Uncertainty or incertitude refers to situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown, and is particularly relevant for decision-making. Uncertainty arises in partially observable or stochastic or complex or dynamic environments, as well as due to ignorance, indolence, or both. It arises in any number of fields, including insurance, philosophy, physics, statistics, economics, entrepreneurship, finance, medicine, psychology, sociology, engineering, metrology, meteorology, ecology and information science.
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:
Uncertainty The lack of certainty, a state of limited knowledge where it is impossible to exactly describe one or more of the following: the existing state or goal or set of options, the full set of possible future states or their probabilities or their values or utilities to stakeholders, the full set of stakeholders involved, or any other piece of information that inhibits the calculation of the full set of expected values for the options available.Hubbard, D. W. (2014). How to measure anything: finding the value of "intangibles" in business. Wiley.Arend, R.J. (2024). Uncertainty in Strategic Decision Making: Analysis, Categorization, Causation and Resolution. Palgrave Macmillan.
Measurement Uncertainty, only when considered as objective or subjective risk, can be measured through 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.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,
Second-order uncertainty In statistics and economics, second-order uncertainty - expressed as the confidence over outcome probability estimates - is represented in probability density functions over (first-order) probabilities.David Sundgren and Alexander Karlsson. Uncertainty levels of second-order probability. Polibits, 48:5–11, 2013.
Opinions in subjective logicAudun Jøsang. Subjective Logic: A Formalism for Reasoning Under Uncertainty. Springer, Heidelberg, 2016. carry this type of uncertainty.
Risk Risk exists when the future realized point value of a distribution of possible outcomes is not known but an estimated value can be calculated, and where some possible outcomes have an undesired effect or significant loss. Measurement of risk includes a set of outcome probabilities along with the valuations of those outcomes, where some possible outcomes involve losses. This also includes loss functions over continuous variables. Because an expected value for a decision can be calculated, some do not consider risk to be a true type of uncertainty (see below).Douglas Hubbard (2010). How to Measure Anything: Finding the Value of Intangibles in Business, 2nd ed. John Wiley & Sons. Description , contents , and preview.Jean-Jacques Laffont (1989). The Economics of Uncertainty and Information, MIT Press. Description and chapter-preview links.Jean-Jacques Laffont (1980). Essays in the Economics of Uncertainty, Harvard University Press. Chapter-preview links.Robert G. Chambers and John Quiggin (2000). Uncertainty, Production, Choice, and Agency: The State-Contingent Approach. Cambridge. Description and preview.
Risk versus variability There is a difference between risk and variability. Risk is quantified by a probability distribution which depends upon knowledge about the likelihood of what the single, true value of the future quantity will be, as in the case of a roll of the dice. Variability is quantified by a distribution of frequencies of multiple instances of the quantity, derived from observed data, as in the case of a batting average.Begg, Steve H., Matthew B. Welsh, and Reidar B. Bratvold. "Uncertainty vs. Variability: What's the Difference and Why is it Important?." SPE Hydrocarbon Economics and Evaluation Symposium. OnePetro, 2014.
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. To Knight, uncertainty is uninsurable while risk is (hypothetically) insurable. Because of the absence of clearly defined statistics in most economic decisions where people face uncertainty, he believed that we cannot measure probabilities in such cases; this is now referred to as Knightian uncertainty.
Knight pointed out that the unfavorable outcome of known risks can be insured during the decision-making process because it has a clearly defined expected probability distribution. Uncertainties have no known expected probability distribution, which can lead to extreme outcomes when borne. Because Knight refers to those who do bear uncertainty as 'entrepreneurs', the field of entrepreneurship includes a stream of research on uncertainty and how it creates opportunities.Arend, R.J. (2024). Uncertainty and Entrepreneurship: A Critical Review of the Research, with Implications for the Field. Foundations and Trends in Entrepreneurship, 20(2): 109-244.
Other taxonomies of uncertainties and decisions include more specific characterizations of uncertainty, specifying what is known, knowable, and unknowable about the phenomenon, as well as how it should be approached from an ethics perspective:
Risk and uncertainty 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 risk 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.
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 among 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.
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.
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. Ambiguity can also refer to a type of uncertainty where the range of a distribution of possible outcomes is known, but not their probabilities. Daniel Ellsberg is famous for his urn experiments that illustrated ambiguity, its delineation from risk, and its separate avoidance.Ellsberg, D. (1961). Risk, Ambiguity, and the Savage Axioms. Quarterly Journal of Economics, 75(4), 643-669.
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.
Radical uncertainty The term 'radical uncertainty' was popularised by John Kay and Mervyn King in their book Radical Uncertainty: Decision-Making for an Unknowable Future, published in March 2020. It is distinct from Knightian uncertainty, by whether or not it is 'resolvable'. If uncertainty arises from a lack of knowledge, and that lack of knowledge is resolvable by acquiring knowledge (such as by primary or secondary research) then it is not radical uncertainty. Only when there are no means available to acquire the knowledge which would resolve the uncertainty, is it considered 'radical'.
In 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 of 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: Type A, those evaluated by statistical methods Type B, those evaluated by other means, e.g., by assigning a probability distribution 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 metrology, physics, engineering, and chemistry, 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. The uncertainty is often the standard uncertainty, which assumes an approximately Gaussian distribution, with the uncertainty expressing one standard deviation. This may be denoted by error bars on a graph, or by the following notations: measured value ± uncertainty measured value measured value (uncertainty)