Error propagation | PhysicsRank

A door is almost as wide as its frame. Both widths look well measured, yet their small difference may be uncertain enough to decide whether the door fits. Here we follow uncertainty through the actual calculation, rather than attach a percentage at the end.

We will first answer that question with intervals and ordinary algebra. Then we will derive the familiar small-uncertainty rules and distinguish them from statistical quadrature. This order matters: a list of formulas cannot tell you whether the uncertainties are bounds, whether two inputs are independent, or whether the same measured quantity has been used twice.

What you will learn to do

Find exact output bounds for simple monotonic models.
Derive first-order relative bounds for products, quotients and powers.
Distinguish a worst-case bound from a combined standard uncertainty.
Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
Handle partially correlated inputs and common calibration parameters.
Use sensitivity budgets to choose an experimental improvement.
Recognize when a first-order uncertainty approximation fails.

Useful before this chapterMeasurement uncertainty →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Why can subtracting two accurately measured large lengths give a poorly known small gap?

Compare your reasoning

The gap is small while the absolute measurement uncertainties may remain substantial. Relative uncertainty compares uncertainty with the result, so a small denominator can make it large.

Revisit the explanation →

01 / Propagation of uncertainty

First write what the experiment actually computes

Start with a measurement model such as v = L/t or ρ = m/(πr²h). The symbols must mean the quantities that were actually measured. A radius inferred from one diameter is not an independent second measurement of that same cylinder.

Also identify the input uncertainty model. In the first examples, x = x0 ± Δx means every value in that interval is admissible and no additional relation between inputs restricts the extreme combinations. A worst-case output interval must then include the largest and smallest values the model can produce.

The simplest method is to ask how the output changes when each input increases. A positive product grows with either factor. A positive quotient grows with its numerator but shrinks when its denominator grows. These physical directions determine which endpoints to combine.

WORKED EXAMPLE

Find a speed interval without an approximation

A path length is (12.0 ± 0.2) m and its elapsed time is (3.00 ± 0.05) s. Both ± terms are bounds. What speed interval is permitted?

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: First write what the experiment actually computes
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Propagation of uncertainty

Why absolute bounds add even in a difference

For z = x + y, deviations combine as δz = δx + δy. If the input deviations may independently reach ±Δx and ±Δy, their signs can align to give the largest magnitude. Thus a safe worst-case bound is Δx + Δy.

For z = x − y, the deviation is δx − δy. The largest positive change occurs when x is high and y is low, so the same sum of absolute bounds applies. The minus sign in the nominal calculation does not automatically subtract uncertainty.

|\delta z|\le |\delta x|+|\delta y|\le\Delta x+\Delta y

This triangle-inequality argument applies to a sum or difference when no extra dependence restricts the input deviations.

Subtraction of nearly equal numbers deserves extra care. The output may be much smaller than either input, while its absolute bound is not. Relative uncertainty can then become large, and even the sign of the result may remain unresolved.

WORKED EXAMPLE

A small nominal difference need not be a precise result

Two lengths have stated bounds: A = (10.0 ± 0.1) cm and B = (9.8 ± 0.1) cm. Determine D = A − B and its relative bound.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Why absolute bounds add even in a difference
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
|\delta z|\le |\delta x|+|\delta y|\le\Delta x+\Delta y
Meaning: This triangle-inequality argument applies to a sum or difference when no extra dependence restricts the input deviations.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Propagation of uncertainty

Derive the small-error rule instead of memorizing it

For a product, write the perturbed values explicitly: (x + δx)(y + δy). Expanding shows two terms that are first order in the input deviations and a product of deviations that is second order. Dividing by the nominal product exposes the relative changes.

When both fractional deviations are small, their product is much smaller and may be neglected for a first-order estimate. That is the approximation behind adding percentage bounds. It is not an exact identity for arbitrarily large input uncertainties.

\frac{\delta(xy)}{xy}=\frac{\delta x}{x}+\frac{\delta y}{y}+\frac{\delta x\,\delta y}{xy}

The final term is the relative second-order correction. Keep it or use endpoint evaluation when it matters.

\frac{\Delta(xy)}{|xy|}\approx\frac{\Delta x}{|x|}+\frac{\Delta y}{|y|}

This is a first-order worst-case bound estimate, using positive uncertainty magnitudes.

