Tags: propagation of uncertainty, error propagation, uncertainty analysis, partial derivatives, ENGR 216, ENGR-PHYS 216, experimental physics, Texas A&M, quadrature, relative uncertainty, absolute uncertainty, independent uncertainties, correlated uncertainties
When you calculate a result from measured quantities, the uncertainties in those measurements carry through ("propagate") into the result. The general method uses partial derivatives and adds contributions in quadrature (square root of the sum of squares). Special shortcut rules exist for addition/subtraction and multiplication/division, and correlated variables require extra care.
Propagation of uncertainty (error propagation)
The process of determining the uncertainty in a calculated result from the uncertainties in the individual measurements used to compute it.
Absolute uncertainty (δx)
The uncertainty expressed in the same units as the measurement itself (e.g. 0.010 m).
Relative (fractional) uncertainty (δx/x)
The uncertainty expressed as a fraction of the measured value. Often multiplied by 100 to give percentage uncertainty.
Quadrature (addition in quadrature)
Combining uncertainties by taking the square root of the sum of their squares. This is the standard approach for independent, random uncertainties.
Independent uncertainties
Uncertainties from separate measurements that have no shared systematic source. Their errors are uncorrelated, so propagation uses the root-sum-of-squares (quadrature) formula.
Correlated uncertainties
Uncertainties that share a common source (e.g. two masses weighed on the same biased scale). Their propagation requires additional covariance terms or, in some cases, combining partial derivatives before squaring.
For a function q = f(x, y, z, ...) where x, y, z are measured independently, the uncertainty in q is:
$$\delta q = \sqrt{ \left(\frac{\partial f}{\partial x} \cdot \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \cdot \delta y \right)^2 + \left(\frac{\partial f}{\partial z} \cdot \delta z \right)^2 + \cdots }$$
Each term is the partial derivative of f with respect to one variable, multiplied by that variable's uncertainty. You then add all terms in quadrature.
This works for any differentiable function, regardless of whether it involves addition, multiplication, powers, or anything else.
For q = x + y or q = x - y:
$$\delta q = \sqrt{(\delta x)^2 + (\delta y)^2}$$
You add the absolute uncertainties in quadrature. Note that subtraction and addition give the same uncertainty formula, because (∂q/∂x) = ±1 in both cases and the sign vanishes when squared.
This means subtracting two nearly equal numbers can produce a result with very large relative uncertainty, even if the absolute uncertainty is modest. This is sometimes called "catastrophic cancellation" in uncertainty terms.
For q = x · y or q = x / y:
$$\frac{\delta q}{|q|} = \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2}$$
You add the relative uncertainties in quadrature. This extends naturally to products of more than two variables.
For q = x^n (where n is an exact constant):
$$\frac{\delta q}{|q|} = |n| \cdot \frac{\delta x}{x}$$
The relative uncertainty is scaled by the absolute value of the exponent. So x² doubles the relative uncertainty, √x halves it, and 1/x leaves it the same as x.
For a function like q = (constant) · x^a · y^b · z^c, the relative uncertainty is:
$$\frac{\delta q}{|q|} = \sqrt{\left(a \cdot \frac{\delta x}{x}\right)^2 + \left(b \cdot \frac{\delta y}{y}\right)^2 + \left(c \cdot \frac{\delta z}{z}\right)^2}$$
This is the workhorse formula for most lab calculations. It covers any expression that is a product of measured quantities raised to powers.
When a measured value is divided by a constant with no uncertainty (e.g. the number of pages in a book, or the number 2 in a formula), the uncertainty simply scales by that same constant:
$$q = \frac{x}{N} \implies \delta q = \frac{\delta x}{N}$$
Constants carry no uncertainty of their own, so they do not add a quadrature term.
