Measurement uncertainty of type A
Type A uncertainties include any uncertainties that, by their nature, can be calculated only statistically. The result of the calculation is law distribution p(q) for which the conditions are met:
∫+∞-∞ p(q)dq = 1
μq = ∫+∞-∞qp(q)dq
σ2q = ∫+∞-∞ (q-μq)2p(q)dq
Statistical estimates
Statistical estimation of the average value of μq for n measurements under the same conditions:
q = 1/n Σnk=1 qk (1)
Experimental variance - statistical estimation of variance σ2:
s2(qk) = 1/(n-1) Σnj=1 (qj - q)2 (2)
Statistical estimation of the variance of the mean value σ(q)2 = σ2/n:
s2(q) = s2(qk)/n (3)
Uncertainty value
Uncertainty u(xi) of statistical estimation of the average value of n measurements of the value Xi equal to s(Xi) (formula 3).
The degree of freedom vi for the value u(xi) equal to n-1 (n is the number of measurements of the value xi) must be specified in the documentation for the definition of uncertainty type A.
Average uncertainty value
The statistical estimate of the desired value Y, denoted y, is calculated based on statistical estimates of the values x1, x2, ..., xn: y = f(x1, x2, ..., xn). Sometimes it is preferable to calculate a statistical estimate of Y using the formula:
y = Y = 1/n Σnk=1Yk = 1/n Σnk=1f(X1,k, X2,k, ..., Xn,k)
Example calculation of uncertainty by type A
The difficulty of calculating type A uncertainty lies in the correct choice of the statistical analysis method, so, for example, a statistical estimate of variance can be obtained by the mathematical expectation formula, or calculated by approximating the distribution law to the normal distribution followed by by choosing a confidence interval.
Consider an example of measuring the diameter of a cylinder with a nominal diameter 19.45see using a micrometer.
| Measurement number | Measurement result |
| 1 | 19.472 |
| 2 | 19.206 |
| 3 | 19.404 |
| 4 | 19.624 |
| 5 | 19.212 |
| 6 | 19.378 |
| 7 | 19.562 |
| 8 | 19.585 |
| 9 | 19.275 |
| 10 | 19.314 |
| 11 | 19.650 |
| 12 | 19.399 |
| 13 | 19.237 |
| 14 | 19.668 |
| 15 | 19.627 |
| 16 | 19.386 |
| 17 | 19.320 |
| 18 | 19.528 |
| 19 | 19.205 |
| 20 | 19.689 |
| 21 | 19.482 |
| 22 | 19.263 |
| 23 | 19.268 |
| 24 | 19.678 |
| 25 | 19.609 |
| 26 | 19.394 |
| 27 | 19.268 |
| 28 | 19.287 |
| 29 | 19.407 |
| 30 | 19.445 |
| 31 | 19.688 |
| 32 | 19.586 |
| 33 | 19.597 |
| 34 | 19.651 |
| 35 | 19.559 |
| 36 | 19.603 |
| 37 | 19.432 |
| 38 | 19.603 |
| 39 | 19.544 |
| 40 | 19.261 |
| 41 | 19.442 |
| 42 | 19.533 |
| 43 | 19.536 |
| 44 | 19.313 |
| 45 | 19.392 |
| 46 | 19.368 |
| 47 | 19.488 |
| 48 | 19.350 |
| 49 | 19.500 |
| Table 1. Result of measuring the cylinder diameter with a micrometer | |
Statistical estimation of the average value 49 independent dimensionsmost easily defined as the arithmetic mean, according to the formula:
q = 1/n (Σnk=1qk)
q = (19.472 + 19.206 + ... + 19.5) / 49 = 19.455
Statistical estimation of the variance of the general population:
s2(qk) = 1/(n-1) Σnj=1(qj - q)2
s2(qk) = [(19.472 - 19.455)2 + (19.206 - 19.455)2 + ... + (19.5 - 19.455)2] / 48 = 0.022
We obtained a statistical estimate of the variance and the value of σ = √s2 is experimental the value of the standard deviation.
The best statistical estimate of the standard deviation of the mean is σ2(q) = σ2/n, which we will get using the standard error formula:
s2(q) = s2(qk)/n
s2(q) = 0.022 / 49 = 0.000449
This value, s2(q), describes the interval, in which the value μq is expected.
Thus, for the diameter value obtained as a result of 49 independent measurements, the uncertainty of type A of the mean value is u(q) = s(q):
uA(q) = 0.021190
Important!
This example is simple and cannot be used as a general case for searching for uncertainty type A in cases with complex measurement models. In many cases, the measurement result is a complex calibration model, for example, based on the least squares method. In such cases it is necessary to perform a statistical analysis of measurements. For quantities dependent on multiple variables, variance analysis is used (ANOVA).
Uncertainty of type A in excel
Download: Неопределенность_А.xls
The implementation in excel is very simple, only the SUM formulas and the ROOT are required here. The parameters are calculated as in the example above:
- Statistical estimation of the average value is the ratio of the sum of the results to their number
- Statistical estimation of the variance of the general population - according to the formula q = 1/n (Σnk=1qk)
- The standard deviation of the mean value, sq is the ratio of variance to the number of results minus one
- The standard uncertainty of type A is the root of the standard deviation of the mean
Measurement uncertainty of type B
Values Xi for which a statistical estimate was obtained not by measurement, but based on some scientific information, called uncertainty type B. The example of such information can serve as: data from previous measurements, experience, manufacturer's specification, calibration data, information from reference books and other sources of a priori values.
The correct definition of type B uncertainty is based only on experience and a general understanding of the process measurements. Uncertainty of type B can be as informative as uncertainty of type A exclusively in situations where type A uncertainty is based on a relatively small number of independent measurements.
Examples of type B uncertainty
Type B uncertainty is a general concept, so the number of examples can be unlimited, but the general idea is - this is an interval, for example, "A confidence interval with a confidence level of 82%", or "Uncertainty within three standard deviations".
Example 1. Uncertainty in standard deviations
The calibration certificate indicates that the actual mass value of the stainless steel sample, nominal weighing 1 kg, is equal to 1000,000325 g and "The uncertainty of the mass is 240 micrograms within three standard deviations."
Thus, the standard uncertainty is: u = 240 micrograms/3 = 80 micrograms. Expected variance: u2 = (80 mcg)2 = 6,4• 10-9 g2.
Example 2. Uncertainty in the confidence interval
The calibration certificate states that the resistance of the sample Rs, with a nominal resistance of 10 ohms, is equal to 10,000742 ohms ± 129 mcOm and the uncertainty of 129 mcOm covers the confidence interval with the level trust 99%.
Standard uncertainty u(Rs) = (129 mcOm)/2.58 = 50 mcOm (pro number 2.58 and confidence the interval is described in article). Relative uncertainty u(Rs)/Rs = 5,0 • 10-6. Expected variance: u2(Rs) = (50 mcOm)2 = 2,5 • 10 -9 Om2.
