# Percent error and uncertainty propagation calculator

> Percent error from measured and accepted values, and uncertainty propagation through sums, products, quotients and powers, in quadrature or worst case.

Interactive version: https://www.calcopenly.com/science/percent-error-propagation-calculator
Subject: Science calculators

Percent error compares a measurement with an accepted value: |measured − accepted| / |accepted| × 100. Uncertainty propagation works out how the ± uncertainties of inputs carry into a calculated result. For sums and differences the absolute uncertainties combine; for products and quotients the relative uncertainties combine; a power aⁿ multiplies the relative uncertainty by |n|. Independent random errors combine in quadrature (the square root of the sum of squares), and the worst-case option adds them linearly.

The default, a measured g of 9.72 m/s² against the accepted 9.81 m/s², is 0.917% low. Multiplying (2.0 ± 0.1) by (3.0 ± 0.2) gives relative uncertainties of 5% and 6.7%, which combine in quadrature to 8.3%, so the product is reported as 6.0 ± 0.5.

Results are rounded as Taylor's An Introduction to Error Analysis recommends: the uncertainty to one significant figure, or two when it starts with 1, and the value to the same decimal place. The formulas are first-order, so they assume each uncertainty is small compared with its value.

## Inputs

- **Calculate** (options: Percent error, Sum or difference a ± b, Product or quotient a × b, a ÷ b, Power aⁿ)
- **Measured value**
- **Accepted (true) value**
- **Operation** (options: a + b or a × b, a − b or a ÷ b)
- **a**
- **Uncertainty in a (±)**
- **b**
- **Uncertainty in b (±)**
- **Exponent n**: Use 0.5 for a square root
- **Combine uncertainties** (options: In quadrature, Worst case)

## Results

- Result q
- Uncertainty δq
- Relative uncertainty
- Report as
- Percent error — main result
- Signed percent error
- Absolute error

## Formula

$$
\%\,\text{error} = \frac{|x_m - x_a|}{|x_a|}\times100;\quad \delta(a\pm b) = \sqrt{\delta a^2 + \delta b^2};\quad \frac{\delta(ab)}{|ab|} = \sqrt{\Big(\frac{\delta a}{a}\Big)^2 + \Big(\frac{\delta b}{b}\Big)^2};\quad \frac{\delta(a^n)}{|a^n|} = |n|\frac{\delta a}{|a|}
$$

## Worked examples

### Measured g = 9.72 against 9.81

- Calculate: Percent error
- Measured value: 9.72
- Accepted (true) value: 9.81
- **Percent error: 0.9174%**
- **Signed percent error: -0.9174%**
- **Absolute error: 0.09**
- Checked against: Python 3.8 decimal: |9.72 − 9.81| / 9.81 × 100 = 0.9174312

### Sum in quadrature: (10.0 ± 0.3) + (5.0 ± 0.4)

- Calculate: Sum or difference a ± b
- Operation: a + b or a × b
- a: 10.0
- Uncertainty in a (±): 0.3
- b: 5.0
- Uncertainty in b (±): 0.4
- Combine uncertainties: In quadrature
- **Result q: 15**
- **Uncertainty δq: 0.5**
- **Report as: 15.0 ± 0.5**
- Checked against: Taylor §3.5: √(0.3² + 0.4²) = 0.5

### Same sum, worst case

- Calculate: Sum or difference a ± b
- Operation: a + b or a × b
- a: 10.0
- Uncertainty in a (±): 0.3
- b: 5.0
- Uncertainty in b (±): 0.4
- Combine uncertainties: Worst case
- **Uncertainty δq: 0.7**
- **Report as: 15.0 ± 0.7**
- Checked against: Taylor §3.3: δa + δb = 0.7

### Product (2.0 ± 0.1) × (3.0 ± 0.2)

- Calculate: Product or quotient a × b, a ÷ b
- Operation: a + b or a × b
- a: 2.0
- Uncertainty in a (±): 0.1
- b: 3.0
- Uncertainty in b (±): 0.2
- Combine uncertainties: In quadrature
- **Result q: 6**
- **Uncertainty δq: 0.5**
- **Report as: 6.0 ± 0.5**
- Checked against: Python 3.8 decimal: 6 × √(0.05² + 0.0667²) = 6 × 0.083333 = 0.5

