# Uncertainty

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Certainty series:

Uncertainty is a term used in subtly different ways in a number of fields, including philosophy, statistics, economics, finance, insurance, psychology, engineering and science. It applies to predictions of future events, to physical measurements already made, or to the unknown.

## Relation between uncertainty, probability and risk

In his seminal work Risk, Uncertainty, and Profit, University of Chicago economist Frank Knight (1921) established the important distinction between risk and 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."

Risk is defined as uncertainty based on a well grounded (quantitative) probability. Formally, Risk = (the probability that some event will occur) X (the consequences if it does occur). Genuine uncertainty, on the other hand, cannot be assigned such a (well grounded) probability. Furthermore, genuine uncertainty can often not be reduced significantly by attempting to gain more information about the phenomena in question and their causes. (Andersen et. al.: Modelling Society’s Capacity to Manage Extraordinary Events, 2004.)

There are other measures of uncertainty:

• In stochastics, risk is an uncertainty for which probability can be calculated (with past statistics for example) or at least estimated (doing projection scenarios) mathematically.
• In insurance, risk deals only with negative uncertainty (those bringing loss or harm)
• In cognitive psychology, uncertainty can be real, or just a matter of perception, such as expectations, threats, etc.

Mathematicians handle uncertainty using probability theory, Dempster-Shafer theory, and fuzzy logic. See also probability.

Surprisal is a measure of uncertainty in information theory.

## Relation between uncertainty, accuracy, precision, standard deviation, standard error, and confidence interval

The uncertainty of a measurement is stated by giving a range of values which are likely to enclose the true value. This may be denoted by error bars on a graph, or as value ± uncertainty, or as decimal fraction(uncertainty). The latter "concise notation" is used for example by IUPAC in stating the atomic mass of elements. There, 1.00794(7) stands for 1.00794 ± 0.00007.

Often, the uncertainty of a measurement is found by repeating the measurement enough times to get a good 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 and the mean is reported, then the averaged measurement has uncertainty equal to the standard error which is the standard deviation divided by the square root of the number of measurements.

When the uncertainty represents the standard error of the measurement, then about 68.2% of the time, the true value of the measured quantity falls within the stated uncertainty range. For example, it is likely that for 31.8% 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 68.2% ("one sigma"), 95.4% ("two sigma"), or 99.7% ("three sigma") confidence intervals.

## Fields of activities or knowledge where uncertainty is important

• Investing in financial markets such as the stock market.
• Uncertainty is used in engineering notation when talking about significant figures. Or the possible error involved in measuring things such as distance.
• Uncertainty is designed into games, most notably in gambling, where chance is central to play.
• In physics in certain situations, uncertainty has been elevated into a principle, the uncertainty principle.
• In weather forcasting it is now commonplace to include data on the degree of uncertainty in a weather forecast.
• Uncertainty is often an important factor in economics. According to economist Frank Knight, it is different from risk, where there is a specific probability assigned to each outcome (as when flipping a fair coin). Uncertainty involves a situation that has unknown probabilities, while the estimated probabilities of possible outcomes need not add to unity.
• In metrology, measurement uncertainty is a central concept quantifying the dispersion one may reasonably attribute to a measurement result. Such an uncertainty can also be referred to as a measurement error. In daily life, measurement uncertainty is often implicit ("He is 6 feet tall" give or take a few inches), while for any serious use an explicit statement of the measurement uncertainty is necessary. The expected measurement uncertainty of many measuring instruments (scales, oscilloscopes, force gages, rulers, thermometers, etc) is often stated in the manufacturers specification.
The most commonly used procedure for calculating measurement uncertainty is described in the Guide to the Expression of Uncertainty in Measurement (often referred to as "the GUM") published by ISO. A derived work is for example the National Institute for Standards and Technology (NIST) publication NIST Technical Note 1297 "Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results". 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 which are evaluated by statistical methods,
• Type B, those which are 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.

## Uncertainty as an artistic theme

Uncertainty has been a common theme in art, both as a thematic device (see, for example, the indecision of Hamlet), and as a quandary for the artist (such as Martin Creed's difficulty with deciding what artworks to make).