Uncertainty Analysis and Propagation of Errors
Uncertainty analysis is the process of estimating the range within which the true value of a measured quantity is likely to lie, written as value ± uncertainty, and propagation of errors is the method of calculating the uncertainty of a result from the uncertainties of the quantities used to calculate it. In brief: for sums and differences, add the absolute uncertainties; for products and quotients, add the percentage (fractional) uncertainties; for a power xn, multiply the percentage uncertainty by |n|. These simple rules give the maximum likely uncertainty. Statistics (adding in quadrature) gives a smaller, more realistic estimate for independent random errors.
A measurement without an uncertainty is incomplete: it cannot be compared with a prediction or another result. This guide explains absolute, fractional and percentage uncertainty, the rules for combining them, the quadrature method, compares the rules in a table, works through numerical examples (including finding g from a pendulum) and corrects common mistakes.
What are absolute, fractional and percentage uncertainty?
The absolute uncertainty Δx has the same units as the quantity: a time of 20.14 s ± 0.25 s. The fractional uncertainty is Δx ÷ x (0.25 ÷ 20.14 = 0.0124), and the percentage uncertainty is Δx ÷ x × 100 percent (1.2 percent). The uncertainty of a single reading is estimated from the resolution of the instrument (± one smallest division, or half a division with care); for repeated readings, from half the range (max − min)/2, or from the standard deviation of the mean. The uncertainty is quoted to one significant figure (occasionally two), and the value is rounded to the same number of decimal places: 20.1 ± 0.3 s, not 20.14 ± 0.25 s with mismatched precision. A measurement of a larger quantity with the same instrument has a smaller percentage uncertainty.
What are the rules for propagating uncertainties?
- Addition and subtraction: for y = a ± b, Δy = Δa + Δb (add the absolute uncertainties, even for subtraction).
- Multiplication and division: for y = a × b or a ÷ b, Δy/y = Δa/a + Δb/b (add the fractional or percentage uncertainties).
- Powers: for y = an, Δy/y = |n| × Δa/a. So a square doubles the percentage uncertainty and a square root halves it.
- Multiplication by an exact constant: for y = ka, Δy = kΔa, with the percentage uncertainty unchanged. Exact numbers (such as 4π2, or the count of oscillations) have no uncertainty.
- Several steps: work through the formula in stages, or combine the percentage uncertainties of all the factors with their powers.
- Other functions (such as sin or ln): the uncertainty can be found from the largest and smallest values: Δy = (ymax − ymin)/2.
When the same quantity appears more than once in a formula, treat the repeats as a power (a × a = a2, so the percentage uncertainty is 2Δa/a). Quadrature must not be applied to repeats of one measurement, because they are not independent.
What is adding in quadrature?
The simple rules assume that all the errors act in the same direction at the same time, which gives the greatest possible uncertainty. If the uncertainties are independent random errors, they partly cancel, and a better estimate for the combined uncertainty is the root of the sum of squares: for a sum, Δy = √(Δa2 + Δb2); for products and quotients, Δy/y = √((Δa/a)2 + (Δb/b)2). For a general function y(a, b, …) the formula is Δy = √((∂y/∂a·Δa)2 + (∂y/∂b·Δb)2 + …). The quadrature result is always less than or equal to the simple sum. Note that when one uncertainty is more than three times the others, it dominates, and the others can be neglected, so it pays to improve the worst measurement first.
Rules compared
| Operation | Simple (maximum) rule | Quadrature (independent) | Example |
|---|---|---|---|
| y = a ± b | Δy = Δa + Δb | Δy = √(Δa² + Δb²) | (12.0 ± 0.1) − (8.0 ± 0.2) = 4.0 ± 0.3 |
| y = a × b or a ÷ b | Add % uncertainties | Root of the sum of squares of the % | V ÷ I, 1.7% + 5% = 6.7% |
| y = an | % uncertainty × |n| | % uncertainty × |n| | T² has twice the % of T |
| y = ka | Δy = kΔa | Δy = kΔa | Constants are exact |
Worked example: uncertainty in g from a pendulum
- Percentage uncertainty of a repeated measurement. The time for 10 oscillations is 20.14 s with a half-range of 0.25 s. The percentage uncertainty is 0.25 ÷ 20.14 × 100 = 1.2 percent.
- Difference. The temperature rise ΔT = 4.0 ± 0.3 K comes from final (12.0 ± 0.1) and initial (8.0 ± 0.2) readings: the absolute uncertainties add (0.1 + 0.2 = 0.3 K), a percentage uncertainty of 0.3 ÷ 4.0 = 7.5 percent. Subtracting two close numbers gives a large percentage uncertainty.
- Quotient. A resistor has V = 6.0 ± 0.1 V (1.7 percent) and I = 2.0 ± 0.1 A (5 percent). R = 6.0 ÷ 2.0 = 3.0 Ω. The percentage uncertainty is 1.7 + 5 = 6.7 percent, so ΔR = 3.0 × 0.067 = 0.2 Ω: R = 3.0 ± 0.2 Ω. In quadrature, √(1.7² + 5²) = 5.3 percent, ΔR = 0.16 Ω. The current’s uncertainty dominates.
- Pendulum. From T = 2π√(L/g), g = 4π2L/T2. Take L = 1.000 ± 0.002 m (0.2 percent), and the time for 10 oscillations 20.1 ± 0.2 s, so T = 2.01 ± 0.02 s (1 percent).
- Value. g = 4π2 × 1.000 ÷ 2.012 = 9.77 m/s2.
- Uncertainty. Percentage uncertainty = 0.2 + 2 × 1 = 2.2 percent (T is squared, so its percentage is doubled). Δg = 9.77 × 0.022 = 0.21 m/s2. Result: g = 9.8 ± 0.2 m/s2. The accepted value 9.81 m/s2 lies inside the range 9.6 to 10.0 m/s2, so the experiment is consistent with it.
- Improvement. The time T gives almost all of the uncertainty (2 of 2.2 percent), so the length measurement is not worth improving; timing more oscillations (20 instead of 10) would halve the uncertainty from T.
How did uncertainty analysis develop?
Gauss and Laplace developed the mathematics of errors in the early 1800s, and the treatment of combined errors followed from their work on least squares and the normal distribution. In 1993 the International Organization for Standardization (ISO) with other bodies published the Guide to the Expression of Uncertainty in Measurement (GUM), which established the standard approach, with standard uncertainties combined in quadrature. Physics results today, such as the values of fundamental constants, are published with an uncertainty, often given in brackets for the last digits.
Which mistakes are common?
Mistake 1: Subtracting uncertainties when subtracting quantities
The absolute uncertainties are added for both sums and differences.
Mistake 2: Forgetting to double the percentage uncertainty for a squared quantity
The power multiplies the percentage uncertainty: T2 has twice the percentage uncertainty of T.
Mistake 3: Mixing absolute and percentage uncertainties in the same rule
Add absolute uncertainties for sums and percentage uncertainties for products; convert one to the other when needed.
Mistake 4: Quoting too many significant figures in the uncertainty
Round the uncertainty to one significant figure and the value to the same decimal place.