How do I calculate uncertainty?
Type two measured values with their uncertainties and how the result is worked out, or a list of repeated readings. The uncertainty calculator gives the combined standard uncertainty by NIST’s rules, the relative uncertainty, and the expanded uncertainty for a coverage factor k.
- Combined standard uncertainty u
- 1.52411
The result is 55.35 with a standard uncertainty of 1.52411 and an expanded uncertainty of ±3.04821 (k = 2).
- Result (or mean)
- 55.35
- Expanded uncertainty U
- 3.04821
- Relative uncertainty (%)
- 2.75358
- Lower end (result − U)
- 52.3018
- Upper end (result + U)
- 58.3982
Combined standard uncertainty u: 1.52411. The result is 55.35 with a standard uncertainty of 1.52411 and an expanded uncertainty of ±3.04821 (k = 2).
How to calculate
Propagates measurement uncertainty through A + B, A − B, A × B, A ÷ B or a power by NIST TN 1297, or finds the standard uncertainty of the mean of repeated readings, with the expanded uncertainty for a coverage factor k.
Example with the default inputs (Uncertainty of A result from A and B, Value A 12.3, Uncertainty of A (±) 0.2, Value B 4.5, Uncertainty of B (±) 0.1, Result A × B, Coverage factor k 2): The result is 55.35 with a standard uncertainty of 1.52411 and an expanded uncertainty of ±3.04821 (k = 2).
Method: u² = u(A)² + u(B)² for A ± B; u² = (B u(A))² + (A u(B))² for A × B; u² = (u(A) ÷ B)² + (A u(B) ÷ B²)² for A ÷ B; u = |n Aⁿ⁻¹| u(A) for Aⁿ; for n readings u = s ÷ √n; U = k × u.
- A and B are independent (uncorrelated), so the covariance term of the law of propagation is 0.
- The law of propagation is first order: it treats the result as linear in A and B near the measured values.
- Uncertainties are standard uncertainties (one standard deviation); the expanded uncertainty is k times the combined one.
Worked examples
Each example is checked against the calculator on every build.
- Uncertainty of A result from A and B, Result A × B, Value A 12.3, Uncertainty of A (±) 0.2, Value B 4.5, Uncertainty of B (±) 0.1, Coverage factor k 2 gives Result (or mean) 55.35, Combined standard uncertainty u 1.524106, Expanded uncertainty U 3.048213, Relative uncertainty (%) 2.75358.Source: NIST Technical Note 1297 (1994), Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Appendix A (law of propagation of uncertainty, Eq. A-3; mean and standard deviation of the mean, Eqs. A-4 and A-5) and §6 (U = k u_c; k = 2 gives about 95%), https://www.nist.gov/physical-measurement-laboratory/nist-tn-1297-appen-law-propagation-uncertainty (retrieved 2026-10-18); NIST, Constants, Units and Uncertainty, Combining uncertainty components (uncorrelated inputs: Y = a₁X₁ + a₂X₂ gives u_c² = a₁²u²(x₁) + a₂²u²(x₂); Y = A X₁^p₁ X₂^p₂ combines relative uncertainties the same way), https://physics.nist.gov/cuu/Uncertainty/combination.html (retrieved 2026-10-18)
- Uncertainty of A result from A and B, Result A + B, Value A 10, Uncertainty of A (±) 0.3, Value B 5, Uncertainty of B (±) 0.4, Coverage factor k 2 gives Result (or mean) 15, Combined standard uncertainty u 0.5, Expanded uncertainty U 1, Lower end (result − U) 14, Upper end (result + U) 16.Source: NIST, Constants, Units and Uncertainty, Combining uncertainty components (uncorrelated inputs: Y = a₁X₁ + a₂X₂ gives u_c² = a₁²u²(x₁) + a₂²u²(x₂); Y = A X₁^p₁ X₂^p₂ combines relative uncertainties the same way), https://physics.nist.gov/cuu/Uncertainty/combination.html (retrieved 2026-10-18)
