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
| Operation | Uncertainty of the result |
|---|---|
| A + B or A − B | √(ΔA² + ΔB²) |
| A × B or A ÷ B | |result| × √((ΔA/A)² + (ΔB/B)²) |
| k × A (k exact) | |k| × ΔA |
| Aⁿ | |n| × |result| × ΔA/|A| |
Inputs explained
| Field | What to enter |
|---|---|
| Calculate the uncertainty from | Choose one: A measured value and its relative uncertainty, Repeated readings of the same quantity, Two measurements combined |
| Measured value (MV) | A number |
| Relative uncertainty (R) | In % |
| Readings | A list of numbers separated by commas, spaces or new lines |
| Uncertainty to report | Choose one: Standard error of the mean (s ÷ √n), Standard deviation of the readings, Half the range (max − min) ÷ 2 |
| Measurement A | A number |
| Uncertainty of A (±) | A number |
| Measurement B | A number |
| Uncertainty of B (±) | A number |
| Combine as | Choose 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])
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)
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.
Frequently asked questions
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.