Errors, uncertainty and instrument accuracy
Measuring 5 V on the 200 V range of a multimeter instead of the 20 V range multiplies the uncertainty by six. The instrument is the same; the decision belongs to whoever is measuring.
01No measurement is exact
To measure is to compare a quantity with a reference, and no comparison is perfect. The value an instrument gives differs from the true one because of the instrument, the method, the environment and the person measuring. What distinguishes an engineering measurement is not that it has no error, but that it says how much it may be wrong: a result without its uncertainty does not allow any decision. “5.00 V” does not say whether the supply meets a specification of 5 V ± 1%; “5.003 V ± 0.007 V” does.
Metrology has a precise vocabulary, fixed in the International Vocabulary of Metrology (VIM):
| Term | What it means |
|---|---|
| Measurand | The quantity to be measured, defined with all its conditions |
| Trueness | Closeness of the average of many measurements to the true value: it is affected by systematic error |
| Precision | Closeness of the measurements to each other: it is affected by random error |
| Accuracy | Closeness of a measurement to the true value: it combines the two previous ones |
| Resolution | The smallest change the instrument can display; it is not accuracy |
How to use the instruments is covered in the technician program, in digital instruments and analog instruments.
02Systematic and random errors
A systematic error repeats identically in every measurement: a shifted calibration, the loading effect of the voltmeter, the resistance of the leads. Averaging does not reduce it; it has to be known and corrected. A random error changes from one measurement to the next: noise, fluctuations, the reading of the last digit. Averaging \( n \) measurements reduces its effect by \( \sqrt{n} \). Gross errors, such as writing down a wrong value or using the wrong range, are not handled with statistics: they are avoided.
When the result is calculated from several measured quantities, the errors propagate. For a product or a quotient, such as \( P = V \cdot I \) or \( R = V / I \), in the worst case the relative errors add: if \( V \) has ±0.5% and \( I \) ±1%, the power has ±1.5%.
03Uncertainty according to the GUM
The Guide to the Expression of Uncertainty in Measurement (GUM) is the method accepted worldwide. Each source of uncertainty is expressed as a standard deviation, the standard uncertainty \( u \), and is evaluated in one of two ways:
The result is reported as \( y \pm U \), with the unit and the value of \( k \), and with the uncertainty rounded to one or two significant figures: “5.0003 V ± 0.0066 V (k = 2)”.
04How to read an instrument’s accuracy
Digital multimeters specify accuracy as \( \pm(\% \text{ of reading} + \text{counts}) \) or \( \pm(\% \text{ of reading} + \% \text{ of range}) \). The fixed term weighs more the smaller the reading is compared with the range: that is why the lowest range that does not overflow is always preferable. Analog instruments specify a class: the maximum error as a percentage of full scale, so a reading at one third of the scale has three times the relative error.
The specification holds under certain conditions: a temperature range (typically 23 °C ± 5 °C), a time since the last calibration (one or two years) and, in AC, a frequency range. Outside them, additional coefficients are added.
A 4½-digit multimeter (19,999 counts) with accuracy ±(0.05% of reading + 3 counts) measures a DC voltage with noise of about 2 counts. \( n \) readings are taken; the dots are the readings, the line is the average and the band is the interval \( \pm U \).
05Traceability and standards
A measurement is traceable when it can be linked to the International System of Units by an unbroken chain of calibrations, each with its uncertainty. The laboratory multimeter is calibrated against a calibrator, which is calibrated against a standard from an accredited laboratory, which is calibrated against the national standard. In Argentina, the national standards are maintained by INTI (the National Institute of Industrial Technology), and Law 19,511 on Metrology establishes the legal system of units, the SIMELA.
| Document | What it establishes |
|---|---|
| JCGM 200 (VIM) | The vocabulary of metrology |
| JCGM 100 (GUM) | The method for evaluating and expressing uncertainty |
| ISO/IEC 17025 | The requirements for a testing or calibration laboratory to be competent |
| IEC 61010 | The safety of measuring instruments, with their overvoltage categories |
06In the laboratory
Measure a 1 kΩ resistor ten times with a multimeter, calculate the Type A uncertainty and the Type B one from the specification, and express the result with its expanded uncertainty.
Measure the same voltage with a 3½-digit multimeter and a 6½-digit one, calculate the two uncertainties and check whether the results are compatible.
Measure voltage and current in a resistor, calculate the power and its combined uncertainty with the propagation law.
07Common mistakes
- Confusing resolution with accuracy. Six digits on the display do not guarantee six correct digits.
- Measuring on a range that is too high. The counts or range term grows in proportion.
- Reporting the uncertainty with many digits. One or two are enough; the value is rounded to the same decimal position.
- Believing that averaging fixes everything. It only reduces the random part; the calibration error remains.
08Self-assessment
What Type B uncertainty does a specification of ±0.1 V with a rectangular distribution give?
\( 0.1 / \sqrt{3} \approx 0.058 \) V.
Ten readings have a standard deviation of 3 mV. What is \( u_A \)?
\( 3 / \sqrt{10} \approx 0.95 \) mV.
What is the maximum error of a class 1 voltmeter with a full scale of 10 V when reading 3 V?
±0.1 V, which is 3.3% of the reading.
What does it mean for a measurement to be traceable?
That it can be related to the SI standards by a documented chain of calibrations, each with its uncertainty.
09Further reading
- JCGM. Evaluation of Measurement Data: Guide to the Expression of Uncertainty in Measurement (JCGM 100:2008). BIPM, 2008. The GUM itself, freely available on the BIPM website.
- Les Kirkup and Robert B. Frenkel. An Introduction to Uncertainty in Measurement. Cambridge University Press, 2006. The GUM explained with examples, for those applying it for the first time.
- John P. Bentley. Principles of Measurement Systems. 4th ed., Pearson, 2005. Measurement systems from the sensor to the display, with their errors.