Error Propagation Calculator
Find the uncertainty of a calculated result from the uncertainties of the measured values that it is built from.
What it does
Measure two sides of a rectangle, multiply them, and you have an area that looks far more solid than the two numbers that produced it. It is not. A rectangle measured to the nearest centimetre carries an uncertainty that grows when the sides are multiplied, and the honest way to report the area is with an uncertainty attached.
The international guide to the expression of uncertainty in measurement, which NIST follows, builds the answer from the law of propagation of uncertainty. Each input has a standard uncertainty, which is a positive square root of an estimated variance. The combined standard uncertainty of the result is found by adding up the squares of the sensitivities of the result to each input, each multiplied by the squared uncertainty of that input, and taking the square root of the sum.
For most working cases the sensitivities are simple. A sum or a difference of two independent values has an uncertainty that adds in quadrature. A product or a quotient carries the relative uncertainties of the inputs. A square or a cube multiplies the sensitivity, so a doubled length doubles the uncertainty of the area.
This page runs those rules for the combinations that turn up most often in a lab notebook or on site: a sum, a difference, a product, a quotient, a square and a cube. It returns the result, the combined standard uncertainty, the relative uncertainty, the expanded uncertainty with a coverage factor of 2 and the interval that goes with it.
The limits are clear. The rule assumes the inputs are independent of each other, so two measurements that share the same instrument error are counted twice when they should not be. It also assumes the standard uncertainties are the right ones, which means knowing how each measurement was actually made.
How to use
- Choose metric or imperial for the measurements and the result.
- Pick how the result is built: a sum, a difference, a product, a quotient, a square or a cube.
- Enter the first measured value and its standard uncertainty, which is the same kind of number as the value itself.
- Enter the second measured value and its uncertainty, if the operation needs one.
- Read the combined standard uncertainty and the relative uncertainty, which says how good the result is as a percentage.
- Read the expanded uncertainty and the interval, which is the form most reports use, and keep the result as the value plus or minus that number.
Formula
- u_c(y)
- Combined standard uncertainty of the result, the same kind of number as the result itself.
- c_i
- Sensitivity of the result to input i, which is the partial derivative of the result with respect to that input.
- u(x_i)
- Standard uncertainty of each measured input.
- U
- Expanded uncertainty, which is the combined standard uncertainty multiplied by a coverage factor, usually 2.
The coverage factor of 2 gives an interval that holds the true value in about 95 percent of cases when the inputs are well behaved, which is the convention the NIST policy follows for reporting results other than fundamental constants. The sensitivity for a product or a quotient is the result itself divided by the input, which is why the relative uncertainties of the inputs are the ones that get added in quadrature. When two inputs share a common cause, such as the same ruler, the independent case overstates the combined uncertainty.
Worked example
Example: the area of a rectangle from two measured sides
Defaults: metric units, a product of two lengths, 5 m with an uncertainty of 0.01 m and 2.5 m with an uncertainty of 0.01 m.
- The result is the product of the two lengths: 5 x 2.5 = 12.5000 m2.
- The relative uncertainty of the first length is 0.01 / 5 = 0.002, and of the second it is 0.01 / 2.5 = 0.004.
- Add the squares: 0.002 squared plus 0.004 squared = 0.000004 + 0.000016 = 0.00002.
- The square root of that is 0.0044721, and multiplying by the result gives 12.5 x 0.0044721 = 0.0559 m2.
- As a percentage of the result the combined standard uncertainty is 0.0559 / 12.5 = 0.4472 %.
- The expanded uncertainty with a coverage factor of 2 is 2 x 0.0559 = 0.1118 m2.
- The tool shows 12.5000 m2, 0.0559 m2, 0.4472 %, 0.1118 m2 and the interval from 12.3882 m2 to 12.6118 m2.