WORKED EXAMPLE

Compare a first-order estimate with exact bounds

A rectangle measures (5.00 ± 0.10) cm by (2.00 ± 0.04) cm. Find its area and compare approximate and exact bounds.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Derive the small-error rule instead of memorizing it
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
\frac{\delta(xy)}{xy}=\frac{\delta x}{x}+\frac{\delta y}{y}+\frac{\delta x\,\delta y}{xy}
Meaning: The final term is the relative second-order correction. Keep it or use endpoint evaluation when it matters.
\frac{\Delta(xy)}{|xy|}\approx\frac{\Delta x}{|x|}+\frac{\Delta y}{|y|}
Meaning: This is a first-order worst-case bound estimate, using positive uncertainty magnitudes.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Propagation of uncertainty

Powers measure sensitivity

If an output depends on a measured radius squared, a small percentage change in radius produces roughly twice that percentage change in the output. A cube gives roughly three times the fractional change. These factors describe sensitivity, not the number of separate independent radius measurements.

For a product of powers, differentiating the logarithm or expanding each factor gives a compact first-order relation. In a worst-case bound, use the absolute values of the exponents because deviations may combine to enlarge the output error. A negative exponent does not create a negative uncertainty.

z=Cx^a y^b\quad\Rightarrow\quad\frac{\Delta z}{|z|}\approx |a|\frac{\Delta x}{|x|}+|b|\frac{\Delta y}{|y|}

C is taken as exact here; include its contribution separately if it is measured. The power-law model must be differentiable over the allowed input region.

WORKED EXAMPLE

Choose which measurement to improve

A cylinder’s density is calculated as ρ = m/(πr²h). The mass, radius and height have small stated relative bounds of 1%, 2% and 1%. Estimate the worst-case percentage bound in density.

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STOP & THINK

A pendulum estimate is g = 4π²L/T². If L and T have small bounds of 1% each, what is the first-order percentage bound in g?

Answer in %.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Powers measure sensitivity
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
z=Cx^a y^b\quad\Rightarrow\quad\frac{\Delta z}{|z|}\approx |a|\frac{\Delta x}{|x|}+|b|\frac{\Delta y}{|y|}
Meaning: C is taken as exact here; include its contribution separately if it is measured. The power-law model must be differentiable over the allowed input region.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Propagation of uncertainty

Quadrature answers a different question

If the inputs are described by standard uncertainties, the goal is to estimate an output standard deviation rather than a maximum excursion. For small deviations and uncorrelated inputs, squared sensitivity-weighted standard uncertainties add. Taking the square root gives the combined standard uncertainty.

The squared terms arise because we are combining variances. Independence is sufficient for zero covariance, but it must be justified rather than assumed from the fact that two symbols look different. Shared calibration, common clocks or reused readings can correlate the inputs.

u_z^2\approx\left(\frac{\partial z}{\partial x}\right)^2u_x^2+\left(\frac{\partial z}{\partial y}\right)^2u_y^2

This linearized form assumes zero covariance. With covariance, an additional term 2(∂z/∂x)(∂z/∂y)cov(x,y) is required.

Suppose a sum contains two uncorrelated standard uncertainties of 0.3 cm and 0.4 cm. Their combined standard uncertainty is √(0.3² + 0.4²) = 0.5 cm. If those same numerical values were instead stated as independently admissible worst-case bounds, the worst-case sum would be 0.7 cm. Neither answer can be selected until the meaning of the inputs is known.

STOP & THINK

Two independent input standard uncertainties in a sum are 3 units and 4 units. What is their combined standard uncertainty?

7 units
5 units
1 unit

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Quadrature answers a different question
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
u_z^2\approx\left(\frac{\partial z}{\partial x}\right)^2u_x^2+\left(\frac{\partial z}{\partial y}\right)^2u_y^2
Meaning: This linearized form assumes zero covariance. With covariance, an additional term 2(∂z/∂x)(∂z/∂y)cov(x,y) is required.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Propagation of uncertainty

Do not count the same uncertainty twice

Imagine defining y = 2x using one measured x. Now calculate y/(2x). The ratio is exactly one within that model for every permitted nonzero x. Treating y and x as independent uncertain measurements would manufacture an uncertainty that the algebra does not contain.

This does not mean all repeated appearances of a symbol magically cancel in real experiments. The cancellation is justified because it is literally the same measured input with an exact relationship. If the numerator and denominator were measured independently on two occasions, they would be different inputs and require their actual uncertainty relationship.

A common additive zero offset can similarly cancel in a difference of two coordinates read on the same unchanged scale. It does not follow that all uncertainty cancels: random endpoint reading errors, scale-factor error and changes in the offset may remain. Write separate terms for separate physical contributions.

