If σ is not known, use the computed standard deviation s based on k measurements as an estimate of σ, where s2 = 1 Σ (;-). The number (k-1) is the degrees of freedom associated with s. k-1 i=1 2 3 For interpretation see Chapter 1, NBS Handbook 91, Experimental Statistics, by M. G. Natrella, 1963. For details see Chapter 5 (straight line), and Chapter 6 (multivariate and polynomial), NBS Handbook 91. * For details see "Notes on the use of propagation of error formulas", by Harry H. Ku, NBS Journal of Research, Vol. 70C, No. 4, October-December, 1966. See "Realistic Evaluation of the Precision and Accuracy of Instrument Calibration Systems" by Churchill Eisenhart, NBS Journal of research, Vol. 67C, No. 2, April-June, 1963, and "Systematic Errors in Physical Constants" by W. J. Youden, Physics Today 14, 1961. FIGURE 1 SUMMARY OF RECOMMENDATIONS ON EXPRESSIONS OF THE UNCERTAINTIES OF FINAL RESULTS SYSTEMATIC ERROR AND IMPRECISION BOTH NEGLIGIBLE (CASE 1) In this case, the reported result should be given correct to the number of significant figures consistent with the accuracy requirements of the situation, together with an explicit statement of its accuracy or correctness. SYSTEMATIC ERROR NOT (a) Qualification of a reported result should be limited to a single quasi-absolute type of statement that places bounds on its inaccuracy; (b) These bounds should be stated to no more than two significant figures; (c) The reported result itself should be given (i.e., rounded) to the last place affected by the stated bounds, unless it is desired to indicate and preserve such relative accuracy or precision of a higher order that the result may possess for certain particular uses; (d) Accuracy statements should be given in sentence form in all cases, except when a number of results of different accuracies are presented, e.g., in tabular arrangement. If it is necessary or desirable to indicate the respective accuracies of a number of results, the results should be given in the form a + b (or ab, if necessary) with an appropriate explanatory remark (as a footnote to the table, or incorporated in the accompanying test) to the effect that the + b -c' + b, or signify bounds to the errors which the a's may be subject. (e) The fact that the imprecision is negligible should be stated explicity. NEITHER SYSTEMATIC ERROR NOR (a) A reported result should be qualified by: (1) a quasi-absolute type of statement that places bounds on its systematic error; and, (2) a separate statement of its standard error or its probable error or of an upper bound thereto, whenever a reliable determination of such value or bound is available - otherwise, a computed value of the standard error or probable error so designated should be given, together with a statement of a number of degrees of freedom on which it is based; (b) The bounds to its systematic error and the measure of its imprecision should be stated to no more than two significant figures; (c) The reported result itself should be stated, at most, to the last place affected by the finer of the two qualifying statements, unless it is desired to indicate and preserve such relative accuracy or precision of a higher order that the result may possess for certain particular uses; (d) The qualification of a reported result, with respect to its imprecision and systematic error, should be given in sentence form, except when results of different precision or with different bounds to their systematic errors are presented in tabular arrangement. If it is necessary or desirable to indicate their respective imprecisions or bounds to their respective systematic errors, such information may be given in a parallel column or columns, with appropriate identification. SYSTEMATIC ERROR NEGLIGIBLE, (a) Qualification of a reported value should be limited to a statement of its standard error or of an upper bound thereto, whenever a reliable determination of such value or bound is available. Otherwise, a computed value of the standard error so designated should be given, together with a statement of the number of degrees of freedom on which it is based; (b) The standard error, or upper bound thereto, should be stated to not more than two significant figures; (c) The reported result itself should be stated, at most, to the last place affected by the stated value or bound to its imprecision, unless it is desired to indicate and preserve such relative precision of a higher order that the result may possess for certain particular uses; (d) The qualification of a reported result with respect to its imprecision should be given in sentence form, except when results of different precision are presented in tabular arrangement and it is necessary or desirable to indicate their respective imprecisions, in which event such information may be given in a parallel column or columns, with appropriate identification. (e) The fact that the systematic error is negligible should be stated explicitly. 2. Design of Experiments in Calibration Papers 2.1. General considerations in planning experiments. Natrella, Mary G. .... 2.2 New experimental designs for paired observations. Youden, W. J., and Connor, W. S. ... 2.3 Design and statistical procedures for the evaluation of an auto Page 81 86 matic gamma-ray point-source calibration. Garfinkel, S., Mann, W. B. and Youden, W. J. 92 2.4 Instrumental drift. Youden, W. J. 103 2.5 Comparison of four national radium standards (Part 2). Connor, W. S., and Youden, W. J. 108 2.6 Physical measurements and experiment design. Youden, W. J. ... 117 Foreword Statistical design deals with the scheduling and the orderly arrangement of the sequence of observations in an experiment. Since each experiment is an individual undertaking, so is its design. Some basic cosiderations, however, are applicable to almost all experiments. These principles are summarized in Chapter 11 of Handbook 91, reprinted here as the first paper (2.1) in this section. In Physical Measurements and Experiment Design (2.6), Youden highlighted the shift of emphasis from the classical designs for agricultural experimentation to that for physical experimentation. He argues that the designs should take advantage of the special features that are characteristic of the class of problems in physical sciences, and gives several examples illustrating his point. His other three papers (2.2, 2.3, 2.5) are coauthored with scientists in various areas of the Bureau, and illustrate the need for tailoring the design to the particular experiment. A variety of other examples are also given in Statistical Design (Selected Reference D2), which is a collection of his bimonthly articles from Industrial and Engineering Chemistry. In calibration work, it is not uncommon for different laboratories to use the same method of comparison for the same type of standards, and hence to use the same type of design. Design for the comparison of groups of standard cells are given in NBS Technical Note 430 (abstracted in 7.2). Current designs used in the comparison of mass standards are illustrated in Technical Note 288 (abstracted in 7.1). It is expected that more of this "standard type" of calibration designs for physical quantities that are routinely measured in Standards Laboratories will be published in the form of Technical Notes. One such publication in preparation is that for the series of mass standards. |