### Quotient (10 ± 0.5) ÷ (4 ± 0.2): leading 1 keeps two figures

- Calculate: Product or quotient a × b, a ÷ b
- Operation: a − b or a ÷ b
- a: 10
- Uncertainty in a (±): 0.5
- b: 4
- Uncertainty in b (±): 0.2
- Combine uncertainties: In quadrature
- **Result q: 2.5**
- **Uncertainty δq: 0.176777**
- **Report as: 2.50 ± 0.18**
- Checked against: Python 3.8 decimal: 2.5 × √(0.05² + 0.05²) = 0.1767767; Taylor §2.2 rounding

### Cube: (2.0 ± 0.1)³

- Calculate: Power aⁿ
- a: 2.0
- Uncertainty in a (±): 0.1
- Exponent n: 3
- **Result q: 8**
- **Uncertainty δq: 1.2**
- **Relative uncertainty: 15%**
- **Report as: 8.0 ± 1.2**
- Checked against: Taylor §3.7: δq/q = 3 × 0.05 = 0.15, δq = 1.2

## Questions

### How do you calculate percent error?

Percent error = |measured − accepted| / |accepted| × 100. Measuring g as 9.72 m/s² against an accepted 9.81 m/s² gives 0.09/9.81 × 100 = 0.917%. Keep the sign when the direction matters: −0.917% says the measurement came out low. The formula fails when the accepted value is 0; report the absolute error instead.

### How do you propagate uncertainty when multiplying or dividing?

Combine the relative uncertainties in quadrature: δq/|q| = √((δa/a)² + (δb/b)²). For (2.0 ± 0.1) × (3.0 ± 0.2) the relative uncertainties are 5% and 6.7%, giving 8.3% of 6.0, so q = 6.0 ± 0.5. Division follows the same rule: (10 ± 0.5) ÷ (4 ± 0.2) = 2.50 ± 0.18.

### How do you add or subtract uncertainties?

Combine the absolute uncertainties: δq = √(δa² + δb²) when the errors are independent and random, or δa + δb for a worst-case bound. (10.0 ± 0.3) + (5.0 ± 0.4) = 15.0 ± 0.5 in quadrature, or 15.0 ± 0.7 worst case. Subtraction uses the same rule, so the difference of two close values can carry a large relative uncertainty.

### When should uncertainties be added in quadrature?

When the errors in the inputs are independent and random, the assumption behind the combined standard uncertainty in NIST Technical Note 1297 and the GUM. Quadrature never exceeds the linear sum: 0.3 and 0.4 combine to 0.5 rather than 0.7. Use the linear worst case when errors may be correlated, such as two readings taken with the same miscalibrated instrument.

### How many significant figures should an uncertainty have?

One, or two when its leading digit is 1, following Taylor's An Introduction to Error Analysis (§2.2); the value is then rounded to the same decimal place. So 0.1768 becomes 0.18 and the result is written 2.50 ± 0.18, while 0.52 becomes 0.5 and the result 6.0 ± 0.5. Writing 6.0 ± 0.5234 claims more precision than the measurement has.

### How accurate is the percent error and uncertainty propagation calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 7 worked examples whose answers come from independent sources; for example, “Measured g = 9.72 against 9.81” is checked against Python 3.8 decimal: |9.72 − 9.81| / 9.81 × 100 = 0.9174312.

### Where does the method come from?

Taylor, An Introduction to Error Analysis (2nd ed., 1997), ch. 2–3 (rules for reporting and propagating uncertainties); NIST Technical Note 1297 — Guidelines for evaluating and expressing the uncertainty of NIST measurement results.

## Sources

- Taylor, An Introduction to Error Analysis (2nd ed., 1997), ch. 2–3 (rules for reporting and propagating uncertainties)
- [NIST Technical Note 1297 — Guidelines for evaluating and expressing the uncertainty of NIST measurement results](https://www.nist.gov/pml/nist-technical-note-1297)