Reference table
| Result | Combined standard uncertainty | Note |
|---|---|---|
| y = a + b | the square root of ua squared plus ub squared | uncertainties add in quadrature |
| y = a - b | the square root of ua squared plus ub squared | the same rule as a sum |
| y = a times b | y times the square root of (ua over a) squared plus (ub over b) squared | relative uncertainties add in quadrature |
| y = a divided by b | y times the square root of (ua over a) squared plus (ub over b) squared | the same rule as a product |
| y = a squared | 2 times a times ua | the sensitivity is 2a |
| y = a cubed | 3 times a squared times ua | the sensitivity is 3a squared |
Limits
- The rule assumes the inputs are independent, and two measurements that share the same instrument error are treated as independent here.
- It is a first order rule, so it loses accuracy when the relative uncertainties are large, and the exact result for a product is not symmetric.
- It cannot tell you whether a standard uncertainty is the right one. That comes from how the measurement was actually made.
- It says nothing about systematic errors, which do not shrink with more measurements and are not covered by any of these rules.
Common mistakes
- Adding the uncertainties instead of adding them in quadrature. Two uncertainties of 0.3 and 0.2 give 0.3606, not 0.5.
- Applying the relative rule to a sum. A sum uses the absolute uncertainties, and only a product or a quotient uses the relative ones.
- Forgetting that the uncertainty of a square is twice the sensitivity times the uncertainty, so it grows with the size of the value.
- Reporting the combined standard uncertainty on its own. Most readers expect the expanded value with a coverage factor of 2.
- Ignoring correlations. If both lengths were measured against the same setting of the same rule, their errors are linked and the combined uncertainty comes out too large.
Try it yourself
Quick checks. Answers sit tucked under each question.
Two measurements of 12.4 m and 7.9 m with uncertainties of 0.3 m and 0.2 m are added. What is the uncertainty of the sum?
The result is 20.3000 m and the combined standard uncertainty is 0.3606 m, which is 1.7761 % of the result. The expanded uncertainty is 0.7211 m, so the sum is usually written 20.3000 m plus or minus 0.7211 m, with an interval from 19.5789 m to 21.0211 m.
What if the same pair of values is turned into a ratio?
The quotient is 1.5696 and the combined standard uncertainty is 0.0550, which is 3.5018 % of the result. The expanded uncertainty is 0.1099, so the interval runs from 1.4597 to 1.6795. A ratio carries the relative uncertainties of both inputs, so it is less precise than either measurement on its own.
How does squaring a single length change the uncertainty?
A length of 2.5 m with an uncertainty of 0.01 m gives an area of 6.2500 m2 with a combined standard uncertainty of 0.0500 m2, which is 0.8000 % of the result. The sensitivity is 2a, so the uncertainty of the area is five times the uncertainty of the length, even though the area is much larger.
Questions
What is the law of propagation of uncertainty?
The rule that the combined standard uncertainty of a result is the square root of the sum of the squared sensitivities times the squared input uncertainties. The NIST policy describes it as the usual root sum of squares method.
What is the difference between a standard uncertainty and an expanded uncertainty?
The standard uncertainty is the estimated standard deviation of the result. The expanded uncertainty multiplies it by a coverage factor, usually 2, to give an interval that covers the true value about 95 percent of the time.
Why do uncertainties add in quadrature rather than directly?
Because the errors in separate measurements are independent random effects, and the variance of a sum of independent random variables is the sum of their variances. Adding the standard deviations directly would assume the two errors always happen in the same direction by the same amount.
What if my two measurements are not independent?
Then the rule has to include the covariance, and the combined uncertainty comes out smaller than the independent case. Two lengths measured against the same rule setting are a common example of a shared error.
How many significant figures should the result carry?
As many as the uncertainty supports, so the result is usually given in the same number of figures as its first or second significant digit in the uncertainty.
Does this cover systematic errors?
No. The rule deals with the random spread of repeated measurements. A calibration error or a zero error affects every reading in the same way, and no amount of propagation will show it.
Sources
- NIST Technical Note 1297: guidelines for evaluating and expressing the uncertainty of NIST measurement results
- BIPM JCGM 100:2008: evaluation of measurement data, guide to the expression of uncertainty in measurement
- NIST publication page for Technical Note 1297, with the citation and the policy summary