Complete Guide

What is Uncertainty
Propagation?

A complete guide for physics and engineering students - from first principles to lab report ready formulas, with worked examples throughout.

⏱ 12 min read🎓 University level🔬 Lab report focused

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What is Uncertainty Propagation?

Every physical measurement has an uncertainty - a range within which the true value is likely to lie. When you measure the length of a rod with a ruler, you might write L = 12.3 ± 0.1 cm. The ± 0.1 cm is the measurement uncertainty.

The problem arises when you use multiple measured quantities to calculate a derived quantity. If you calculate the area of a rectangle from its length and width - both of which have uncertainties - what is the uncertainty in the area?

Uncertainty propagation (also called error propagation or propagation of errors) is the mathematical method for answering exactly this question. It tells you how the individual measurement uncertainties combine - propagate - through your formula to produce the total uncertainty in the final result.

Type of Measurement Error

Before propagating uncertainties, you need to understand the two fundamentally different types of error in experimental physics.

Random errors
Unpredictable fluctuations that vary from one measurement to the next. They scatter readings randomly around the true value. Causes include electrical noise, vibration, human reaction time, and air currents. Random errors can be reduced by taking repeated measurements and averaging - the uncertainty in the mean decreases as 1/n1/\sqrt{n}.
Systematic errors
Consistent biases that shift every reading in the same direction by the same amount. Causes include miscalibrated instruments, parallax, zero-offset errors, and environmental factors. Systematic errors cannot be reduced by averaging - they must be identified and corrected. They do not appear in uncertainty propagation unless explicitly included as a variable.

The General Propagation Formula

For a function f that depends on n independently measured variables x1,x2,…,xnx_1, x_2, \dots, x_n, each with uncertainty Δx1,Δx2,…,Δxn\Delta x_1, \Delta x_2, \dots, \Delta x_n, the propagated uncertainty is:

Δf=(∂f∂x1⋅Δx1)2+(∂f∂x2⋅Δx2)2+⋯+(∂f∂xn⋅Δxn)2\Delta f = \sqrt{\left(\frac{\partial f}{\partial x_1} \cdot \Delta x_1\right)^2 + \left(\frac{\partial f}{\partial x_2} \cdot \Delta x_2\right)^2 + \cdots + \left(\frac{\partial f}{\partial x_n} \cdot \Delta x_n\right)^2}
The quadrature formula for uncertainty propagation (assuming independent variables)

This formula is derived from a first-order Taylor expansion of f around the measured values, keeping only the linear terms. It assumes:

  • The uncertainties are small compared to the values themselves
  • The variables are independent (not correlated)
  • The uncertainties are random, not systematic

The key operation is the partial derivative ∂f∂xi\frac{\partial f}{\partial x_i} - this tells you how sensitive the result is to a change in variable xix_i while all other variables are held constant. A large partial derivative means that variable contributes heavily to the total uncertainty.

Special Cases and Rules

The general formula simplifies beautifully for the most common mathematical operations. These rules are worth memorising for lab reports.

Addition and Subtraction

For f=x+yf = x + y or f=x−yf = x - y, the partial derivatives are both ±1\pm 1, so:

Δf=(Δx)2+(Δy)2\Delta f = \sqrt{(\Delta x)^2 + (\Delta y)^2}
Add absolute uncertainties in quadrature

Notice that subtraction does not reduce uncertainty - you still add in quadrature. If you subtract two nearly-equal numbers with similar uncertainties, the relative uncertainty of the result can become very large. This is called catastrophic cancellation.

Multiplication and Division

For f=x⋅yf = x \cdot y or f=x/yf = x / y:

Δf∣f∣=(Δxx)2+(Δyy)2\frac{\Delta f}{|f|} = \sqrt{\left(\frac{\Delta x}{x}\right)^2 + \left(\frac{\Delta y}{y}\right)^2}
Add relative (fractional) uncertainties in quadrature

The key insight: for products and quotients, use relative uncertainties (Δx/x\Delta x/x), not absolute ones. The variable with the largest relative uncertainty dominates the total.

