Chemistry · Lesson
Accuracy, precision and measurement uncertainty
Compare repeated readings with a reference, then separate closeness, repeat agreement and the uncertainty attached to a result.
Back to the Chemistry mapBy the end, you can
- Distinguish accuracy from precision.
- Recognise systematic effects that repeated readings do not remove.
- Interpret an uncertainty statement without treating it as a guarantee about every possible result.
Close together is different from close to a reference
Imagine a teaching example with a well-characterised reference value of 100.0 units and negligible reference uncertainty for the comparison. Three instrument readings are 105.0, 105.1 and 104.9. They are close together, yet all lie about 5 units above the reference. A second set, 99.0, 100.0 and 101.0, is more spread out but centred near the reference. These invented sets answer different questions.
Accuracy concerns closeness to the true or accepted reference value in the relevant measurement context. It is a qualitative concept. Precision concerns agreement among repeated results under specified conditions. A measure such as a standard deviation can quantify their dispersion, but “accurate” and “precise” are not interchangeable labels.
Illustrative set
Reading / arbitrary unit
- Tight group, offset
- Wider group, centred
- ┊ Comparison reference: 100
Underlying data
| Reading / arbitrary unit | Illustrative set |
|---|---|
| 104.9 | 1 · Tight group, offset |
| 105 | 1 · Tight group, offset |
| 105.1 | 1 · Tight group, offset |
| 99 | 2 · Wider group, centred |
| 100 | 2 · Wider group, centred |
| 101 | 2 · Wider group, centred |
Invented data against a 100.0-unit reference, with negligible reference uncertainty for this illustration. A tightly grouped set can retain a large offset.
A stable offset can survive repetition
Random effects contribute to variation between readings. A systematic effect can shift many readings in the same direction, such as an uncorrected calibration offset. Taking more readings can improve an estimate affected by random variation, but repeating the same biased method does not automatically remove its bias.
The reference itself may have uncertainty, and the sample or procedure may change. Record those conditions when comparing results. Do not classify every unexpected reading as random or every disagreement as a broken instrument. Investigate the measurement process and retain the original data.
Describe the uncertainty of the result
Measurement uncertainty characterises the dispersion of values that could reasonably be attributed to the measured quantity, using the available information. It can include repeat variation, calibration information, resolution, sample preparation and other relevant effects. A display increment tells you how the instrument reports a reading; it does not describe every source of uncertainty.
A report such as 12.45 g with an expanded uncertainty of 0.04 g needs a stated coverage factor or coverage information and the method used. The range 12.41 to 12.49 g follows by subtraction and addition, but its statistical interpretation depends on that information. Without it, do not claim a particular confidence level or a guarantee that a true value lies inside the range.
Use uncertainty in the question you actually asked
Two measurements may differ without providing evidence of a meaningful difference in the underlying quantities. Compare them with their uncertainties, conditions and any shared sources of error. Simple overlap of two quoted intervals is not a universal statistical test. Later data-analysis lessons develop suitable comparison methods.
For a beginner, the practical habit is to ask what was measured, how uncertain each result is, and what conclusion the evidence can support. Rounding should preserve the meaning of the reported result. Additional decimal places in a calculated average do not create extra measurement information.
Worked examples
Read a stated uncertainty interval
A report gives 12.45 g and expanded uncertainty 0.04 g. What endpoints follow, and what still needs to be stated?
- Lower endpoint:12.45−0.04=12.41 g.
- Upper endpoint:12.45+0.04=12.49 g.
- Request the coverage factor or coverage information and evaluation method.
- Keep the result, uncertainty and measured quantity together.
The arithmetic endpoints are 12.41 g and 12.49 g. The numbers alone do not specify a confidence level or guarantee coverage.
Try it yourself
Repeated readings cluster around 105.0 when the comparison reference is 100.0. Does the tight cluster establish accuracy?
Show the answer and reasoning
No.
It shows repeat agreement under the stated conditions, while the reference comparison reveals an offset.
Does averaging more readings from the same uncorrected biased instrument automatically remove its offset?
Show the answer and reasoning
No.
Repetition can reduce some random effects in an average, but shared systematic effects require separate evaluation or correction.
A display changes in 0.01 g increments. Is the full measurement uncertainty necessarily 0.01 g?
Show the answer and reasoning
No.
Resolution is one contribution. Calibration, repeatability, sampling and other effects may also matter.
A result is written as a value plus or minus an uncertainty. May you infer 95% coverage without further information?
Show the answer and reasoning
No.
Coverage depends on how the uncertainty was evaluated and reported. Request the stated coverage information.
What to carry forward
- Precision concerns agreement among results; accuracy concerns closeness to the relevant true or reference value.
- Repetition does not automatically remove systematic effects.
- Uncertainty is broader than display resolution or repeated scatter alone.
- Keep coverage information and conditions attached to an uncertainty statement.
References
Teaching resources and reference material supporting this lesson’s explanations and examples.
- International Vocabulary of Metrology (VIM3)Joint Committee for Guides in Metrology (JCGM)Reference guideVIM3, 2.13 measurement accuracyVIM3, 2.15 measurement precisionVIM3, 2.26 measurement uncertaintyVIM3, 4.14 resolution
- Guide to the expression of uncertainty in measurement — Part 1: Introduction, JCGM GUM-1:2023Guide to the expression of uncertainty in measurement — Part 1: Introduction, JCGM GUM-1:2023Reference guide · PDF
- NIST Technical Note 1297, section 7: Reporting UncertaintyNIST Technical Note 1297, section 7: Reporting UncertaintyReference guide