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How measured values differ from references, how systematic and random errors differ, and why uncertainty and mistakes need separate treatment.
What is measurement error?
Measurement error is a measured value minus a reference value. It tells us the signed difference in a particular comparison. VIM: measurement error.
Suppose a reference mass has an assigned value of 100.0 g and a balance indicates 100.3 g:
Error = 100.3 g − 100.0 g = +0.3 g.
The positive sign means the indication is high relative to the reference. A reading of 99.7 g would give an error of −0.3 g. These are illustrative comparisons; the reference and the comparison method also have uncertainty.
Can we always know the error?
We need an appropriate reference to assess an error. A display alone cannot tell us its own error.
In everyday measurement, the exact true value is generally unknown. A calibration compares indications with measurement standards under specified conditions and provides information for interpreting later readings. Its results also have uncertainty.
An instrument's disagreement with a design target is a different matter. If a table should be 2 m long but measures 2.01 m, the 0.01 m difference is a measured departure from the target. It does not establish that the ruler is wrong.
What are the main components of error?
| Component | Behaviour in repeated measurements | An illustrative cause |
| Systematic error | Stays constant or changes predictably under the stated conditions | A balance has an uncorrected zero offset |
| Random error | Changes unpredictably between repetitions | Fluctuations in a sensor's readings |
The distinction describes how an effect behaves in the measurement procedure. “Instrument error” or “human error” names a possible source, not a separate behaviour. An operator can introduce a consistent offset or varying differences.
One measurement may contain both components. Repetition can reveal scatter while leaving a shared offset unnoticed. See Systematic Error and Random Error.
What about gross errors and mistakes?
Writing 103.0 g when the display says 100.3 g is a recording mistake. Selecting the wrong unit or measuring the wrong object can also invalidate a result.
Some introductory texts call such blunders gross errors. Treat them as problems to investigate and correct, rather than as a third component to average together with systematic and random error.
An unusual reading is a reason to check the procedure, equipment and record. It is not, by itself, proof of a mistake. Do not discard an inconvenient result simply because it is far from the others. The formal vocabulary distinguishes measurement error from a mistake. VIM: measurement error, note 2.
How can an error be expressed?
Using the 100.0 g reference example:
| Expression | Calculation | Result |
| Signed error | 100.3 g − 100.0 g | +0.3 g |
| Magnitude of error | Absolute value of +0.3 g | 0.3 g |
| Relative error | +0.3 g / 100.0 g | +0.003 |
| Percentage error | +0.003 × 100% | +0.3% |
Terminology around “absolute error” varies, so state whether you mean a signed difference in the original unit or its unsigned magnitude. Relative error requires a non-zero reference and a quantity scale for which the ratio is meaningful; percentages based on Celsius temperatures, for example, can mislead.
These are ways of expressing a comparison. They are not additional causes of error.
Is error the same as uncertainty?
No. Error concerns a difference from a reference. Uncertainty describes what values can reasonably be attributed to the quantity using the available information.
A reported uncertainty of 0.3 g does not mean an error of +0.3 g has been found. A correction for an estimated offset can improve a result, but the correction itself has uncertainty. NIST: uncertainty glossary.
How should we deal with errors?
Define the quantity and conditions, use suitable standards, repeat measurements where useful and investigate effects that may shift the results. Keep records so another person can understand the comparison.
Use the pattern to guide the response: investigate predictable offsets, assess random spread and correct documented mistakes. Report the remaining uncertainty with the result.
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Ai disclosure: written with the help of AI (ChatGPT). You are encouraged to point out errors and omissions.
