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How to define a quantity, choose a measuring method, understand error and uncertainty, and report a useful result.
What is measurement?
Measurement is the process of finding a value for a quantity through an experiment. We measure a table's length, a bag's mass, water temperature or the time taken to run a race.
A number needs a unit to make sense: 2 metres, 5 kilograms or 10 seconds. It also needs context. “The temperature is 25 °C” leaves a question: the temperature of what, where and when?
The quantity we intend to measure is called the measurand. Defining it clearly is the first step towards a useful result.
How does measurement work?
We choose a method that relates the quantity to an established unit. Sometimes the comparison is direct, as when we place a ruler beside a table. Sometimes we calculate the result from other measurements, as when we divide distance by elapsed time to find average speed.
| Quantity | An ordinary method | Unit |
| Length | Compare a table with a ruler | metre (m) |
| Mass | Place rice on a balance | kilogram (kg) |
| Temperature | Put a suitable thermometer in water | degree Celsius (°C) |
| Time interval | Time a race with a stopwatch | second (s) |
An instrument gives an indication, such as a display reading. Obtaining a measurement result may also require a calculation, a correction and an assessment of uncertainty. NIST: measurement equations.
Does measuring tell us the exact value?
An ordinary measurement gives an estimate. Its usefulness depends on how well the method supports the decision we want to make.
A kitchen balance may be suitable for weighing rice for dinner. The same balance may be unsuitable for weighing a very small laboratory sample. More displayed digits cannot make an unsuitable method adequate.
Several ideas help us judge a result:
| Term | The question it answers |
| Resolution | What small change can the instrument distinguish? |
| Accuracy | How close is the measured value to the true value? |
| Precision | How closely do repeated results agree under stated conditions? |
| Error | What is the measured value minus a reference value? |
| Uncertainty | What dispersion of values is attributable to the quantity, given the available information? |
These terms describe different aspects of measurement. Accuracy and precision need to be considered together.
What is resolution?
Resolution is the smallest change in the quantity that produces a perceptible change in the indication.
A thermometer displaying 0.01 °C increments does not thereby establish accuracy to 0.01 °C. Calibration, noise, placement and the instrument's response also matter. In some conditions, useful resolution is poorer than the last displayed digit suggests. VIM: resolution.
What is measurement error?
Measurement error = measured value − reference value.
Suppose a length standard has an assigned value of 50.000 mm. An instrument reads 50.020 mm. Its indicated error relative to that reference is +0.020 mm. The reference and the comparison also have uncertainty.
For an ordinary object, the exact true value is generally unknown. We cannot find its exact measurement error merely by looking at the display.
Error can include systematic and random components. Their differences, and the distinction between an error and a mistake, are explained in Measurement Error.
What is measurement uncertainty?
Uncertainty describes the dispersion of values attributable to the quantity, based on the information available.
It may include contributions from calibration, resolution, repeated readings, temperature, alignment and the way the measurand is defined. Contributions associated with systematic effects matter alongside those associated with random variation. VIM: measurement uncertainty.
| Term | Meaning |
| Standard uncertainty, u | Uncertainty expressed as a standard deviation |
| Combined standard uncertainty, u_c | The standard uncertainty obtained by combining relevant contributions |
| Expanded uncertainty, U | A stated multiple of the combined standard uncertainty: U = k × u_c |
| Coverage factor, k | The multiplier used to obtain expanded uncertainty |
Type A evaluations use statistical analysis of observations. Type B evaluations use other information, such as a calibration certificate. These labels describe methods of evaluating uncertainty; they do not mean “random error” and “systematic error”. NIST: uncertainty glossary.
How should a result be reported?
An illustrative report is:
Length = (50.100 ± 0.010) mm, with expanded uncertainty U = 0.010 mm and coverage factor k = 2.
The ± term describes expanded uncertainty. It is not a manufacturing tolerance or a guaranteed limit on the error. A factor of 2 may support approximately 95% coverage when the distribution and uncertainty evaluation justify it.
A useful report identifies the quantity, method and relevant conditions, explains the uncertainty and states its coverage basis. The numbers above illustrate a reporting format; they were not calculated from the two-piece example in the accuracy article. GUM: reporting uncertainty.
Keep useful digits during calculation, then round the reported value consistently with its uncertainty. See Significant Figures and Rounding.
Why can repeated readings differ?
The quantity itself may change, or the measurement may vary. These possibilities need separate investigation.
Repeatedly measuring the same stable quantity helps assess measurement repeatability. Measuring different objects also includes real differences between those objects.
For example, cutting metal produces pieces whose lengths may vary. Repeated measurements of one stable piece investigate the measuring method. Measurements of many pieces also reflect production variation. A 50 mm design target is a requirement; it is not a reference standard for testing the instrument.
Averaging independent repeated measurements can reduce the contribution from random variation. The standard uncertainty of the mean is estimated by s/√n, where s is the sample standard deviation and n is the number of observations. Averaging does not remove a common measurement offset. NIST: Type A evaluation.
What makes a measurement useful?
Define the quantity, choose a suitable method and ask how much uncertainty the decision can tolerate. Report what the evidence supports.
For a production decision, keep the measurement result separate from the design tolerance. Whether an item passes near a limit also depends on the uncertainty and the agreed decision rule.
Explore measurement
- Measurement Error: errors, their components and common mistakes.
- Accuracy and Precision: closeness, consistency and tolerance.
- Significant Figures: reporting digits and rounding.
Ai disclosure: written with the help of AI (ChatGPT). You are encouraged to point out errors and omissions.
