Effective degrees of freedom

Source of the problem

The result of the measurement analysis is recorded as the value of the measurement result and uncertainty. Uncertainty can be presented in a standard and expanded form. The extended uncertainty gives an explicit coverage of the values, which facilitates its subsequent interpretation, the use of frequency analysis or subjective evaluation of the result is no longer relevant.

Extended uncertainty corresponds to some level of trust. To build this level of confidence, we use the Student's distribution calculated using a standard distribution and a coefficient selected depending on the variables of the measurement model that affect the total uncertainty. For the chi-square distribution, the coefficient is chosen based on the Student's distribution with the number of degrees of freedom of the chi-square distribution.

The variable does not always have a chi-squared distribution, and therefore the statistical estimate has a distribution other than the Student's distribution. This burden does not allow us to calculate the confidence level for the measurement result, and as a result, calculate the exact value of the extended uncertainty.

In the description of the method for calculating the extended uncertainty according to the iso 98-1 norm, any distribution approaches the Student's distribution, while the number of degrees of freedom is calculated by the formula Welch-Satterthwaite. Such a number of degrees of freedom is called effective degrees of freedom.

Formula

ν_eff = u_c⁴(y) / Σⁿ_i=1u_i⁴(y)/ν_i
If u(x_i) is an uncertainty of type B, then, as a rule, the number of degrees of freedom tends to infinity, otherwise, the number of degrees of freedom for n dimensions is calculated using the following algorithm: if the statistical estimate The average is calculated using the arithmetic mean formula, then v = n - 1, if the statistical estimate is determined by the least squares method using m independent factors, then v = n - m.

You can see an example of using the Welch-Satterthwaite formula on the page example of uncertainty calculation.