- Uncertainty of A result from A and B, Result A ÷ B, Value A 100, Uncertainty of A (±) 2, Value B 4, Uncertainty of B (±) 0.1, Coverage factor k 2 gives Result (or mean) 25, Combined standard uncertainty u 0.800391, Relative uncertainty (%) 3.201562.Source: NIST Technical Note 1297 (1994), Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Appendix A (law of propagation of uncertainty, Eq. A-3; mean and standard deviation of the mean, Eqs. A-4 and A-5) and §6 (U = k u_c; k = 2 gives about 95%), https://www.nist.gov/physical-measurement-laboratory/nist-tn-1297-appen-law-propagation-uncertainty (retrieved 2026-10-18)
- Uncertainty of A result from A and B, Result Aⁿ (A to a power), Value A 2, Uncertainty of A (±) 0.1, Exponent n 3, Coverage factor k 2 gives Result (or mean) 8, Combined standard uncertainty u 1.2, Expanded uncertainty U 2.4, Relative uncertainty (%) 15.Source: NIST Technical Note 1297 (1994), Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Appendix A (law of propagation of uncertainty, Eq. A-3; mean and standard deviation of the mean, Eqs. A-4 and A-5) and §6 (U = k u_c; k = 2 gives about 95%), https://www.nist.gov/physical-measurement-laboratory/nist-tn-1297-appen-law-propagation-uncertainty (retrieved 2026-10-18); NIST, Constants, Units and Uncertainty, Combining uncertainty components (uncorrelated inputs: Y = a₁X₁ + a₂X₂ gives u_c² = a₁²u²(x₁) + a₂²u²(x₂); Y = A X₁^p₁ X₂^p₂ combines relative uncertainties the same way), https://physics.nist.gov/cuu/Uncertainty/combination.html (retrieved 2026-10-18)
- Uncertainty of A result from A and B, Result Aⁿ (A to a power), Value A 9, Uncertainty of A (±) 0.3, Exponent n 0.5, Coverage factor k 2 gives Result (or mean) 3, Combined standard uncertainty u 0.05, Relative uncertainty (%) 1.666667.Source: NIST, Constants, Units and Uncertainty, Combining uncertainty components (uncorrelated inputs: Y = a₁X₁ + a₂X₂ gives u_c² = a₁²u²(x₁) + a₂²u²(x₂); Y = A X₁^p₁ X₂^p₂ combines relative uncertainties the same way), https://physics.nist.gov/cuu/Uncertainty/combination.html (retrieved 2026-10-18)
- Uncertainty of Repeated readings, Readings 9.81, 9.79, 9.83, 9.8, 9.82, Coverage factor k 2 gives Result (or mean) 9.81, Standard deviation of the readings, s 0.015811, Combined standard uncertainty u 0.007071, Expanded uncertainty U 0.014142, Number of readings 5.Source: NIST Technical Note 1297 (1994), Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Appendix A (law of propagation of uncertainty, Eq. A-3; mean and standard deviation of the mean, Eqs. A-4 and A-5) and §6 (U = k u_c; k = 2 gives about 95%), https://www.nist.gov/physical-measurement-laboratory/nist-tn-1297-appen-law-propagation-uncertainty (retrieved 2026-10-18)
How it works
A result from A and B. A and B are measured values with standard uncertainties u(A) and u(B), and they are independent (uncorrelated). By the law of propagation of uncertainty (NIST TN 1297, Eq. A-3, with the covariance term 0), u² = (∂y/∂A)² u(A)² + (∂y/∂B)² u(B)²:
| Result y | Combined standard uncertainty u |
|---|---|
| A + B or A − B | u² = u(A)² + u(B)² |
| A × B | u² = (B × u(A))² + (A × u(B))² |
| A ÷ B | u² = (u(A) ÷ B)² + (A × u(B) ÷ B²)² |
| Aⁿ | u = |n × Aⁿ⁻¹| × u(A) |
For A × B and A ÷ B these are the same as adding the relative uncertainties in quadrature, (u ÷ y)² = (u(A) ÷ A)² + (u(B) ÷ B)², but they also work when A or B is 0. For Aⁿ the relative uncertainty is |n| × u(A) ÷ |A|.
Repeated readings. For n readings x₁ … xₙ of one quantity (TN 1297, Eqs. A-4 and A-5):
- Mean = Σx ÷ n
- Standard deviation s = √[Σ(x − mean)² ÷ (n − 1)]
- Standard uncertainty of the mean u = s ÷ √n
Then, in both cases:
- Expanded uncertainty U = k × u (TN 1297, §6), with k = 2 by default.
- Relative uncertainty = u ÷ |result| × 100 (not shown when the result is 0).
- Lower and upper ends = result − U and result + U.