(x_2+b)-(x_1+b)=x_2-x_1

A common additive b cancels exactly. An uncertain multiplicative scale factor would behave differently.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Do not count the same uncertainty twice
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
(x_2+b)-(x_1+b)=x_2-x_1
Meaning: A common additive b cancels exactly. An uncertain multiplicative scale factor would behave differently.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

07 / Propagation of uncertainty

Correlated inputs can reinforce or cancel each other

The previous shared-input example was exact cancellation. A more general situation is partial correlation: two readings tend to wander in the same or opposite directions without being identical. Correlation is a statement about how their deviations vary together, not about whether their displayed values happen to be similar.

For two inputs, write the small output deviation as δy ≈ c₁δx₁ + c₂δx₂, where each c is a derivative of the measurement model at the reported inputs. Squaring and averaging gives two variance terms and one covariance term. The sign of that last term depends on both the correlation and the signs of the sensitivities.

For a sum, positive correlation increases the combined standard uncertainty. For a difference, the sensitivities have opposite signs, so positive correlation reduces it. Independent random errors have zero covariance, but shared zero offsets, gains or timing references can prevent independence. Do not assign a correlation coefficient merely to make an uncertainty smaller.

u_y^2\approx c_1^2u_1^2+c_2^2u_2^2+2c_1c_2\rho u_1u_2,\qquad -1\le\rho\le1

This is a first-order standard-uncertainty relation. c₁ and c₂ carry the units needed to convert input deviations to output units. It is not a worst-case bound or automatically a 95% confidence interval.

WORKED EXAMPLE

Calculate a difference from partially correlated readings

Two coordinate estimates have standard uncertainties 0.30 cm and 0.40 cm, with correlation coefficient +0.50 from their measurement model. Find the standard uncertainty of the second minus the first.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Correlated inputs can reinforce or cancel each other
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
u_y^2\approx c_1^2u_1^2+c_2^2u_2^2+2c_1c_2\rho u_1u_2,\qquad -1\le\rho\le1
Meaning: This is a first-order standard-uncertainty relation. c₁ and c₂ carry the units needed to convert input deviations to output units. It is not a worst-case bound or automatically a 95% confidence interval.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

08 / Propagation of uncertainty

Write shared offsets and gains as shared variables

A calibration uncertainty is not automatically an independent error added to every reading. If two coordinates share the same unchanged zero correction, introduce one variable b for it. If they share a scale factor, introduce one variable s. Only after simplifying the measurement equation should you propagate the independent inputs.

For coordinates x₁ = sr₁ + b and x₂ = sr₂ + b, their separation is s(r₂ − r₁). The shared additive offset cancels exactly. The shared scale factor remains because it stretches the separation itself. Separate random reading errors remain as well.

This formulation is often clearer than constructing a covariance matrix of the final coordinates. It also makes the experimental condition visible: if the instrument is reset between readings, or its zero drifts, the two offsets may no longer be the same variable. Cancellation must describe the procedure actually used.

WORKED EXAMPLE

A ruler’s zero cancels, but its scale does not

Raw coordinates r₁ = 12.0 cm and r₂ = 30.0 cm have independent standard reading uncertainties of 0.030 cm each. The shared scale factor is s = 1 with standard uncertainty 0.002; a shared additive offset has standard uncertainty 0.20 cm. Assume the scale uncertainty is uncorrelated with the reading errors. Find the separation and its combined standard uncertainty.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Write shared offsets and gains as shared variables
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

09 / Propagation of uncertainty

Use an uncertainty budget to choose the next measurement improvement

An uncertainty calculation should help design the experiment, not only add a ± sign at the end. For each independent input, calculate the squared sensitivity-weighted contribution. The largest contribution shows which improvement could reduce the total most, if cost and other conditions are comparable.

The role of an exponent is physical sensitivity. For g = 4π²L/T² in the small-angle pendulum model, a fractional period error affects g twice as strongly as the same fractional length error. The sign is negative, but its square enters an uncorrelated standard-uncertainty budget.

This model also assumes an appropriate pivot-to-centre-of-mass length, small oscillation angle and negligible corrections from drag or support motion. Excellent stopwatch precision does not eliminate an omitted model correction. Include significant corrections and their uncertainty when the required accuracy makes them relevant.

