Tags: propagation of error, error analysis, uncertainty, error propagation rules, partial derivatives, quadrature, independent uncertainties, random error, physics lab, PHYS 206
Every measured quantity carries an uncertainty, and when you use measurements in a calculation, those uncertainties propagate through to the final result. Propagation of error gives you the rules for computing how big the uncertainty in your answer is, based on the uncertainties in your inputs. The two core cases are addition/subtraction (absolute uncertainties add in quadrature) and multiplication/division (fractional uncertainties add in quadrature).
Uncertainty (δx)
The estimated range within which the true value of a measurement is expected to lie. Written as x ± δx.
Absolute uncertainty
The uncertainty expressed in the same units as the measurement itself (e.g. 0.005 inches).
Fractional (relative) uncertainty
The ratio of the absolute uncertainty to the measured value: δx / x. Often expressed as a percentage. Sometimes called "relative error."
Propagation of error (propagation of uncertainty)
The set of rules for calculating the uncertainty in a result that depends on one or more measured quantities, each with their own uncertainty.
Addition in quadrature
Combining uncertainties by taking the square root of the sum of their squares: √(δa² + δb² + ...). This is the standard method when uncertainties are independent and random.
Independent uncertainties
Uncertainties that arise from separate, unrelated sources, so knowing one tells you nothing about the other. This is the assumption that lets you add in quadrature rather than simply adding.
When a quantity q depends on measured variables x, y, z, ..., the uncertainty in q (written δq) depends on how sensitive q is to changes in each variable, combined with each variable's uncertainty.
The general formula uses partial derivatives:
δq = √[ (∂q/∂x · δx)² + (∂q/∂y · δy)² + (∂q/∂z · δz)² + ... ]
This assumes all uncertainties are independent and random. If they are not independent, you must account for correlations (not covered at this level).
If q = c · x, where c is an exact constant with no uncertainty:
δq = |c| · δx
The constant simply scales the uncertainty. This is the simplest case and appears whenever you divide a measurement by an exact number (e.g. dividing a deck thickness by 52 cards).
If q = x + y or q = x − y:
δq = √(δx² + δy²)
Absolute uncertainties add in quadrature regardless of whether the operation is addition or subtraction. Note that subtraction does not cancel uncertainty; the uncertainty in x − y is the same size as in x + y.
If q = x^a · y^b · z^c (a product of powers), the fractional uncertainty is:
δq / |q| = √[ (a · δx/x)² + (b · δy/y)² + (c · δz/z)² ]
Each exponent multiplies the corresponding fractional uncertainty before you add in quadrature. This covers:
Simple multiplication (a = b = 1)
Division (one exponent is negative)
Squares, square roots, and other powers
For any function q = f(x, y, z, ...) that does not fit neatly into the above categories, fall back to the general formula:
δq = √[ (∂f/∂x · δx)² + (∂f/∂y · δy)² + ... ]
This is the most powerful and general approach. You compute the partial derivative of f with respect to each variable, evaluate it at the measured values, multiply by that variable's uncertainty, and add everything in quadrature.
Measuring N identical items together and dividing by N reduces the uncertainty in each individual item by a factor of N:
Measure a stack of N items with uncertainty δ (from the instrument)
Thickness of one item = total / N
Uncertainty of one item = δ / N
To achieve a target uncertainty δ_target for one item, you need N = δ / δ_target items in the stack. This works because the instrument uncertainty δ is fixed regardless of how many items you stack.
To decide whether a measured value agrees with a predicted value (or another measurement), check whether their ranges overlap. Two values x ± δx and y ± δy are consistent if:
|x − y| ≤ δx + δy
If the difference between the central values is larger than the sum of their uncertainties, the values are inconsistent (they disagree). A more sophisticated version uses the combined uncertainty √(δx² + δy²) and checks whether the discrepancy is within about 2 standard deviations, but the simple overlap test is standard at this level.
General propagation formula:
δq = √[ (∂q/∂x · δx)² + (∂q/∂y · δy)² + ... ]
Addition/subtraction: q = x ± y → δq = √(δx² + δy²)
Product of powers: q = x^a · y^b · z^c → δq/|q| = √[(a·δx/x)² + (b·δy/y)² + (c·δz/z)²]
Constant multiple: q = c·x → δq = |c|·δx
Reducing uncertainty by stacking: δ_single = δ_instrument / N
⚠️ Subtraction does not reduce uncertainty. Students often assume x − y has less uncertainty than x + y. It does not; the formula is the same.
⚠️ Know when to use absolute uncertainties (addition/subtraction) versus fractional uncertainties (multiplication/division/powers). Mixing them up is the most common error propagation mistake.
⚠️ The quadrature rule (square root of sum of squares) only applies when uncertainties are independent. If they share a common source, you must add linearly or account for correlation.
⚠️ When checking consistency, compare the discrepancy to the sum (or quadrature sum) of both uncertainties, not just one of them.
⚠️ Exponents multiply fractional uncertainty. A quantity that appears squared (like R² in the moment of inertia) contributes twice its fractional uncertainty. This is easy to overlook.
Q: You measure a length as 5.00 ± 0.02 m and a time as 2.00 ± 0.01 s. You calculate speed = length/time. What is the fractional uncertainty in the speed?
A: Fractional uncertainty = √[(0.02/5.00)² + (0.01/2.00)²] = √[(0.004)² + (0.005)²] = √[0.000016 + 0.000025] = √0.000041 ≈ 0.0064, or about 0.64%.
Q: Why do you add uncertainties in quadrature rather than just adding them directly?
A: Because independent random uncertainties are unlikely to all push in the same direction at once. Adding in quadrature reflects the statistical reality that random errors partially cancel. Straight addition gives the worst-case (maximum possible) uncertainty, which is overly conservative.
Q: If q = x², what is the fractional uncertainty in q?
A: δq/q = 2 · (δx/x). The exponent 2 doubles the fractional uncertainty.
Q: You measure 10 identical sheets stacked together with an instrument that has ±0.01 mm uncertainty. What is the uncertainty in the thickness of a single sheet?
A: δ_single = 0.01 mm / 10 = 0.001 mm.
Q: Two results are 4.5 ± 0.3 and 5.0 ± 0.2. Are they consistent?
A: The difference is |5.0 − 4.5| = 0.5. The sum of uncertainties is 0.3 + 0.2 = 0.5. Since the discrepancy equals the sum of uncertainties, they are just barely consistent (on the boundary). Most would say marginally consistent.
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