Estimating measurement uncertainties

The number of degrees of freedom ν is equal to in the case of the direct measurement of a quantity estimated by the arithmetic mean of N independent observations. If these N observations are used to determine the slope observations are used to determine the slope of a straight line by the least-squares method (case of a calibration straight line such as ), the number of degrees of freedom respectively associated with the standard uncertainties and is . For an adjustment by the least-squares method of p parameters for N data, the number of degrees of freedom of the standard uncertainty of each parameter is . Table 4 sums up the different enumerated cases.

Tableau 4 : Degrés de liberté et niveaux de confiance
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Table 5 shows a selection of values of for different values of the confidence level and of the number of degrees of freedom ν.

Table 5
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When ν → ∞, Student's t-distribution tends towards normal law and where k is the necessary coverage factor to obtain a confidence interval of level for a normally distributed variable. Thus in table 5 the value of for a given level of confidence is equal to the value of k for the same value of in table 3. From the expression of , we can also evaluate the number of degrees of freedom νL for which enabling us to evaluate the number of replicate measurements beyond which Student' s t-distribution is less than 10% from a normal law, that is to say :

hence

that is to say

Therefore, at least 10 replicate measurements are needed to approach with a 10% accuracy a normal law describing the distribution of the values of the measured quantity around its average value.

Student's t-distribution does not generally describe the law of the variable if u2 c (y) is the sum of several components of variance estimated even if each xi is the estimate of a normally distributed input quality Xi . However it is still possible to use Student's t-distribution with an effective number of degrees of freedom νeff obtained by the Welch-Satterthwaite formula [3a][3b][4] .

where ν i is the number of degrees of freedom of each component of the combined standard uncertainty uc(y) for . For a component obtained from a Type A evaluation, ν i is associated with the number of independent replicate observations of the corresponding input quantity and with the number of parameters determined from these observations (cf. table 4). For a component obtained from a Type B evaluation, ν i is evaluated from the reliability that can be given to the value of this component following the expression

where is the relative uncertainty of u(xi). It is a subjective quantity which value is obtained from a scientific judgement based on all the available information. Note that if u(xi) can be considered as exactly known then νi→∞.