Two ideas that look alike
A figure written as 10.00 ± 0.05 mm can mean two quite different things.
- A tolerance is a requirement. It states how far a dimension, resistance or other property may differ from its nominal value and still be accepted. A part measuring anywhere from 9.95 to 10.05 mm meets this requirement.
- An uncertainty is a statement about knowledge. It describes how much doubt remains about a measured value, because every measurement has limits.
A resistor marked 1 kΩ ± 5% has a tolerance: its real value is expected to lie between 950 Ω and 1,050 Ω (1,000 × 0.05 = 50). When you measure a real resistor with a meter, the reading has its own uncertainty, from the meter and the method. Both matter, and they combine in any conclusion you draw.
Where uncertainty comes from
- Resolution: the smallest step the instrument displays. A ruler marked in millimetres cannot show tenths directly.
- Accuracy and calibration: how close readings are to the true value. An instrument can have fine resolution and still read consistently high.
- Repeatability: how much repeated readings scatter, because of noise, handling or small changes in conditions.
- Method and environment: temperature, alignment, loading effects of a meter on a circuit, and reading technique.
Resolution is not accuracy. A digital display with many digits may be neither accurate nor stable, which is another reason to be wary of reporting more digits than the evidence supports; see significant figures.
Repeat measurements
If you measure the same thing several times, the readings scatter. The average of them is usually a better estimate than a single reading, and the spread gives a sense of the random part of the uncertainty. The mean and standard deviation are the usual summary. Spread from repeated readings does not reveal a consistent offset, such as a badly zeroed gauge, which can only be found by comparison with a trusted reference.
Combining tolerances: a simple stack-up
Suppose three parts, each 20.0 ± 0.1 mm, are stacked end to end. These numbers are illustrative.
Worst case. All three could be at their limits in the same direction. The nominal total is 60.0 mm, and the extreme variation is 3 × 0.1 = ±0.3 mm. So the stack lies between 59.7 and 60.3 mm. This method is safe in the sense that it includes every possible case, but it can be conservative, since all parts being at the same extreme together is unlikely for many parts.
Root-sum-square (RSS). If the variations are independent and random, they partly cancel. The combined variation is the square root of the sum of squares:
√(0.1² + 0.1² + 0.1²) = √0.03 ≈ 0.17 mm.
So the statistical estimate is about ±0.17 mm, smaller than the worst-case ±0.3 mm. RSS depends on assumptions, such as independence and the way the values are distributed, and it does not guarantee that every assembly stays inside the range. Which method is appropriate depends on the consequences of failure and the practice in your industry, so check what applies to your work.
Percentage uncertainties in products
When values are multiplied or divided, relative (percentage) uncertainties are what combine.
Illustration: a rectangle measures 200 mm ± 1 mm by 100 mm ± 1 mm. The area is 200 × 100 = 20,000 mm².
- Relative uncertainties: 1 ÷ 200 = 0.5%, and 1 ÷ 100 = 1%.
- Worst case: add them, 0.5% + 1% = 1.5%. That is 0.015 × 20,000 = 300 mm². Check by extremes: the largest area is 201 × 101 = 20,301 and the smallest is 199 × 99 = 19,701, which are indeed about 300 above and below 20,000.
- RSS: √(0.5² + 1²) % = √1.25 % ≈ 1.1%, or about 224 mm².
The smaller dimension, with its larger relative error, dominates the result. This is a helpful lesson: improving the already-good measurement achieves little.
For sums and differences, it is absolute uncertainties that combine, as in the stack-up above.
Subtraction can magnify relative error
If you subtract two nearly equal measured values, the absolute uncertainties add while the difference itself is small, so the relative uncertainty can become large. Measuring 100.2 ± 0.1 and 100.0 ± 0.1 gives a difference of 0.2, with a worst-case uncertainty of ±0.2, which is as large as the difference itself. Where possible, measure the difference directly.
Tolerances in design
Tolerances are a trade-off. Tight tolerances usually cost more to make and inspect, while loose ones risk parts that do not fit or function. Designers choose tolerances based on function, process capability and cost. Datasheets show similar ideas in their tables, where a part’s typical, minimum and maximum values reflect variation between units; the distinction between typical versus worst-case values is part of the same thinking. Building design margins into a circuit or assembly is a way of living with that variation.
Inspection and quality control then check whether produced parts fall inside the limits. A measuring method should normally be considerably better than the tolerance it checks, since otherwise the measurement uncertainty uses up much of the allowed range. How much better depends on the standards and practice that apply to you.
Reporting results
State the value, the uncertainty and the unit together, and say what the uncertainty represents: a worst-case bound, a standard deviation, or something else. “25.3 ± 0.2 mm” is clearer with a note such as “estimated from five readings”. If you are not sure how a published uncertainty was defined, avoid assuming it is a hard limit.
Methods for evaluating uncertainty formally exist and are used in calibration and testing laboratories. This article covers the basics only. Where the result bears on safety, compliance or contracts, follow the applicable standards and consult a qualified metrologist or engineer.