# Analyzing inconsistent measurement data

## New procedure provides improved, more robust results with more reliable uncertainties

PTB-News 2.2015
10.06.2015
Especially interesting for

metrology

statistics

At PTB, a new statistical method has been developed to adjust inconsistent measurement data. Its advantages: more reliable results and a more realistic evaluation of the associated uncertainties. The procedure can, for example, be applied to determine fundamental constants.

Data measured for the Planck constant (left of the dashed line). The error bars show standard uncertainties. As a result of the new adjustment method, the probability distribution for the value of the Planck constant is shown, together with the derived estimate and the corresponding standard uncertainty (right of the dashed line).

Adjusting inconsistent measurement data plays an important role in metrology, for example, to determine reference values when analyzing interlaboratory comparison measurements or when determining values for fundamental constants. The figure shows the example of current values measured for the Planck constant. The data are inconsistent with respect to the quoted uncertainties. Sim-ple application of the weighted mean of the measured values would not be advisable, since this could lead to unreliable results. In such cases, metrology often resorts to the so-called “Birge ratio” method, in which all measurement uncertain-ties are enlarged in the same way so that consistency of the measurement data is achieved. This procedure is, however, only admissible on the basis of a very specific statistical model. If this – very restrictive – assumption is, however, violated, the Birge ratio method might yield unreliable results.

A new procedure based on Bayesian statistics has been developed at PTB; it allows inconsistent measurement data to be adjusted under c on sid er a bly weaker assumptions about the underlying statistical model. The procedure is based on a more general class of statistical models the so-called elliptically contoured distributions. It was possible to demonstrate that the results obtained with the new procedure are better and more robust than those obtained with alternative procedures and, especially, that the uncertainties determined by the new method are more reliable. The figure illustrates the application of the method. The result obtained is a probability distribution for the value of the Planck constant, from which an estimate and its associated uncertainty can be determined.

### Contact:

Olha Bodnar
Department 8.4 Mathematical Modelling and Data Analysis
Phone: +49 (0)30 3481-7414
E-mail: olha.bodnar(at)ptb.de

### Scientific publication:

O. Bodnar, A. Link, C. Elster: Objective Bayesian inference for a generalized marginal random effects model. Bayesian Analysis,
doi: 10.1214/14-BA933 (2015)