Library / Foundations: Maths / FMA-50
Foundations: Maths
Relative uncertainty of a product or power
Combined relative uncertainty of a result built from inputs multiplied together and raised to powers, when the input errors are small and independent.
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- FMA-50
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- v1
- Field
- Foundations: Maths
- Result
- uy/|y| [%]
For a result y = C x1a1 x2a2 ⋯, the relative uncertainty of y is, at first order and for small independent errors, √Σ(ai ui/xi)². An input that is squared counts twice as much as one that is not, and an input in the denominator counts the same as one in the numerator because the exponent is squared. This is the standard-uncertainty (root-sum-square) convention, not the worst-case sum sometimes used in school practical work. Correlated inputs need covariance terms. The three terms here are set to zero by giving an exponent of zero.
Combined relative uncertainty of a result built from inputs multiplied together and raised to powers, when the input errors are small and independent. Each input counts with its relative uncertainty multiplied by its exponent, and the terms add in quadrature.
Calculator
- Contribution of the first input to the variance4 %²
- Contribution of the second input to the variance4 %²
- Contribution of the third input to the variance9 %²
- Worst-case sum for comparison, Σ|ai| ui/xi7 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Inputs with weighted uncertainties of 2 %, 2 % and 3 % combine to 4.123 %. Adding them directly would give 7 %, an overestimate that assumes every error goes the same way at once.
- The third input supplies 52.9 % of the variance, so improving the measurement of that input gains the most.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| u2/x2 Relative uncertainty of the second input | 0 to 20 % | 3.606 | 40.16 | 36.56 | +10% | +2.44 % |
| u3/x3 Relative uncertainty of the third input | 0 to 20 % | 2.828 | 20.2 | 17.37 | +10% | +5.41 % |
| u1/x1 Relative uncertainty of the first input | 0 to 20 % | 3.606 | 20.32 | 16.72 | +10% | +2.44 % |
| a3 Exponent of the third input | −4 to 4 | 2.828 | 12.33 | 9.5 | +0.16 | −8.12 % |
| a1 Exponent of the first input | −4 to 4 | 3.606 | 8.775 | 5.169 | +0.16 | +3.99 % |
| a2 Exponent of the second input | −4 to 4 | 3.606 | 5.385 | 1.78 | +0.16 | +1.94 % |
Sweep of Relative uncertainty of the first input
| Relative uncertainty of the first input (%) | Relative uncertainty of the result (%) |
|---|---|
| 0 | 3.606 |
| 2.5 | 4.387 |
| 5 | 6.164 |
| 7.5 | 8.322 |
| 10 | 10.63 |
| 12.5 | 13.01 |
| 15 | 15.43 |
| 17.5 | 17.87 |
| 20 | 20.32 |
Sweep one variable
Variables and how each one contributes
Relative uncertainty of the result
What the equation gives you. Everything else feeds it.
Exponent of the first input
The exponent of the first input in the product, a plain number. Use 1 for a factor, 2 for a squared quantity, −1 for a divisor and 0 to leave the term out. Raises the combined uncertainty, because a larger exponent magnifies the first input’s relative uncertainty. Raising it by 0.5 (to 1.5) moves the result by +13.8 %, from 4.123 to 4.69 %.
Relative uncertainty of the first input
The relative uncertainty u₁/x₁ of the first input, as a percentage. Raises the combined uncertainty in proportion to its share of the total, because the terms add in quadrature. Raising it by 1 % (to 3 %) moves the result by +13.8 %, from 4.123 to 4.69 %.
Exponent of the second input
The exponent of the second input, a plain number. Raises the combined uncertainty, because a larger exponent magnifies the second input’s relative uncertainty. Raising it by 0.5 (to 2.5) moves the result by +6.41 %, from 4.123 to 4.387 %.
Relative uncertainty of the second input
The relative uncertainty u₂/x₂ of the second input, as a percentage. Raises the combined uncertainty in proportion to its share of the total. Raising it by 1 % (to 2 %) moves the result by +30.6 %, from 4.123 to 5.385 %.
Exponent of the third input
The exponent of the third input, a plain number. The sign does not matter, because it is squared. Raises the combined uncertainty with the size of the exponent, in either direction, because only its square counts. Raising it by 0.5 (to −0.5) moves the result by −22.4 %, from 4.123 to 3.202 %.
Relative uncertainty of the third input
The relative uncertainty u₃/x₃ of the third input, as a percentage. Raises the combined uncertainty in proportion to its share of the total. Raising it by 1 % (to 4 %) moves the result by +18.8 %, from 4.123 to 4.899 %.
Worked example with the values above
- Weight each relative uncertainty by its exponent: 1 × 2 = 2 %, 2 × 1 = 2 %, −1 × 3 = −3 %.
- Square and add: 4 + 4 + 9 = 17.
- Take the square root: 4.123 %.
Assumptions and limits
- A result built as a constant times a product of powers of the inputs.
- Small, independent errors, so the first-order approximation is good. Correlated inputs need covariance terms.
- The nonzero inputs and result.
Common mistakes
- Forgetting to multiply each relative uncertainty by its exponent.
- Adding the uncertainties directly, the worst-case sum, where the quadrature sum is intended.
- Using this for sums and differences, which combine absolute, not relative, uncertainties.
Statistics and uncertainty
The root-sum-square rule gives a standard uncertainty (about one standard deviation) for independent errors. A coverage factor of 2 gives roughly a 95 % interval when the errors are normally distributed. Because a few large terms dominate, reducing the biggest weighted contributor matters far more than polishing small ones.
Where it is used
- Uncertainty of a calculated density, power, stress or resistance from measured quantities.
- Deciding which measurement to improve.
Maths you need
- Relative uncertainty: the error as a fraction of the value
- Root sum of squares: square, add, take the root