Core concepts and evaluation logic of measurement uncertainty
I. The essence of measurement uncertainty: Probabilistic quantification of dispersion
Measurement uncertainty is a probabilistic description of the "reasonable value range of the measured quantity" based on existing information, and it is a parameter bound to the measurement result. Its core connotation needs to be broken down into three points:
"Reasonable assignment": It is not a subjective guess but the best estimate that combines all available information such as measurement data, calibration certificates, and technical manuals.
Dispersion: It describes the distribution range of the possible values of the measured quantity (for example, The mass is approximately 100 g, and it is most likely to be between 99.8 g and 100.2 g), rather than the deviation of the measured value from the true value (Error is absolute and unknowable, while uncertainty is probabilistic and knowable).
"Linked to the result": Uncertainty cannot exist independently. When talking about "uncertainty", it must be clear that it is "the uncertainty of a certain measurement result" (for example, "the uncertainty of 100g ± 0.2g is 0.2g").
II. Standard uncertainty: A basic parameter defined by the standard deviation
The standard uncertainty is the measurement uncertainty using the standard deviation as the quantification tool, with the symbol `u`. It is the "minimum unit" of uncertainty. The key points for understanding are:
- The standard deviation is a classic indicator in statistics to describe the "degree of data dispersion" (under the normal distribution, 68% of the data falls within ±1 standard deviation of the mean, and 95% falls within ±2 standard deviations).
- Standard uncertainty ≠ "Uncertainty of a measurement standard" — A "measurement standard" refers to a specific measuring instrument (such as a reference balance for calibration), while standard uncertainty is a way of characterizing a parameter (regardless of what instrument is used for measurement, as long as the dispersion is described by the standard deviation, it is the standard uncertainty).
III. Components of standard uncertainty: Complementarity between Type A and Type B
Measurement results are usually calculated from multiple "input quantities" (such as mass, size, current). The uncertainty of each input quantity is called the standard uncertainty component. The evaluation of components is divided into two categories, and the only difference lies in the "information source", with no distinction between superiority and inferiority:
(I) Type A evaluation: Using the statistical laws of the observation series
Type A evaluation involves conducting statistical analysis on "independently observed data under the same conditions" to obtain a component, denoted by the symbol `u_A`. Its essence is "inferring the population from the sample" – estimating the dispersion of the population through a small sample obtained from repeated measurements.
1. Premise: "Same conditions + Independent observations"
Same conditions: the same instrument, the same operator, the same environment (stable temperature/humidity), excluding the interference of systematic changes.
Independent observations: There is no correlation between each measurement result (for example, the 5th measurement does not depend on the result of the 4th measurement).
2. Calculate: The experimental standard deviation of the arithmetic mean
Measure the input quantity `Q` `n` times (`n
$$\bar{q} = \frac{1}{n}\sum_{j=1}^n q_j$$
The experimental standard deviation of the mean value (i.e., the Type A component) is:
$$u_A(\bar{q}) = \frac{s(q)}{\sqrt{n}}$$
Among them, `s(q)` is the sample standard deviation of the observed values.
$$s(q) = \sqrt{\frac{1}{n-1}\sum_{j=1}^n (q_j - \bar{q})^2}$$
3. Improvement of small samples: Pooled sample standard deviation `s_p`
When `n`When `< 10`, the reliability of statistical results of small samples is low (for example, when measuring the mass 3 times, the sample standard deviation fluctuates greatly). at this time, the pooled sample standard deviation `s_p` can be used — it is the square root of the average of the within - group variances of long - term repeated measurements (for example, for 100 sets of measurement data within half a year on a production line, with 5 measurements in each set, `s_p` is the weighted average of the sample standard deviations of these 100 sets). `s_p` is more stable and suitable for routine measurements under statistical control (such as quality inspection of batch products).
(II) Type B evaluation: Inference based on experience from external information
Type B evaluation estimates the components using "information from non - repeated observations", with the symbol `u_B`. It solves the problem of "inability to conduct repeated measurements" (for example, the instrument calibration certificate only provides a single result). The core is to "transform experience and documents into standard deviation".
