## 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

μ_{q}= ∫^{+∞}_{-∞}qp(q)dq

σ^{2}_{q}= ∫^{+∞}_{-∞}(q-μ_{q})^{2}p(q)dq

#### Statistical estimates

Statistical estimation of the average value of μ_{q}for n measurements under the same conditions:

q = 1/n Σ^{n}_{k=1}q_{k}(1)

Experimental variance - statistical estimation of variance σ^{2}:

s^{2}(q_{k}) = 1/(n-1) Σ^{n}_{j=1}(q_{j}- q)^{2}(2)

Statistical estimation of the variance of the mean value σ(q)^{2}= σ^{2}/n:

s^{2}(q) = s^{2}(q_{k})/n(3)

#### Uncertainty value

Uncertainty u(x_{i}) of statistical estimation of the average value of n measurements of the value X_{i}
equal to s(X_{i}) (formula 3).

*The degree of freedom v _{i} for the value u(x_{i}) equal to n-1 (n is the number of measurements of the value x_{i})
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 x_{1}, x_{2}, ..., x_{n}: y = f(x_{1}, x_{2}, ..., x_{n}).
Sometimes it is preferable to calculate a statistical estimate of Y using the formula:

y = Y = 1/n Σ^{n}_{k=1}Y_{k}= 1/n Σ^{n}_{k=1}f(X_{1,k}, X_{2,k}, ..., X_{n,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 23.45see using a micrometer.

Measurement number | Measurement result |

1 | 23.334 |

2 | 23.491 |

3 | 23.748 |

4 | 23.594 |

5 | 23.608 |

6 | 23.695 |

7 | 23.470 |

8 | 23.558 |

9 | 23.296 |

10 | 23.476 |

11 | 23.452 |

12 | 23.537 |

13 | 23.649 |

14 | 23.280 |

15 | 23.237 |

16 | 23.300 |

17 | 23.708 |

18 | 23.536 |

19 | 23.659 |

20 | 23.282 |

21 | 23.133 |

22 | 23.622 |

23 | 23.678 |

24 | 23.728 |

25 | 23.136 |

26 | 23.758 |

27 | 23.115 |

28 | 23.733 |

29 | 23.626 |

30 | 23.590 |

31 | 23.440 |

32 | 23.759 |

33 | 23.266 |

34 | 23.717 |

35 | 23.424 |

36 | 23.363 |

37 | 23.428 |

38 | 23.314 |

39 | 23.346 |

40 | 23.711 |

41 | 23.539 |

42 | 23.674 |

Table 1. Result of measuring the cylinder diameter with a micrometer |

Statistical estimation of the average value 42 independent dimensionsmost easily defined as the arithmetic mean, according to the formula:

q = 1/n (Σ^{n}_{k=1}q_{k})

q = (23.334 + 23.491 + ... + 23.674) / 42 =23.500

Statistical estimation of the variance of the general population:

s^{2}(q_{k}) = 1/(n-1) Σ^{n}_{j=1}(q_{j}- q)^{2}

s^{2}(q_{k}) = [(23.334 - 23.500)^{2}+ (23.491 - 23.500)^{2}+ ... + (23.674 - 23.500)^{2}] / 41 =0.036

We obtained a statistical estimate of the variance and the value of σ = √s^{2} is experimental
the value of the standard deviation.

The best statistical estimate *of the standard deviation of the mean* is
σ^{2}(q) = σ^{2}/n,
which we will get using the standard error formula:

s^{2}(q) = s^{2}(q_{k})/n

s^{2}(q) = 0.036 / 42 =0.000857

This value, s^{2}(q), describes the interval,
in which the value μ_{q} is expected.

Thus, for the diameter value obtained as a result of 42 independent measurements, the uncertainty of type A of the mean value is u(q) = s(q):

u_{A}(q) = 0.029275

#### 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 (Σ
^{n}_{k=1}q_{k}) - The standard deviation of the mean value, s
_{q}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 X_{i} 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: u^{2} =
(80 mcg)^{2} = 6,4• 10^{-9} g^{2}.

#### Example 2. Uncertainty in the confidence interval

The calibration certificate states that the resistance of the sample R_{s}, 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(R_{s}) = (129 mcOm)/2.58 = 50 mcOm (pro number 2.58 and confidence
the interval is described in article). Relative uncertainty
u(R_{s})/R_{s} = 5,0 • 10^{-6}. Expected variance: u^{2}(R_{s})
= (50 mcOm)^{2} = 2,5 • 10 ^{-9} Om^{2}.