contingent Approach. Cambridge. Description and preview. ISBN   0-521-62244-1
  10. Knight, F. H. (1921). Risk, Uncertainty, and Profit. Boston: Hart, Schaffner & Marx.
  11. Tannert C, Elvers HD, Jandrig B (2007). "The ethics of uncertainty. In the light of possible dangers, research becomes a moral duty". EMBO Rep. 8 (10): 892–6. doi:10.1038/sj.embor.7401072. PMC   2002561 . PMID   17906667.
  12. "Standard Uncertainty and Relative Standard Uncertainty". CODATA reference. NIST. Archived from the original on 16 October 2011. Retrieved 26 September 2011.Cite uses deprecated parameter |deadurl= (help)
  13. 1 2 3 4 5 Zehr, S. C. (1999). Scientists' representations of uncertainty. In Friedman, S.M., Dunwoody, S., & Rogers, C. L. (Eds.), Communicating uncertainty: Media coverage of new and controversial science (3–21). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
  14. Nisbet, M.; Scheufele, D. A. (2009). "What's next for science communication? Promising directions and lingering distractions". American Journal of Botany . 96 (10): 1767–1778. doi:10.3732/ajb.0900041. PMID   21622297.
  15. Shackley, S.; Wynne, B. (1996). "Representing uncertainty in global climate change science and policy: Boundary-ordering devices and authority". Science, Technology, & Human Values. 21 (3): 275–302. doi:10.1177/016224399602100302.
  16. Somerville, R. C.; Hassol, S. J. (2011). "Communicating the science of climate change". Physics Today . 64 (10): 48–53. Bibcode:2011PhT....64j..48S. doi:10.1063/pt.3.1296.
  17. 1 2 3 4 5 6 7 8 9 Stocking, H. (1999). "How journalists deal with scientific uncertainty". In Friedman, S. M.; Dunwoody, S.; Rogers, C. L. (eds.). Communicating Uncertainty: Media Coverage of New and Controversial Science. Mahwah, NJ: Lawrence Erlbaum. pp. 23–41. ISBN   978-0-8058-2727-9.
  18. 1 2 Nisbet, M.; Scheufele, D. A. (2007). "The Future of Public Engagement". The Scientist . 21 (10): 38–44.
  19. Gregory, Kent J.; Bibbo, Giovanni; Pattison, John E. (2005). "A Standard Approach to Measurement Uncertainties for Scientists and Engineers in Medicine". Australasian Physical and Engineering Sciences in Medicine. 28 (2): 131–139. doi:10.1007/BF03178705.
  20. "Archived copy". Archived from the original on 2015-09-26. Retrieved 2016-07-29.Cite uses deprecated parameter |deadurl= (help)CS1 maint: archived copy as title (link)

Further reading