Uncertainty is a situation which involves
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.
It arises in any number of fields, including
insurance, philosophy, physics, statistics,
economics, finance, 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 the existing state, a future outcome,
or more than one possible outcome.
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.
Second order uncertainty
In statistics and economics, second-order
uncertainty is represented in probability
density functions over (first-order) probabilities..
Opinions in subjective logic 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.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.
Other taxonomies of uncertainties and decisions
include a broader sense of uncertainty and
how it should be approached from an ethics
perspective:
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.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.
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.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.
== 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:
Type A, those evaluated by statistical methods
Type B, those evaluated by other means, e.g.,
by assigning a probability distributionBy
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:
measured value ± uncertainty
measured value +uncertainty−uncertainty
measured value (uncertainty)
In the last notation, parentheses are the
concise notation for the ± notation.
For example, applying 10 ​1⁄2 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.
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.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.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.
Notice that 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.
== Uncertainty and the media ==
Uncertainty in science, and science in general,
may be interpreted differently in the public
sphere than in the scientific community.
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.
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.
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.
This interpretation may have even been intentionally
promoted, as scientific uncertainty may be
managed to reach certain goals.
For example, global warming contrarian activists
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."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."
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.
Conversely, uncertainty is often interpreted
by the public as ignorance.
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).
One way that journalists inflate uncertainty
is by describing new research that contradicts
past research without providing context for
the change.
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.
In the same vein, journalists may give non-scientists
the same amount of attention and importance
as scientists.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".
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.
There is often a "product over process" approach
to science journalism that aids, too, in the
downplaying of uncertainty.
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".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.
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.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.
== Applications ==
Uncertainty is designed into games, most notably
in gambling, where chance is central to play.
In scientific modelling, in which the prediction
of future events should be understood to have
a range of expected values
In optimization, uncertainty permits one to
describe situations where the user does not
have full control on the final outcome of
the optimization procedure, see scenario optimization
and stochastic optimization.
In weather forecasting, it is now commonplace
to include data on the degree of uncertainty
in a weather forecast.
Uncertainty or error is used in science and
engineering notation.
Numerical values should only be expressed
to those digits that are physically meaningful,
which are referred to as significant figures.
Uncertainty is involved in every measurement,
such as measuring a distance, a temperature,
etc., the degree depending upon the instrument
or technique used to make the measurement.
Similarly, uncertainty is propagated through
calculations so that the calculated value
has some degree of uncertainty depending upon
the uncertainties of the measured values and
the equation used in the calculation.
In physics, the Heisenberg uncertainty principle
forms the basis of modern quantum mechanics.
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' specifications.
In engineering, uncertainty can be used in
the context of validation and verification
of material modeling.
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).
Uncertainty is 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).
Knightian uncertainty involves a situation
that has unknown probabilities.
Investing in financial markets such as the
stock market involves Knightian uncertainty
when the probabiliy of a rare but catastrophic
event is unknown.
In entrepreneurship: New products, services,
firms and even markets may be created in the
absence of probability estimates.
According to entrepreneurship research, expert
entrepreneurs use experience based heuristics
called effectuation (as opposed to causality)
to overcome uncertainty.
== See also ==
== References ==
== Further reading ==
Lindley, Dennis V. (2006-09-11).
Understanding Uncertainty.
Wiley-Interscience.
ISBN 978-0-470-04383-7.
Gilboa, Itzhak (2009).
Theory of Decision under Uncertainty.
Cambridge: Cambridge University Press.
ISBN 9780521517324.
Halpern, Joseph (2005-09-01).
Reasoning about Uncertainty.
MIT Press.
ISBN 9780521517324.
Smithson, Michael (1989).
Ignorance and Uncertainty.
New York: Springer-Verlag.
ISBN 0-387-96945-4.
== External links ==
Measurement Uncertainties in Science and Technology,
Springer 2005
Proposal for a New Error Calculus
Estimation of Measurement Uncertainties — an
Alternative to the ISO Guide
Bibliography of Papers Regarding Measurement
Uncertainty
Guidelines for Evaluating and Expressing the
Uncertainty of NIST Measurement Results
Strategic Engineering: Designing Systems and
Products under Uncertainty (MIT Research Group)
Understanding Uncertainty site from Cambridge's
Winton programme
Bowley, Roger (2009).
"∆ – Uncertainty".
Sixty Symbols.
Brady Haran for the University of Nottingham.