Powers

For f=xnf = x^n (where n is a constant):

Δf∣f∣=∣n∣⋅Δxx\frac{\Delta f}{|f|} = |n| \cdot \frac{\Delta x}{x}
The relative uncertainty is multiplied by the power

This is why small uncertainties in a radius become large uncertainties in a volume (r³) - the power multiplier amplifies them. A 1% uncertainty in r becomes a 3% uncertainty in r³.

Multiplication by a Constant

For f=c⋅xf = c \cdot x (where c is an exact constant):

Δf=∣c∣⋅Δx\Delta f = |c| \cdot \Delta x
Constants simply scale the uncertainty linearly

Quick Reference Table

OperationFormula fUncertainty rule
Additionf = x + yΔf = √(Δx² + Δy²)
Subtractionf = x − yΔf = √(Δx² + Δy²)
Multiplicationf = x · yΔf/f = √((Δx/x)² + (Δy/y)²)
Divisionf = x / yΔf/f = √((Δx/x)² + (Δy/y)²)
Powerf = xⁿΔf/f = |n| · Δx/x
Constant × varf = c · xΔf = |c| · Δx
Log (natural)f = ln(x)Δf = Δx / |x|
Exponentialf = eˣΔf = eˣ · Δx

Worked Example: Kinetic Energy

Let's walk through a complete lab report calculation. We want to find the kinetic energy of a moving cart:

KE=12mv2KE = \frac{1}{2} m v^2

We measure:

m2.50 ± 0.05 kgMass - weighed on balance
v4.00 ± 0.10 m/sVelocity - from timing gate

Step 1 - Compute the result

KE=12×2.50×(4.00)2=20.0 JKE = \frac{1}{2} \times 2.50 \times (4.00)^2 = 20.0 \text{ J}

Step 2 - Compute partial derivatives

∂KE∂m=12v2=12×16=8.0 m2/s2\frac{\partial KE}{\partial m} = \frac{1}{2} v^2 = \frac{1}{2} \times 16 = 8.0 \text{ m}^2/\text{s}^2
∂KE∂v=mv=2.50×4.00=10.0 kg⋅m/s\frac{\partial KE}{\partial v} = m v = 2.50 \times 4.00 = 10.0 \text{ kg}\cdot\text{m}/\text{s}

Step 3 - Compute each term

(∂KE∂m⋅Δm)2=(8.0×0.05)2=(0.40)2=0.160 J2\left(\frac{\partial KE}{\partial m} \cdot \Delta m\right)^2 = (8.0 \times 0.05)^2 = (0.40)^2 = 0.160 \text{ J}^2
(∂KE∂v⋅Δv)2=(10.0×0.10)2=(1.00)2=1.000 J2\left(\frac{\partial KE}{\partial v} \cdot \Delta v\right)^2 = (10.0 \times 0.10)^2 = (1.00)^2 = 1.000 \text{ J}^2

Step 4 - Combine in quadrature

ΔKE=0.160+1.000=1.160≈1.08 J\Delta KE = \sqrt{0.160 + 1.000} = \sqrt{1.160} \approx 1.08 \text{ J}

Step 5 - Write the final result

KE = 20.0 ± 1.1 J

Relative uncertainty: 5.4%

Correlated Variables

The standard quadrature formula assumes all variables are independent. When variables are correlated - when an error in one tends to accompany an error in another - a correction term must be added.

The full formula including correlations is:

Δf2=∑i(∂f∂xi)2Δxi2+2∑i<j∂f∂xi∂f∂xjcov⁡(xi,xj)\Delta f^2 = \sum_i \left(\frac{\partial f}{\partial x_i}\right)^2 \Delta x_i^2 + 2\sum_{i<j} \frac{\partial f}{\partial x_i}\frac{\partial f}{\partial x_j} \operatorname{cov}(x_i, x_j)
Full covariance formula - reduces to the simple form when cov(xᵢ,xⱼ) = 0

In most undergraduate lab experiments, variables are measured independently and correlations can be ignored. However, if you compute intermediate quantities and then use them together - for example, computing a ratio of two quantities that both depend on the same raw measurement - you must account for the correlation.

