1. User’s Guide to the ‘QDE Toolkit Pro’ National ResearchConseil national Council Canadade recherches Excel Tools for Presenting Metrological Comparisons by B.M. Wood, R.J. Douglas & A.G. Steele Chapter 6. Degrees of Freedom and Student Distributions This chapter reviews some basic information about Student distributions and effective degrees of freedom, and discusses how the default options are set by the QDE Toolkit Pro, and how easily it can cope with a participant who has chosen to express an effective degrees of freedom as is suggested in the ISO Guide to the expression of Uncertainty in Measurement. It also presents an improved method for determining the effective degrees of freedom, aimed specifically at the coverage factors. May 2, 2002 Ch 6: 83

2. QDE Toolkit Pro - Probability Density Functions (PDFs) The PDF of the measurand underlies the ideas of the standard uncertainty developed in the ISO Guide to the Expression of Uncertainty in Measurement. The QDE Toolkit Pro uses PDFs that are symmetric about their mean value, so it is straightforward to use the PDF of the measurand as the PDF of the (repeated) measurement. Left-right reflection of the PDF, and explicit repeatability extrapolation of the uncertainty are not addressed. QDE Toolkit Pro uses  Normal (or Gaussian) PDFs  Student PDFs with effective degrees of freedom Ch 6: 84

3. QDE Toolkit Pro - Normal and Student PDFs  Normal (or Gaussian) PDFs PDF(x) = a N exp(-(x-x 0 ) 2 /(2u 2 )), a N from  PDF(x) dx = 1  Student PDFs with effective degrees of freedom PDF(x) = a S [1+ ((x-x 0 )/u) 2 / ] -( +1)/2, a S from  PDF(x) dx = 1 where is used to parameterize the ‘excess tails’ of a PDF: in the limit as , the Student distribution converges to the Normal distribution. Note that does not have to be an integer. Ch 6: 85

4. QDE Toolkit Pro - Type B Estimates of for Student PDFs is used by QDE Toolkit Pro for its graphing of PDFs, and for calculating coverage factors. The basic thought is that u is not known perfectly, but is only estimated by u 0. For >10, the traditional Type B method is based on the variance of the chi distribution from its mean, and can harness your expert opinion on “how well u is known - within ±  u”:  0.5[u 0 /  u] 2 QDE Toolkit Pro function: nu_from_chi_variance. However, for smaller ’s, the (reduced) chi distribution is very asymmetric (either about its mean or about 1 - which is how it is used). An improved coupling of the Student distribution tails to your experience can be made through the inverse-chi distribution, using the related question “What  u would have u 0 +  u larger than 84% of the possible u’s?” ( 1-.84) = (1-.68)/2 If you, as the responsible metrologist, have an answer for this question, an improved Type B estimate for the effective degrees of freedom, aimed specifically at describing the tails of the most appropriate Student distribution (good for S >1), is S  0.5[u 0 /  u] 2 [1 + 3[  u/u 0 ] + 1.2[  u/u 0 ] 2 ] QDE Toolkit Pro function: nu_sub_S. Ch 6: 86

5. QDE Toolkit Pro - Improved Estimate for : the details The Student distribution is derived from the chi-square distribution of estimates u 0 2 for the unknown variance of the underlying normal curve. The variance of the reduced-chi distribution is used in the ISO Guide to check that  0.5[u 0 /  u] 2 Tails here make broad tails in the Student distribution Tails here make broad tails in the Student distribution For <10, asymmetry in the distribution casts doubt on this variance-based approach. To link to expert opinion, we use the inverse-reduced-chi distribution, and ask for expert insight about the value of [1+  u/u] needed to have 84% of the distribution below that value. Using inverse-chi distributions, the derivation of the Student distribution becomes intuitive and straightforward. Ch 6: 87

6. QDE Toolkit Pro - Improved Estimate for : the details The usual derivation of the Student distribution (independent normal and chi- square distributions) can be transformed, as shown here for =4, into a bivariate distribution of normal curves (width  u) having a marginal distribution that is an inverse- reduced-chi distribution. To choose the ‘best’ degrees of freedom from this family of Student distributions, based on an expert’s opinion about how well u is known, he or she needs only to express that opinion as a  u/u 0, such that u is expected to exceed (u 0 +  u) 16% (~1/6) of the time. (Easier than mental variance calculations!!) Normal Curves Student Distribution Inverse-chi Distribution  u/u 0 16% Ch 6: 88

