# Absolute Uncertainty Calculator

URL: https://speedycalculator.com/calculators/statistics/absolute-uncertainty/
Last updated: 2026-10-30

> The absolute uncertainty calculator finds the ± uncertainty of a measurement in its own units – from a relative (percentage) uncertainty, from repeated readings (standard error, standard deviation or half-range) or by combining two measurements in a sum, difference, product or quotient.

## Inputs

- Calculate the uncertainty from
- Measured value (MV)
- Relative uncertainty (R) (%)
- Readings
- Uncertainty to report
- Measurement A
- Uncertainty of A (±)
- Measurement B
- Uncertainty of B (±)
- Combine as

## Results

- Best value
- Absolute uncertainty (±)
- Relative uncertainty
- Lower limit (value − AU)
- Upper limit (value + AU)
- Number of readings
- Standard deviation of the readings
- Worst-case uncertainty (simple addition)

## Formula

```
AU = MV × R ÷ 100 · repeated readings: AU = s ÷ √n (standard error) · A ± B: AU = √(ΔA² + ΔB²) · A × B or A ÷ B: AU = |result| × √((ΔA/A)² + (ΔB/B)²)
```

## Explanation

Every measurement has some doubt attached. The absolute uncertainty expresses that doubt in the same units as the measurement: a length of 52.6 ± 0.8 cm means the true value most likely lies between 51.8 and 53.4 cm. The relative uncertainty expresses the same doubt as a fraction or percentage of the value (here about 1.5%), which makes it easy to compare measurements of very different sizes.

Where does the number come from? Sometimes an instrument's specification gives a percentage accuracy, so the absolute uncertainty is that percentage of the reading. When you repeat a measurement several times, the scatter of the readings is the evidence: the standard deviation describes the uncertainty of a single reading, and the standard error of the mean, s/√n, the uncertainty of their average. For a quick estimate, half the range of the readings is also common in school laboratories.

Results are often calculated from several measurements. If the uncertainties are independent, they add in quadrature: for sums and differences the absolute uncertainties combine as √(ΔA² + ΔB²); for products and quotients the relative uncertainties do. The calculator handles all of these and reports the result as value ± uncertainty with its relative size.

### How to use the Absolute Uncertainty Calculator

- **Calculate the uncertainty from**
- **Measured value (MV)**
- **Relative uncertainty (R)**
- **Readings**
- **Uncertainty to report**
- **Measurement A**
- **Uncertainty of A (±)**
- **Measurement B**
- **Uncertainty of B (±)**
- **Combine as**

Results update as you type, and everything is calculated in your browser.

### How it works

Choose how the uncertainty is known. From a relative uncertainty, AU = |MV| × R/100. From repeated readings the best value is the mean and AU is the standard error, standard deviation or half-range. For two measurements the uncertainties are combined in quadrature (absolute for + and −, relative for × and ÷).

### Worked example

A length of 52.6 cm is measured with an instrument rated at ±1.5%. The absolute uncertainty is 52.6 × 1.5 ÷ 100 = **0.789 cm**, so the result is 52.6 ± 0.8 cm (51.8 to 53.4 cm). Eight pendulum measurements of g – 9.81, 9.79, 9.84, 9.80, 9.77, 9.83, 9.82, 9.78 m/s² – have mean 9.805 and s = 0.0245, so g = 9.805 ± 0.009 m/s² (standard error). Adding two lengths (12.4 ± 0.2) + (7.9 ± 0.15) gives 20.3 ± 0.25.

### Propagation rules for independent uncertainties

OperationUncertainty of the resultA + B or A − B√(ΔA² + ΔB²)A × B or A ÷ B|result| × √((ΔA/A)² + (ΔB/B)²)k × A (k exact)|k| × ΔAAⁿ|n| × |result| × ΔA/|A|

### Inputs explained

FieldWhat to enterCalculate the uncertainty fromChoose one: A measured value and its relative uncertainty, Repeated readings of the same quantity, Two measurements combinedMeasured value (MV)A numberRelative uncertainty (R)In %ReadingsA list of numbers separated by commas, spaces or new linesUncertainty to reportChoose one: Standard error of the mean (s ÷ √n), Standard deviation of the readings, Half the range (max − min) ÷ 2Measurement AA numberUncertainty of A (±)A numberMeasurement BA numberUncertainty of B (±)A numberCombine asChoose one: A + B, A − B, A × B, A ÷ B

### What each result means

- **Relative uncertainty** = [Absolute uncertainty (±)]÷abs([Best value])×100
- **Lower limit (value − AU)** = [Best value] - [Absolute uncertainty (±)]
- **Upper limit (value + AU)** = [Best value] + [Absolute uncertainty (±)]
- **Number of readings** = count([Readings])
- **Standard deviation of the readings** = stdev([Readings])

## Worked example

Inputs — Calculate the uncertainty from: A measured value and its relative uncertainty; Measured value (MV): 52.6; Relative uncertainty (R): 1.5 %.

Results — Best value = 52.6; **Absolute uncertainty (±) = 0.789**; Relative uncertainty = 1.5 %; Lower limit (value − AU) = 51.811; Upper limit (value + AU) = 53.389.

## Key terms

- **Absolute uncertainty**: Uncertainty in the units of the measurement.
- **Relative uncertainty**: Absolute uncertainty divided by the value, often as a percentage.
- **Standard error of the mean**: s ÷ √n; uncertainty of an average of n readings.
- **Quadrature**: Combining independent uncertainties as the square root of the sum of squares.

## FAQ

**What is absolute uncertainty?**
The ± range around a measurement in the measurement's own units, e.g. 52.6 ± 0.8 cm.

**How do you calculate absolute uncertainty from a percentage?**
Multiply the value by the percentage: 1.5% of 52.6 cm is 0.789 cm, so 52.6 ± 0.79 cm.

**What is the difference between absolute and relative uncertainty?**
Absolute uncertainty has units (± 0.79 cm); relative uncertainty is a fraction or percentage of the value (± 1.5%).

**How do I get the uncertainty from repeated measurements?**
Use the standard error of the mean, s ÷ √n. Eight readings with s = 0.0245 give 0.0245 ÷ √8 = 0.0087.

**How do uncertainties combine when adding?**
In quadrature: √(0.2² + 0.15²) = 0.25 for (12.4 ± 0.2) + (7.9 ± 0.15) = 20.3 ± 0.25.

**How do uncertainties combine when multiplying?**
Add the relative uncertainties in quadrature, then multiply by the result: 12.4 × 7.9 = 97.96 ± 2.44.

**How many significant figures should the uncertainty have?**
Usually one, or two if the first digit is 1 or 2; round the value to the same decimal place (52.6 ± 0.8 cm).

**Why does adding in quadrature give less than simple addition?**
Independent errors are unlikely to be at their maximum in the same direction at the same time; simple addition gives a worst-case bound.


## Sources

- [NIST Technical Note 1297 – Guidelines for evaluating and expressing the uncertainty of NIST measurement results](https://www.nist.gov/pml/nist-technical-note-1297)
- [BIPM – Guide to the expression of uncertainty in measurement (GUM)](https://www.bipm.org/en/committees/jc/jcgm/publications)
- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)