Propagation of uncertainty

Key takeaways

Propagation of uncertainty belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Propagation of uncertainty to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Propagation of uncertainty from memory before moving on to harder problems.

Reference excerpt

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 (e.g., instrument precision) which propagate due to the combination of variables in the function. The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval x ± u. However, the most general way of characterizing uncertainty is by specifying its probability distribution. If the probability distribution of the variable is known or can be assumed, in theory it is possible to get any of its statistics. In particular, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases. If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated. In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the Monte Carlo method family. For very large datasets or complex functions, the calculation of the error propagation may be very expensive so that a surrogate model or a parallel computing strategy may be necessary. In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.

Linear combinations Let { f k ( x 1 , x 2 , … , x n ) } {\displaystyle \{f_{k}(x_{1},x_{2},\dots ,x_{n})\}} be a set of m functions, which are linear combinations of n {\displaystyle n} variables x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} with combination coefficients A k 1 , A k 2 , … , A k n , ( k = 1 , … , m ) {\displaystyle A_{k1},A_{k2},\dots ,A_{kn},(k=1,\dots ,m)} :

f k = ∑ i = 1 n A k i x i , {\displaystyle f_{k}=\sum _{i=1}^{n}A_{ki}x_{i},}

or in matrix notation,

f = A x . {\displaystyle \mathbf {f} =\mathbf {A} \mathbf {x} .}

Also let the variance–covariance matrix of x = (x1, ..., xn) be denoted by Σ x {\displaystyle {\boldsymbol {\Sigma }}^{x}} and let the mean value be denoted by μ {\displaystyle {\boldsymbol {\mu }}} :

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Propagation of uncertainty

Start with the simplest possible case. Write down what Propagation of uncertainty claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Propagation of uncertainty before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Propagation of uncertainty ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Propagation of uncertainty

In research: Propagation of uncertainty appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Propagation of uncertainty in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Propagation of uncertainty is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebra of random variables, Numerical analysis, Statistical approximations, so understanding it makes those chapters shorter.
In everyday life: Look for Propagation of uncertainty outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Propagation of uncertainty in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Propagation of uncertainty means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Propagation of uncertainty out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Propagation of uncertainty in simple terms?

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 (e.g., instrument precision) which propagate due…

Why does Propagation of uncertainty matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Propagation of uncertainty?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Propagation of uncertainty.