Source: ENGR 216 Practice Challenges, Texas A&M
Tags: propagation of error, error propagation, uncertainty, partial derivatives, RSS, root sum of squares, ENGR 216, experimental physics
When a result depends on multiple measured quantities, each with its own uncertainty, those uncertainties combine into the uncertainty of the result. Propagation of error tells you how to calculate the total uncertainty using partial derivatives and the root-sum-of-squares (RSS) method. This is a core mid-term topic.
Propagated error (propagated uncertainty)
The uncertainty in a calculated result that arises from the individual uncertainties in each measured input variable.
Partial derivative
The derivative of a multivariable function with respect to one variable while holding all others constant. Used to determine how sensitive the result is to each input.
Root sum of squares (RSS)
The method of combining independent, random errors: square each component error, sum them, and take the square root.
Independent and random errors
Errors in the measured quantities that are uncorrelated, meaning one measurement's error does not influence another's. This is the assumption that allows the RSS method.
Absolute error
The uncertainty expressed in the same units as the measurement (e.g., ±0.1 kg).
Relative error
The uncertainty expressed as a fraction or percentage of the measured value (e.g., ±5%).
For a result R that depends on measured variables x₁, x₂, ..., xₙ:
R = f(x₁, x₂, ..., xₙ)
The propagated error is:
δR = √[ (∂R/∂x₁ · δx₁)² + (∂R/∂x₂ · δx₂)² + ... + (∂R/∂xₙ · δxₙ)² ]
Each term in the sum is the partial derivative of R with respect to one variable, multiplied by that variable's uncertainty, and then squared.
Write the equation relating the result to the measured variables.
Take the partial derivative of the result with respect to each variable.
Multiply each partial derivative by the corresponding uncertainty.
Square each of those products.
Sum all squared products.
Take the square root of the sum.
For R = a · b:
∂R/∂a = b
∂R/∂b = a
δR = √[ (b · δa)² + (a · δb)² ]
For R = x · y · z:
∂R/∂x = y · z
∂R/∂y = x · z
∂R/∂z = x · y
δR = √[ (yz · δx)² + (xz · δy)² + (xy · δz)² ]
General propagation of error (RSS method):
δR = √[ Σ (∂R/∂xᵢ · δxᵢ)² ]
For R = a · b:
δR = √[ (b · δa)² + (a · δb)² ]
For R = x · y · z:
δR = √[ (yz · δx)² + (xz · δy)² + (xy · δz)² ]
E = mgh, where m = 2.0 ± 0.1 kg, g = 9.8 ± 0.2 m/s², h = 5.0 ± 0.3 m.
Step 1: Partial derivatives
∂E/∂m = gh = 9.8 × 5.0 = 49.0
∂E/∂g = mh = 2.0 × 5.0 = 10.0
∂E/∂h = mg = 2.0 × 9.8 = 19.6
Step 2: Multiply by uncertainties
∂E/∂m · δm = 49.0 × 0.1 = 4.9
∂E/∂g · δg = 10.0 × 0.2 = 2.0
∂E/∂h · δh = 19.6 × 0.3 = 5.88
Step 3: Square each
4.9² = 24.01
2.0² = 4.00
5.88² = 34.5744
Step 4: Sum and take the square root
δE = √(24.01 + 4.00 + 34.5744) = √62.5844 = 7.91 J
Answer: 7.91
T = a · b, where a = 25 ± 0.5, b = 4.0 ± 0.2.
Step 1: Partial derivatives
∂T/∂a = b = 4.0
∂T/∂b = a = 25
Step 2: Multiply by uncertainties
4.0 × 0.5 = 2.0
25 × 0.2 = 5.0
Step 3: Square, sum, square root
δT = √(2.0² + 5.0²) = √(4 + 25) = √29 = 5.39
Answer: 5.39
⚠️ The partial derivative step is where most mistakes happen. Write the function clearly first, then differentiate one variable at a time while treating every other variable as a constant.
⚠️ Plug in the nominal (central) values of the other variables when evaluating the partial derivative, not the uncertainties.
⚠️ Do not simply add the uncertainties together. The RSS method accounts for the statistical independence of random errors. Straight addition would overestimate the total error.
⚠️ Watch for problems that give you a product of three variables (like E = mgh). Each variable contributes its own partial derivative term.
⚠️ On the exam: round only the final answer to two decimal places.
Q: For R = a · b, what are the partial derivatives used in error propagation?
A: ∂R/∂a = b and ∂R/∂b = a. Each partial derivative is the other variable.
Q: Why do we square the error terms before summing?
A: Squaring ensures that positive and negative errors do not cancel each other out. Taking the square root at the end returns the result to the original units.
Q: If one variable's uncertainty is much larger than the others, what happens?
A: That variable dominates the total propagated error. The other terms contribute relatively little. This tells you which measurement to improve for better overall precision.
Q: What assumption must hold for the RSS method to be valid?
A: The errors in the individual measurements must be independent and random (uncorrelated).
Q: For E = mgh with m = 3.0 ± 0.2, g = 9.8 ± 0.1, h = 10.0 ± 0.5, what is δE?
A: ∂E/∂m = gh = 98.0, ∂E/∂g = mh = 30.0, ∂E/∂h = mg = 29.4. Components: (98.0 × 0.2)² + (30.0 × 0.1)² + (29.4 × 0.5)² = 384.16 + 9.0 + 216.09 = 609.25. δE = √609.25 = 24.68.
propagation of error, error propagation, uncertainty propagation, partial derivatives, RSS, root sum of squares, quadrature, absolute uncertainty, relative uncertainty, independent errors, random errors, ENGR 216, experimental physics lab, Texas A&M