# How do I calculate uncertainty?

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.

- Page: https://www.acalculator.org/physics/uncertainty-calculator
- JSON spec: https://www.acalculator.org/physics/uncertainty-calculator.json
- Version: c090ce8abfb7

## Default answer

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).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | Uncertainty of | A result worked out from two measured values, or the mean of repeated readings of one quantity. |
| a | Value A | The first measured value. |
| ua | Uncertainty of A (±) | The standard uncertainty of A, 0 or more. |
| b | Value B | The second measured value. |
| ub | Uncertainty of B (±) | The standard uncertainty of B, 0 or more. |
| op | Result | How the result is worked out from A and B. |
| n | Exponent n | The power A is raised to, such as 2 for a square or 0.5 for a square root. |
| x | Readings | Two or more readings of the same quantity, separated by commas, spaces or new lines. |
| k | Coverage factor k | The multiplier for the expanded uncertainty: k = 2 gives about 95% confidence, k = 3 more than 99%. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| value | Result (or mean) | The value worked out from A and B, or the mean of the readings. |
| u | Combined standard uncertainty u | The standard uncertainty of the result, u_c. |
| big | Expanded uncertainty U | k × u: the ± to quote for the chosen coverage. |
| rel | Relative uncertainty (%) | u ÷ \|result\| × 100. |
| low | Lower end (result − U) | The result minus the expanded uncertainty. |
| high | Upper end (result + U) | The result plus the expanded uncertainty. |
| sd | Standard deviation of the readings, s | √[Σ(x − mean)² ÷ (n − 1)]. |
| count | Number of readings | How many readings were typed. |

## 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.

## Assumptions

- 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

1. mode = combine, op = mul, a = 12.3, ua = 0.2, b = 4.5, ub = 0.1, k = 2 gives value = 55.35, u = 1.524106, big = 3.048213, rel = 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).
2. mode = combine, op = add, a = 10, ua = 0.3, b = 5, ub = 0.4, k = 2 gives value = 15, u = 0.5, big = 1, low = 14, high = 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).
3. mode = combine, op = div, a = 100, ua = 2, b = 4, ub = 0.1, k = 2 gives value = 25, u = 0.800391, rel = 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).
4. mode = combine, op = pow, a = 2, ua = 0.1, n = 3, k = 2 gives value = 8, u = 1.2, big = 2.4, rel = 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).
5. mode = combine, op = pow, a = 9, ua = 0.3, n = 0.5, k = 2 gives value = 3, u = 0.05, rel = 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).
6. mode = repeat, x = 9.81 or 9.79, k = 2 gives value = 9.81, sd = 0.015811, u = 0.007071, big = 0.014142, count = 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).

## FAQ

### 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.

## Sources

- NIST Technical Note 1297 (1994 edition), Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results (B. N. Taylor and C. E. Kuyatt), Appendix A: Law of Propagation of Uncertainty (Eq. A-3: u_c²(y) = Σ (∂f/∂x_i)² u²(x_i) + 2 Σ Σ (∂f/∂x_i)(∂f/∂x_j) u(x_i, x_j); Eqs. A-4 and A-5: mean of n observations and its standard deviation). https://www.nist.gov/physical-measurement-laboratory/nist-tn-1297-appen-law-propagation-uncertainty (retrieved 2026-10-18)
- NIST Technical Note 1297, Section 6: Expanded Uncertainty (U = k u_c(y); k = 2 at NIST by convention, about 95 percent; k = 3, more than 99 percent). https://www.nist.gov/physical-measurement-laboratory/nist-tn-1297-6-expanded-uncertainty (retrieved 2026-10-18)
- NIST, Constants, Units and Uncertainty, Combining uncertainty components: with uncorrelated inputs the covariance term vanishes; for Y = a₁X₁ + … + a_NX_N, u_c² = a₁²u²(x₁) + … ; for a product of powers the relative uncertainties combine the same way. https://physics.nist.gov/cuu/Uncertainty/combination.html (retrieved 2026-10-18)