Exact arithmetic. Typed values are read as exact decimals. The result and every square under a root (u², k²u², and 10,000 u² ÷ y² for the relative uncertainty) are exact fractions, so a root that comes out whole is exact. A power with a whole-number exponent is exact. With a fractional exponent, Aⁿ is worked in floating point and then read as its shortest decimal, and u = (|n| × u(A) ÷ A) × Aⁿ is exact from there. At A = 0 a fractional exponent above 1 gives Aⁿ = 0 and u = |n × Aⁿ⁻¹| × u(A) = 0. The ends are exact when U is. Otherwise the end nearer 0 is worked as (result² − U²) ÷ (the other end), with result² − U² exact, so it keeps its 6 figures when U is close to the result.
Rules. A and B are from −10¹⁵ to 10¹⁵ and their uncertainties from 0 to 10¹⁵; n is from −100 to 100; k is above 0 and at most 10; 2 to 10,000 readings. There is no answer for A ÷ B with B = 0, for a negative A with a fractional exponent, for A = 0 with an exponent below 1, or for a result too large or too small to hold in a 64-bit float (beyond about 10³⁰⁸, or a nonzero value below about 10⁻³²³).
Output format. Each value shows 6 significant figures.
Worked examples by hand
An area (the default). A = 12.3 ± 0.2, B = 4.5 ± 0.1, y = A × B = 55.35. u² = (4.5 × 0.2)² + (12.3 × 0.1)² = 0.81 + 1.5129 = 2.3229, so u = 1.52411. U = 2 × 1.52411 = 3.04821. Relative uncertainty 1.52411 ÷ 55.35 = 2.75358%.
A sum. A = 10 ± 0.3, B = 5 ± 0.4, y = 15. u = √(0.09 + 0.16) = √0.25 = 0.5; U = 1, so the result runs from 14 to 16.
A quotient. A = 100 ± 2, B = 4 ± 0.1, y = 25. u² = (2 ÷ 4)² + (100 × 0.1 ÷ 16)² = 0.25 + 0.390625 = 0.640625, so u = 0.800391 and the relative uncertainty is 3.20156%.
A cube. A = 2 ± 0.1, n = 3: y = 8, u = |3 × 2²| × 0.1 = 1.2, U = 2.4, relative uncertainty 1.2 ÷ 8 = 15% (three times A’s 5%).
A square root. A = 9 ± 0.3, n = 0.5: y = 3; relative uncertainty 0.5 × 0.3 ÷ 9 = 1/60 = 1.66667%, so u = 3 ÷ 60 = 0.05.
Five readings of g. 9.81, 9.79, 9.83, 9.80, 9.82 m/s²: mean 9.81. The deviations 0, −0.02, 0.02, −0.01 and 0.01 square to a sum of 0.001, so s² = 0.001 ÷ 4 = 0.00025 and s = 0.0158114. u = √(0.00025 ÷ 5) = 0.00707107; U = 0.0141421.
Other questions people ask
How do I calculate uncertainty?
For repeated readings, the standard uncertainty of the mean is the standard deviation of the readings divided by √n. For a result worked out from measured values, combine their uncertainties with the law of propagation of uncertainty: for A + B or A − B, u = √(u(A)² + u(B)²).
How do uncertainties combine when I multiply or divide?
The relative uncertainties combine in quadrature: (u ÷ |y|)² = (u(A) ÷ A)² + (u(B) ÷ B)². For an area of 12.3 ± 0.2 cm by 4.5 ± 0.1 cm: u = √((4.5 × 0.2)² + (12.3 × 0.1)²) = 1.52 cm², so the area is 55.35 cm² with a standard uncertainty of 1.52 cm².
What is the uncertainty of a power or a square root?
For y = Aⁿ, the relative uncertainty is |n| times that of A: u ÷ |y| = |n| × u(A) ÷ |A|. Squaring doubles the relative uncertainty; a square root halves it. For √(9 ± 0.3): 0.5 × 0.3 ÷ 9 = 1.67%, so 3 ± 0.05.
What is expanded uncertainty?
It is the combined standard uncertainty times a coverage factor k: U = k × u. NIST uses k = 2 by convention, which gives an interval with about 95% confidence when the result is close to normally distributed; k = 3 gives more than 99%.
What is percent (relative) uncertainty?
The uncertainty divided by the size of the result, times 100: u ÷ |y| × 100. It lets you compare the quality of measurements of different sizes. The page shows it for the combined standard uncertainty.
Why add uncertainties in quadrature instead of adding them?
When the errors in A and B are independent, they partly cancel, so the combined uncertainty is the square root of the sum of squares, not the sum. Adding them directly gives an upper bound that overstates the uncertainty.