WORKED EXAMPLE

Decide whether to improve timing or length

A small-angle pendulum experiment uses L = 1.000 m with uL = 0.006 m and T = 2.000 s with uT = 0.008 s. Treat these as uncorrelated standard uncertainties and neglect other corrections. Find the fractional standard uncertainty of g and compare halving uL with halving uT.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: Use an uncertainty budget to choose the next measurement improvement
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

10 / Propagation of uncertainty

A zero derivative does not guarantee an exact answer

First-order propagation approximates a nonlinear function by its tangent. It can fail if the uncertainty spans a large change of slope, crosses a singularity, or is centred where the first derivative vanishes. Before trusting a small standard-uncertainty number, inspect the input range and the curvature of the actual measurement model.

For bounded inputs and a monotonic function, evaluate the appropriate endpoints to obtain an output interval. When the function is not monotonic over the range, also check stationary points. For probabilistic inputs, an interval alone does not determine a standard uncertainty; a distributional calculation or justified higher-order treatment may be needed.

Keep the distinction between reporting and inference. Significant figures communicate a chosen resolution. They cannot certify that a nonlinear approximation is valid or reveal whether common errors cancel. Extra digits in intermediate calculations are useful, but they cannot compensate for a missing uncertainty model.

WORKED EXAMPLE

Test the tangent approximation at a stationary point

A bounded coordinate x lies anywhere from −0.20 m to +0.20 m. The reported nominal value is zero. What range is possible for y = x²?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Propagation of uncertainty
Focus: A zero derivative does not guarantee an exact answer
Learning goals:
- Find exact output bounds for simple monotonic models.
- Derive first-order relative bounds for products, quotients and powers.
- Distinguish a worst-case bound from a combined standard uncertainty.
- Recognize cancellation, correlation and large-uncertainty cases where a memorized rule fails.
- Handle partially correlated inputs and common calibration parameters.
- Use sensitivity budgets to choose an experimental improvement.
- Recognize when a first-order uncertainty approximation fails.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Worst-case bounds describe admissible extremes; standard uncertainties describe standard-deviation scales.
For a sum or difference, independent admissible absolute bounds add.
For products and powers, weighted relative bounds add only as a small-uncertainty approximation.
Quadrature needs the standard-uncertainty model and appropriate covariance assumptions.
Exact cancellation and shared errors must be resolved in the measurement model before using a formula.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

For z = x − y, the input bounds are ±0.07 m and ±0.03 m, with extremes freely admissible. What worst-case absolute bound applies to z?

Answer in m.

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SELF CHECK 2

A sphere radius has a small 2% bound. What is the first-order percentage bound in its volume?

Answer in %.

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SELF CHECK 3

The same measured nonzero x is used to define y = 3x exactly. What uncertainty arises from x in y/x?

Its relative uncertainty doubles
It cancels from the ratio
Use quadrature on y and x as independent inputs

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SELF CHECK 4

A denominator’s permitted interval includes zero. What should you do first?

Apply a percentage-error formula anyway
Inspect the domain and possible divergence
Replace the denominator uncertainty by zero

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SELF CHECK 5

Two inputs each have standard uncertainty 0.20 m and correlation ρ = 0.50. What is the standard uncertainty of their difference?

Answer in m.

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SELF CHECK 6

For y = x² near x = 0, a first-order calculation predicts zero output uncertainty. Which next step is justified?

Declare y exactly known
Inspect the nonlinear model and the input range or distribution
Round x to fewer digits and stop

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

A diameter bound becomes an area boundIntermediate · Hints and guided reasoning When a large zero uncertainty cancelsAdvanced · Hints and guided reasoning A zero derivative hides a finite rangeAdvanced · Hints and guided reasoning How uncertain is the rectangle?Intermediate · Hints and guided reasoning Which uncertainty rule applies?Intermediate · Hints and guided reasoning The same formula, a different uncertaintyIntermediate · Hints and guided reasoning When more readings reach a limitAdvanced · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

Two coordinates share the same additive zero offset. Explain what happens to that offset in their difference and in their sum.

Compare your reasoning

It cancels in the difference because the same variable appears with opposite signs. In the sum it appears twice. First write the measurement model; the word “error” alone does not determine the combination rule.

Revisit the explanation →

A few days later: change the situation

If x lies between −2 and +1 cm, what range can x² occupy? Why is the derivative at zero insufficient?

Compare your reasoning

The minimum is zero and the maximum is 4 cm². A zero tangent at x = 0 does not remove curvature over the interval. These bounds do not specify a standard deviation.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