7. QDE Toolkit Pro - Improved Estimate for : more details This graph shows the variance- based link between expert opinion and the effective degrees of freedom, referred to in the ISO Guide:  0.5[u 0 /  u] 2 [1] It also shows the corresponding limits of the proper inverse-chi distribution, for an interval with the same 68% confidence, where the low-u and high-u tails are each expected 16% of the time: these limits are labeled as CDF=0.16 and CDF=0.84. The upper curve also shows excellent agreement with the numerical fit S  0.5[u 0 /  u’] 2 [1 + 3[  u’/u 0 ] + 1.2[  u’/u 0 ] 2 ] [2] that is used in the QDE Toolkit Pro function : nu_sub_S, and its inverse Toolkit Pro function delta_u_by_u_from_nu_sub_S, to link to expert opinion about the limit on u that is expected to be exceeded only 16% of the time. Ch 6: 89

8. QDE Toolkit Pro - two definitions for ??? Ch 6: 90 Do these Two Equations Imply Two Definitions for Degrees of Freedom? No. In the context of determining coverage factors, the variance-based method is simply a bad approximation for degrees of freedom < 10, if the degrees of freedom is to be used for evaluating coverage factors from Student distributions. Nonetheless, we suggest using the symbol S for a degrees of freedom aimed at describing the tails of the Student distribution. Fortunately, in precision metrology, usually > 10 and there is no difficulty. Should we ever “translate” from one Equation to the Other? No, we think that in high-level Metrology it is almost never appropriate to “correct” another Lab’s opinion in this way. If they have used this bad approximation for, it is rather like them having over-estimated their uncertainty, and in the context of Key Comparisons they should expect to live with the consequences derived from any inaccurate estimates that they make.

9. QDE Toolkit Pro - Getting the Effective Degrees of Freedom Ch 6: 91 If you are convinced by the foregoing discussion, and want to apply the improved estimator for the degrees of freedom, then often only the evaluation of your own Lab’s degrees of freedom would be affected, and only occasionally would any changes be really significant. If you are the Pilot Lab, it could also affect your evaluation of the effective degrees of freedom for the travel uncertainty assigned to each Lab, and for other effects being evaluated by the Pilot Lab. Most often, the effective degrees of freedom will simply be an input parameter provided, along with their uncertainty, by some of the Labs. Sometimes these will have to be combined with other uncertainties, and their effective degrees of freedom, as decided by the Pilot Lab, adding the uncertainties in quadrature and combining the degrees of freedom using the Welch-Satterthwaite formula.

10. QDE Toolkit Pro - Using the Effective Degrees of Freedom Ch 6: 92 From this point on, the ‘QDE Toolkit Pro’ can really help to simplify the use of the effective degrees of freedom. There are two equivalent paths (red and blue): { i } can be edited. After the first Toolkit macro has run to completion, the { i } have the ‘normal’ default, but can be edited for re-running a Toolkit macro. { i } can be added initially, before any Toolkit macro is run. After the macro has run to completion, the input { i } are used with the default correlation coefficients. Ready for editing default correlation coefficients and running any Toolkit macros. Edit Run

11. QDE Toolkit Pro - infinite ?? Ch 6: 93 Normal Distributions - infinite The normal or gaussian distribution is formally a Student distribution with infinite degrees of freedom. This is the default option in the ISO Guide to the Expression of Uncertainty in Measurement: if someone has not told you about any departures from the normal distribution, then they have implicitly told you that it is normal. If there are significant departures from the normal distribution, this is their mistake for not telling you so! Uniform Distributions - infinite The ISO Guide to the Expression of Uncertainty in Measurement argues that the degrees of freedom for a uniform distribution should be infinite. In the context of a variance-based understanding of the degrees of freedom, this seems wrong: the uniform distribution is sometimes used to represent a possible range of values that might include a delta-function distribution as well as a double-delta-function of the maximum width (a “goalpost distribution”). With this variance a variance-based degrees of freedom could be determined. However, on our context of honing the degrees of freedom to help determine the most appropriate coverage factor, the bounded uniform distribution cannot produce large “tails”, and an infinite degrees of freedom seem more appropriate.