1. Source of information (must be reliable)
The information for Type B evaluation shall be traceable. Common sources are as follows:
- Calibration/verification certificate (providing expanded uncertainty and coverage factor);
- Instrument technical manual (e.g., Repeatability: ±0.1%).
- Industry standards/specifications (e.g., "The allowable error of the burette is ±0.02 mL").
- Experimental experience (e.g., The error of manual reading is approximately ±0.05mm).
2. Typical calculation methods
(1) Given the expanded uncertainty `U` and the coverage factor `k`
If the calibration certificate states "Expanded uncertainty `U = 0.2 mg`, `k = 2`", then:
$$u_B = \frac{U}{k} = \frac{0.2}{2} = 0.1\ mg$$
- `U` is the half-width of the confidence interval (when `k = 2`, it contains the true value with a probability of 95%).
- `k` is the "expansion factor" (under the normal distribution, `k = 2` corresponds to a 95% confidence level, and `k = 3` corresponds to 99%).
(2) Uniform distribution with known upper and lower limits
If the input quantity is uniformly distributed between `[a₋, a₊]` (for example, the accuracy of a vernier caliper is `±0.02mm`, and the values appear with equal probability between `-0.02` and `+0.02`), then:
$$u_B = \frac{a}{\sqrt{3}}$$
Among them, `a = (a₊ - a₋)/2` (half-width of the interval). The variance of the uniform distribution is `a²/3`, so the standard deviation is `a/√3`.
(3) Calculate from the repeatability limit `r`
The repeatability limit `r` is the "maximum allowable difference between two measurements under the same conditions" (for example, the standard method stipulates that "r = 0.3%"). The variance of the difference between two measurements is `2u_B²` (the variances of independent variables are added), so:
$$u_B = \frac{r}{\sqrt{2}}$$
(4) Normal distribution with a known confidence level
If the document only states that "the value is between `100±0.5` at a 95% confidence level", then by looking up the normal distribution table, we get `k = 1.96` (corresponding to a 95% confidence level). Calculate:
$$u_B = \frac{0.5}{1.96} \approx 0.255$$
IV. Combined standard uncertainty: Integration of multiple components
When the measurement result `y` is calculated from multiple input quantities `X₁, X₂, …, Xₙ` through the function `y = f(X₁, X₂, …, Xₙ)`, the combined standard uncertainty `u_C(y)` represents the "combined effect" of these components, with the symbol `u_C`.
1. Synthesis of independent input quantities (the most common)
If the input quantities are independent of each other (for example, the mass `m` has no relation with the dimension `r`), the synthesis formula is:
$$u_C(y) = \sqrt{\sum_{i = 1}^n \left(\frac{\partial f}{\partial X_i}\right)^2 u(X_i)^2}$$
Among them, `∂f/∂X_i` is the sensitivity coefficient (which describes the degree of influence of the change in `X_i` on `y`. For example, in `y = IR`, `∂y/∂I = R`, and the uncertainty of the current `I` will affect the voltage `y` after being magnified by a factor of `R`).
2. Synthesis of relevant input quantities
If the input quantities are correlated (for example, both the current `I` and the resistance `R` come from the same power supply and have the same changing trend), the covariance term needs to be added:
$$u_C(y) = \sqrt{\sum_{i = 1}^n \left(\frac{\partial f}{\partial X_i}\right)^2 u(X_i)^2 + 2\sum_{i}$$It seems there is a formatting or symbol issue in the given content. The correct and interpretable part might be something like $\sum_{i,j} \frac{\partial f}{\partial X_i}\frac{\partial f}{\partial X_j} \text{Cov}(X_i,X_j)$. But based on the exact text you provided " The covariance `Cov(X_i, X_j)` describes the degree of correlation (the covariance is positive when there is a positive correlation and negative when there is a negative correlation). V. Expanded uncertainty: Add a "confidence interval" to the result
The expanded uncertainty is the half-width of the interval obtained by multiplying the combined standard uncertainty by a factor `k`, denoted by the symbol `U`. The formula is:
$$U = k \cdot u_C(y)$$
Its function is to make the measurement results "understandable" - for example:
The meaning of this sentence is: Based on the existing information, there is a 95% probability that the true density of the metal block falls between 63.5 and 63.9 g/cm³.