Common Mistakes to Avoid

✕

Adding uncertainties linearly instead of in quadrature

Writing Δf = Δx + Δy instead of Δf = √(Δx² + Δy²) overestimates the uncertainty by assuming errors always act in the worst-case direction simultaneously. Use quadrature for independent random errors.

✕

Using absolute uncertainties where relative are needed

For products and quotients, you must combine relative uncertainties (Δx/x), not absolute ones. Then convert back to absolute at the end by multiplying by the result.

✕

Ignoring the dominant error source

After propagating, always check which variable contributes the most variance. There is no point spending effort to reduce a 0.1% uncertainty if another variable contributes 5%. Focus experimental effort on the dominant term.

✕

Forgetting to square root at the end

The formula gives Δf², not Δf. Students frequently report the sum of squares directly. Always take the square root of the combined sum.

✕

Applying propagation to systematic errors

Uncertainty propagation assumes random, independent errors. Systematic errors (calibration offsets, zero errors) must be corrected separately - propagating them through the formula does not remove them.

✕

Over-reporting significant figures in the uncertainty

Uncertainties should generally be reported to 1 or 2 significant figures. The result should then be rounded to match the last significant figure of the uncertainty. Writing KE = 20.0000 ± 1.0823 J is incorrect.

Reporting Results Correctly

A result is only meaningful if it is reported with the appropriate number of significant figures. The rule of thumb for experimental physics is:

  1. Round the uncertainty to 1 or 2 significant figures
  2. Round the result to match the last decimal place of the uncertainty
  3. Always include units on both the result and the uncertainty
✕ WrongKE = 20.0000 ± 1.0823 J
✕ WrongKE = 20 ± 1.0823 J
✓ CorrectKE = 20.0 ± 1.1 J
✓ Also correctKE = (20.0 ± 1.1) J

Frequently Asked Questions

Q1What is the difference between uncertainty and error?

In modern metrology, 'error' refers to the difference between a measured value and the true value - which is generally unknowable. 'Uncertainty' is the quantified doubt about the result, expressed as a range. The terms are often used interchangeably in undergraduate physics, but technically uncertainty is the preferred term in scientific reporting.

Q2Can I just add uncertainties directly instead of in quadrature?

You can, and it gives a conservative (upper bound) estimate. Linear addition assumes all errors act in the same direction simultaneously - which is a worst-case scenario. Quadrature addition is statistically correct for independent random errors. Most physics labs require quadrature. Some engineering applications use linear addition for safety margins.

Q3Does uncertainty propagation work for trigonometric functions?

Yes. For f = sin(θ), the partial derivative is cos(θ), so Δf = |cos(θ)| · Δθ. For f = cos(θ), Δf = |sin(θ)| · Δθ. Note that angles must be in radians for these derivatives to be correct. Our calculator handles trig functions automatically.

Q4What if my variables are correlated?

You need to include the covariance terms in the propagation formula. This is common when, for example, you use the same ruler to measure both length and width of an object - any calibration error in the ruler affects both. In practice, redesigning the experiment to use independent measurements is often preferable to dealing with correlations.

Q5How many significant figures should I keep during intermediate steps?

Keep at least 2–3 extra significant figures during intermediate calculations to avoid rounding errors. Only round to the final number of significant figures when writing the final result. This is called 'carrying guard digits'.

Q6What is the difference between standard deviation and standard error?

The standard deviation σ describes the spread of individual measurements. The standard error of the mean σ/√n describes the uncertainty in the average of n measurements. For reporting a mean value from multiple trials, use the standard error - it decreases as you take more measurements.

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