Measurement uncertainty is a parameter that characterizes the dispersion of the possible values of a measured quantity. It quantifies the doubt about the result and makes it possible to judge whether two measurements are compatible. It is a fundamental concept of the high school physics and chemistry curriculum, required for the practical exams of the baccalaureate. For a complete and progressive presentation of the approach, see the guide Understanding Measurement Uncertainty with a Smartphone.
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How to measure it in class
With the FizziQ app, it is possible to evaluate measurement uncertainties.
Steps:
- Repeat a measurement several times under the same conditions
- Record the values with FizziQ
- Calculate the mean and the standard deviation s of the measurements
- Deduce the standard uncertainty: it is approximately s for an individual observation, and s/√n for the mean value of n independent observations (see the Formula section)
- Express the result in the form: x = x̄ ± u(x), where u(x) is the standard uncertainty
- Discuss the sources of uncertainty: sensor, operator, conditions
Scientific activities on this topic
- Measurement uncertainty
- Sensor accuracy: standard deviation
- Measuring g by video analysis
- Free fall: measuring g
Learn more
Types of evaluation:
| Type | Method | Example |
|---|---|---|
| Type A | Statistical (repetition) | Standard deviation of n measurements |
| Type B | Non-statistical | Instrument resolution, specifications |
Expressing the result:
In high school, a measurement result is expressed with its standard uncertainty: x = x̄ ± u(x). The value placed after the ± sign is a standard uncertainty (one standard deviation of doubt): it does not automatically mean a 95% confidence interval. The expanded uncertainty U = k·u (for example k = 2) and the confidence interval belong to post-secondary studies and are not required for the baccalaureate.
Writing rules:
- The standard uncertainty has at most 2 significant figures
- The value is rounded to the same decimal place as the uncertainty
- Example: m = 4.37 ± 0.05 kg (correct)
- Example: m = 4.3742 ± 0.05 kg (incorrect)
Propagation of uncertainties:
For a sum: u(A+B) = √[u(A)² + u(B)²] For a product: u(A×B)/(A×B) = √[(u(A)/A)² + (u(B)/B)²]
Formula
Experimental standard deviation of a series of n measurements: s = √[Σ(xᵢ - x̄)²/(n-1)]
Standard uncertainty (Type A evaluation):
- for an individual observation: u(x) ≈ s
- for the mean of n independent observations: u(x̄) = s/√n
The standard deviation s describes the dispersion of the measurements and does not decrease as n increases; the standard uncertainty on the mean u(x̄) = s/√n, on the other hand, does decrease as n increases. Caution: on a continuous, heavily sampled signal, successive values may be correlated (filtering, smoothing), which makes s/√n too optimistic.
Comparison with a reference value (normalized deviation, final-year curriculum): z = |x_mes - x_ref| / u(x)
If z < 2: the result is considered compatible with the reference; if z ≥ 2: the deviation deserves examination (overlooked sources of uncertainty, bias, protocol).
Application examples
- Measurement of g = 9.78 ± 0.15 m/s²: compatible with 9.81 m/s²
- Comparison of two titration methods
- Validation of an experimental protocol for practical exams
- Acceptance or rejection of a hypothesis
FAQ
Q: Why are uncertainties required for practical exams? A: They make it possible to judge the quality of a result and to reach rigorous conclusions. A result without uncertainty has no scientific meaning.
Q: How does FizziQ help evaluate uncertainties? A: By automatically repeating measurements and calculating the mean and standard deviation. The data can be exported for analysis.
Q: Do you always need to take 10 measurements? A: The more measurements you take, the smaller the standard uncertainty becomes (as 1/√n). In practice, 5 to 10 measurements are often sufficient. The sensor’s specifications also give a Type B uncertainty.
Q: How do you distinguish uncertainty from error? A: The error is the difference between a measured value and a reference value; this difference is generally unknown, since the sought value is unknown too. Uncertainty does not designate this error: it characterizes the doubt associated with the result, that is to say the set of values that can reasonably be attributed to the measured quantity. The error should also not be confused with a handling mistake.
Related concepts
Systematic vs Random Error - Standard Deviation - Significant Figures - Reproducibility