VI. Complete evaluation process: From model to report
The evaluation of measurement uncertainty is a "logical closed - loop" and shall be carried out in accordance with the following steps:
1. Build a model: Clarify the relationship between inputs and outputs
The measurement model is `y = f(X₁, X₂, …, Xₙ)`, based on physical laws or experimental experience (such as:
- Density: `ρ = m/(πr²h)` (`m` is mass, `r` is radius, `h` is height);
- Voltage: `U = IR` (`I` is current, `R` is resistance).
The model is not unique (for example, the volume can be obtained through the "water displacement method" or "dimension calculation", corresponding to different models).
2. Rating quantity: Category A or Category B
For each input quantity `X_i`, evaluate `u(X_i)` using Type A (repeatable measurements) or Type B (external information).
3. Calculate the synthesis: Use the propagation formula
Calculate `u_C(y)` based on the functional relationship of the model (for independent inputs, use the square root of the sum of variances; for correlated inputs, add the covariance).
4. Expand the interval: Select the value of `k`
Select the factor `k` (usually `k = 2`, corresponding to a 95% confidence level), and calculate `U = k·u_C(y)`.
5. Write a report: Be clear and complete
The report should include:
- The measured value (e.g., `63.7 g/cm³`);
- Expanded uncertainty (e.g., `±0.2 g/cm³`);
- Include coverage factors and probabilities (e.g., `k=2`, 95%).
VII. Example: Uncertainty evaluation of the density of a metal block
1. Tasks and Methods
Measure the density `ρ` of the cylindrical metal block. Method: Use a balance to measure the mass `m`, use a vernier caliper to measure the diameter `d` (`r = d/2`) and the height `h`, and calculate with the formula `ρ = m/(πr²h)`.
2. Measured data
- Measure `m` 5 times: 100.1 g, 100.2 g, 100.0 g, 100.1 g, 100.2 g → Average `m = 100.12 g`;
- Measure `d` three times: 10.00 mm, 10.02 mm, 9.98 mm → Average `d = 10.00 mm` (`r = 5.00 mm`);
- Measure `h` three times: 20.00 mm, 20.02 mm, 19.98 mm → The average `h = 20.00 mm`.
3. Evaluate the uncertainty of the input quantity
The `u(m)` of `m`: For Type A, `u_A = 0.036g` (sample standard deviation `0.08g`, `n = 5`), for Type B, `u_B = 0.05g` (calibration certificate `U = 0.1g`, `k = 2`) → Combined `u(m)=√(0.036² + 0.05²)=0.0616g`;
For \(u(r)\) of \(r\): Type A, \(u_A = 0.00575\mathrm{mm}\) (sample standard deviation of \(d\) is \(0.02\mathrm{mm}\), \(n = 3\)), Type B, \(u_B = 0.0115\mathrm{mm}\) (uniform distribution of vernier caliper is \(\pm0.02\mathrm{mm}\)) → Combined \(u(r)=0.0128\mathrm{mm}\);
The `u(h)` of `h`: It is the same as `r`, and `u(h) = 0.0128 mm`.
4. Calculate the combined standard uncertainty
Model `ρ = m/(πr²h)`, sensitivity coefficient:
- `∂ρ/∂m = 1/(πr²h) = 1/1.5708 ≈ 0.6366 cm³/g`;
- `∂ρ/∂r = -2ρ/r = -2*63.74/5 ≈ -25.5 g/(cm³·mm)`;
- `∂ρ/∂h = -ρ/h = -63.74/20 ≈ -3.187 g/(cm³·mm)`.
Synthesize `u_C(ρ)=√[(0.6366*0.0616)² + (-25.5*0.0128)² + (-3.187*0.0128)²]≈0.1 g/cm³`.
5. Expansion and Reporting
`k = 2`, `U = 2 * 0.1 = 0.2 g/cm³`, Report:
Summary
The core of measurement uncertainty is the "probabilistic description of dispersion". Type A and Type B evaluations are complementary. The combination and expansion integrate multi - source information into an understandable interval. It is not "negating the accuracy of the measurement result", but "telling users the reliability of the result". Scientific measurement has never been "giving a value", but "giving a range with a confidence level".