Source: Assignment 1, Experimental Physics and Engineering Lab, Texas A&M University
Tags: error propagation, uncertainty, ENGR 216, propagation of error, measurement uncertainty, partial derivatives, quadrature, error analysis, Taylor series uncertainty
Error propagation tells you how uncertainties in your measured quantities carry through into a calculated result. The core rule: for independent, random uncertainties, you combine them in quadrature (square root of the sum of squares of each partial-derivative contribution). Dividing or multiplying a measurement by a constant scales the uncertainty by that same constant.
Uncertainty (δx)
The estimated range within which the true value of a measurement is expected to lie. Written as ± after the best estimate, e.g. 0.590 ± 0.005 in.
Propagation of error (error propagation)
The process of calculating the uncertainty in a result that depends on one or more measured quantities, each with its own uncertainty.
Quadrature (addition in quadrature)
Combining uncertainties by taking the square root of the sum of their squares. For a function q(x, y, ...):
δq = √[ (∂q/∂x · δx)² + (∂q/∂y · δy)² + ... ]
This applies when the individual uncertainties are independent and random.
Independent uncertainties
Uncertainties that are not correlated with one another. Errors in one measurement do not influence errors in another. This is the standard assumption unless stated otherwise.
Fractional (relative) uncertainty
The uncertainty divided by the best estimate: δx / x. Useful for products and quotients, where fractional uncertainties add in quadrature.
If you have a measured quantity x with uncertainty δx, and your result is q = Bx (where B is an exact constant), then:
δq = |B| · δx
The constant just scales the uncertainty directly. No quadrature needed because there is only one uncertain variable.
For q = x + y or q = x − y:
δq = √[ (δx)² + (δy)² ]
Uncertainties add in quadrature regardless of whether the quantities are added or subtracted. This is one of the most commonly tested points.
For q = x · y or q = x / y, the fractional uncertainties add in quadrature:
δq/|q| = √[ (δx/x)² + (δy/y)² ]
This extends naturally to any number of factors.
For q = xⁿ:
δq/|q| = |n| · (δx/|x|)
The exponent multiplies the fractional uncertainty.
For any function q(x, y, z, ...) with independent uncertainties:
δq = √[ (∂q/∂x · δx)² + (∂q/∂y · δy)² + (∂q/∂z · δz)² + ... ]
This is the master formula. Every other rule above is a special case of this one.
General error propagation formula:
δq = √[ (∂q/∂x · δx)² + (∂q/∂y · δy)² + ... ]
Constant multiple: q = Bx → δq = |B| · δx
Sum/difference: q = x ± y → δq = √(δx² + δy²)
Product/quotient fractional uncertainty: δq/|q| = √[ (δx/x)² + (δy/y)² ]
Power rule: q = xⁿ → δq/|q| = |n| · δx/|x|
Setup: Calipers with uncertainty ±0.005 in. measure a deck of 52 cards at 0.590 in.
Part (a) – Thickness of one card:
Thickness of one card = 0.590 / 52 = 0.01135 in.
Since dividing by 52 (an exact constant), the uncertainty is:
δ(one card) = 0.005 / 52 ≈ 0.0001 in. (rounded to 1 sig fig of uncertainty)
Answer: 0.0113 ± 0.0001 in.
Part (b) – How many decks for δ = 0.00002 in.?
If you measure N decks together, you have 52N cards total, so:
δ(one card) = 0.005 / (52N)
Set this equal to 0.00002:
0.005 / (52N) = 0.00002
52N = 0.005 / 0.00002 = 250
N = 250 / 52 ≈ 4.8
Since you need a whole number of decks, round up.
Answer: 5 decks
Setup:
M = 1.10 ± 0.01 kg
R = 0.250 ± 0.005 m
ω = 21.5 ± 0.4 rad/s
L = ½MR²ω
Step 1 – Calculate L (best estimate):
L = 0.5 × 1.10 × (0.250)² × 21.5 L = 0.5 × 1.10 × 0.0625 × 21.5 L = 0.7391 kg·m²/s
Round to: L ≈ 0.739 kg·m²/s
Step 2 – Calculate δL using fractional uncertainties:
Since L = ½MR²ω is a product of M, R², and ω (with ½ as an exact constant), and R appears squared:
δL/L = √[ (δM/M)² + (2·δR/R)² + (δω/ω)² ]
Compute each fractional contribution:
δM/M = 0.01/1.10 = 0.00909
2·δR/R = 2 × 0.005/0.250 = 0.0400
δω/ω = 0.4/21.5 = 0.0186
δL/L = √(0.00909² + 0.0400² + 0.0186²) δL/L = √(0.0000826 + 0.001600 + 0.000346) δL/L = √(0.002029) δL/L = 0.04504
Step 3 – Convert to absolute uncertainty:
δL = 0.04504 × 0.7391 = 0.0333 kg·m²/s
Answer: L = 0.74 ± 0.03 kg·m²/s
Note how the R term dominates the uncertainty (0.0400 fractional vs. 0.0186 and 0.00909). The exponent of 2 on R doubles its fractional contribution, making it the largest source of error.
⚠️ When a variable is raised to a power (like R² in problem 2), the exponent multiplies the fractional uncertainty. This is the single most common slip on propagation problems.
⚠️ Division by a constant (like 52 cards) divides the uncertainty by that same constant. This is the principle behind averaging and repeated measurement.
⚠️ Uncertainties in a difference (x − y) add in quadrature, they do not subtract. Students frequently think subtracting quantities means subtracting uncertainties. The uncertainty always grows.
⚠️ Rounding: the uncertainty should typically be rounded to one significant figure, then the best estimate rounded to match the same decimal place.
Q: You measure a rod's length as 2.50 ± 0.02 m. You cut it into 5 equal pieces. What is the length and uncertainty of one piece?
A: Length = 2.50/5 = 0.500 m. Uncertainty = 0.02/5 = 0.004 m. Answer: 0.500 ± 0.004 m.
Q: If q = x²y and the fractional uncertainties in x and y are 3% and 2% respectively, what is the fractional uncertainty in q?
A: δq/q = √[(2 × 0.03)² + (0.02)²] = √[0.0036 + 0.0004] = √0.0040 = 0.063, so about 6.3%.
Q: Why do we add uncertainties in quadrature rather than simply adding them?
A: Because independent random errors are unlikely to all be at their maximum simultaneously. Quadrature gives a statistically realistic estimate of the combined uncertainty, while simple addition gives an overly pessimistic upper bound.
Q: A measurement of 4.0 ± 0.5 is subtracted from 10.0 ± 0.3. What is the result with uncertainty?
A: Result = 6.0. δ = √(0.5² + 0.3²) = √(0.25 + 0.09) = √0.34 ≈ 0.6. Answer: 6.0 ± 0.6.
error propagation, propagation of uncertainty, uncertainty analysis, quadrature, error in quadrature, partial derivative uncertainty, ENGR 216 HW1, fractional uncertainty, relative uncertainty, absolute uncertainty, Taylor series uncertainty, independent errors, random errors, measurement uncertainty, caliper uncertainty, angular momentum uncertainty, significant figures in uncertainty