When two variables share a common uncertainty source, you cannot simply add their contributions in quadrature. The full formula includes a covariance term:
$$\delta q = \sqrt{ \left(\frac{\partial f}{\partial x}\right)^2 (\delta x)^2 + \left(\frac{\partial f}{\partial y}\right)^2 (\delta y)^2 + 2 \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} \text{cov}(x,y) }$$
A practical red flag for hidden correlation: when the same measured variable appears in more than one place in your formula (e.g. m₁ appearing in both numerator and denominator), you must differentiate with respect to that variable as a single entity, not treat its appearances separately.
Even when two different variables (say m₁ and m₂) are measured with the same instrument, they may share a systematic bias. In that case, the measurements are correlated, and the covariance term must be considered. This is the "hidden correlation" that examiners like to test.
To decide whether a measured value agrees with a theoretical prediction (or another measurement), combine the uncertainties of both values and check the discrepancy:
$$\text{discrepancy} = |q_{\text{measured}} - q_{\text{predicted}}|$$
$$\text{combined uncertainty} = \sqrt{(\delta q_{\text{meas}})^2 + (\delta q_{\text{pred}})^2}$$
If the discrepancy is less than roughly 2 times the combined uncertainty, the values are generally considered consistent. If it is larger, they likely disagree.
General formula (independent variables):
δq = √[ (∂f/∂x · δx)² + (∂f/∂y · δy)² + ... ]
Addition/subtraction rule:
δq = √[ (δx)² + (δy)² ]
Multiplication/division rule (relative uncertainties):
δq/|q| = √[ (δx/x)² + (δy/y)² ]
Power rule:
δq/|q| = |n| · (δx/x)
Consistency test:
|difference| vs. 2 × √[ (δq₁)² + (δq₂)² ]
⚠️ The partial derivative method is the most general and always works. Learn to take partial derivatives confidently, then fall back on shortcuts when the function is a simple product or sum.
⚠️ Subtraction of similar values blows up relative uncertainty. Problems often test this by giving you two close measurements and asking for their difference.
⚠️ Watch for "hidden correlation" when a single measured variable appears multiple times in one formula, or when two variables share the same instrument.
⚠️ Constants (exact numbers like 2, π, or a page count) contribute zero uncertainty. Only measured quantities carry uncertainty.
⚠️ Consistency checks require combining uncertainties from both values, not just one. A common mistake is comparing the discrepancy only to the uncertainty of one measurement.
⚠️ Always report your final answer rounded to match the precision of the uncertainty (typically 1-2 significant figures in the uncertainty).
Q: When propagating uncertainty through q = A - B, does the sign of the subtraction affect the uncertainty formula?
A: No. The partial derivatives are +1 and -1, but both are squared, so the uncertainty formula is identical to addition: δq = √(δA² + δB²).
Q: You measure x = 5.00 ± 0.10 and compute q = x³. What is the relative uncertainty in q?
A: The relative uncertainty in q is 3 × (0.10/5.00) = 3 × 0.02 = 0.06, or 6%. The power multiplies the relative uncertainty.
Q: Two students measure the same length. Student A gets 10.2 ± 0.3 cm, Student B gets 10.9 ± 0.4 cm. Are these consistent?
A: Discrepancy = |10.9 - 10.2| = 0.7 cm. Combined uncertainty = √(0.3² + 0.4²) = 0.5 cm. The ratio 0.7/0.5 = 1.4, which is less than 2, so yes, they are consistent.
Q: Why can you not treat the numerator and denominator of (m₁ - m₂)/(m₁ + m₂) as independent when propagating uncertainty?
A: Because m₁ and m₂ each appear in both the numerator and denominator. They are the same physical measurements, so their contributions are correlated. You must take partial derivatives ∂a/∂m₁ and ∂a/∂m₂ treating the entire expression as one function of m₁ and m₂.
error propagation, uncertainty analysis, propagation of errors, quadrature addition, partial derivative uncertainty, fractional uncertainty, percentage uncertainty, relative uncertainty, absolute uncertainty, combined uncertainty, independent errors, correlated errors, systematic error, random error, uncertainty budget, consistency check, agreement of measurements, ENGR 216, ENGR-PHYS 216, experimental physics lab, Texas A&M