Math · Measurement uncertainty
Uncertainty propagation calculator.
Turn an absolute uncertainty into a relative or percent one, or combine the uncertainties of several independent measurements for a sum, difference, product, quotient or power. Independent uncertainties are added in quadrature (root sum of squares). Runs in your browser; nothing is sent.
(3.50 ± 0.05) × (2.00 ± 0.04) = 7.00. The relative uncertainties are 1.43% and 2.00%; added in quadrature they give 2.46%, so the uncertainty is 7 × 0.02458 = 0.17 and the result is 7.00 ± 0.17.
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Absolute, relative and percent uncertainty
The absolute uncertainty has the units of the measurement: 25.0 ± 0.5 cm has an absolute uncertainty of 0.5 cm. The relative uncertainty divides it by the size of the value, 0.5 ÷ 25.0 = 0.02, and the percent uncertainty is that times 100, here 2%. Relative and percent uncertainty have no units, which is what lets you compare the precision of different measurements.
Combining independent uncertainties
The standard method treats each uncertainty as an estimated standard deviation and combines them by the “root-sum-of-squares” rule, the special case of the law of propagation of uncertainty with no correlation between inputs. For the common cases this gives: sum or difference, u = √(u1² + u2² + …). Product or quotient, relative uncertainty of the result = √((u1/x1)² + (u2/x2)² + …), then multiply by the size of the result. Power f = xn with n exact, relative uncertainty = |n| × u/|x|. These are the first-order formulas; they assume the inputs are independent and the uncertainties are small compared with the values.
Worked examples
Sum: (12.5 ± 0.3) + (4.2 ± 0.4) = 16.7 with u = √(0.3² + 0.4²) = √0.25 = 0.5, so 16.7 ± 0.5. The difference 12.5 − 4.2 = 8.3 has the same u, 8.3 ± 0.5. Product: (3.50 ± 0.05) × (2.00 ± 0.04): relative uncertainties 0.05/3.50 = 1.43% and 0.04/2.00 = 2.00%; √(1.43² + 2.00²) = 2.46%; 7.00 × 2.46% = 0.17, so 7.00 ± 0.17. Quotient: (10.0 ± 0.2) ÷ (4.0 ± 0.1) = 2.50 with relative uncertainty √(2.00² + 2.50²) = 3.20%, so 2.50 ± 0.08. Power: (4.00 ± 0.04)² = 16.0; relative uncertainty 2 × 1% = 2%, so u = 0.32 and the result is 16.0 ± 0.3.
Reporting the result
A common textbook habit, not a published standard, is to round the uncertainty to one significant figure (two if its first digit is 1) and to round the value to the same decimal place. The result line follows it: 0.32 becomes 0.3 and 16.0 is shown to the tenths. The unrounded numbers are also shown. If your uncertainty came from a standard deviation, a ± one standard deviation interval corresponds to roughly 68% confidence when the distribution is approximately normal. For the spread of repeated readings see the statistics calculator, and divide by √n for the standard error of the mean.
Sources, opened 8 October 2026: NIST Technical Note 1297, Section 5: Combined Standard Uncertainty (combined standard uncertainty as the estimated standard deviation of the result, obtained by the root-sum-of-squares method, and the approximately 68 percent interval); NIST Technical Note 1297, Appendix A: Law of Propagation of Uncertainty (the first-order law of propagation of uncertainty, Eq. A-3, from which the sum, product and power formulas above are derived for independent inputs); Wikipedia, Propagation of uncertainty (relative error defined as the absolute uncertainty divided by the value, usually written as a percentage).
Note: Educational calculator for independent measurements. It does not handle correlated inputs, large uncertainties, or full uncertainty budgets (Type A and Type B components, coverage factors). For calibration or compliance work use the method your standard requires.
Frequently asked questions
How do you calculate percent uncertainty?
Divide the absolute uncertainty by the measured value and multiply by 100. A reading of 25.0 cm with an uncertainty of 0.5 cm has a percent uncertainty of 0.5 / 25.0 x 100 = 2%.
How do you add uncertainties?
For independent measurements, add the absolute uncertainties in quadrature when you add or subtract the values: u = square root of (u1 squared + u2 squared). For 12.5 +/- 0.3 plus 4.2 +/- 0.4, u is the square root of 0.25, which is 0.5.
How do you multiply or divide measurements with uncertainty?
Convert each uncertainty to a relative uncertainty, add those in quadrature, then multiply the result by the size of the answer. For 3.50 +/- 0.05 times 2.00 +/- 0.04 the relative uncertainty is 2.46%, so the answer is 7.00 +/- 0.17.
What happens to uncertainty when you raise a value to a power?
The relative uncertainty is multiplied by the absolute value of the power. Squaring 4.00 +/- 0.04 doubles the relative uncertainty from 1% to 2%, giving 16.0 +/- 0.3.
Why add in quadrature instead of just adding?
Independent errors are as likely to partly cancel as to reinforce, so the combined uncertainty is smaller than the plain sum. Adding in quadrature, the root-sum-of-squares method, is the standard way to combine independent standard uncertainties.
How many significant figures should the uncertainty have?
A common textbook habit is one significant figure, or two if the first digit is 1, with the value rounded to the same decimal place. It is a convention rather than a published standard, so follow your instructor.