publications

NBSIR 81-2426

NUMERICAL COMPARISONS OF SEVERAL

ALGORITHMS FOR TREATING

INCONSISTENT DATA IN A

LEAST-SQUARES ADJUSTMENT OF

THE FUNDAMENTAL CONSTANTS

NAT'L INST. OF STAND & TECH

AlllDt.

B. N. Taylor

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Absolute Physical Quantities

Electrical Measurements and Standards Division

Washington, DC 20234

Issued January 1982

Final Report

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 81-2426

<«• NUMERICAL COMPARISONS OF SEVERAL

ALGORITHMS FOR TREATING

INCONSISTENT DATA IN A

LEAST-SQUARES ADJUSTMENT OF

THE FUNDAMENTAL CONSTANTS

HATIOIIAI- BUP.EAU OF STANDARDS UBRARY

JUN 14 1983

f h'JTa •

B. N. Taylor

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Absolute Physical Quantities

Electrical Measurements and Standards Division

Washington, DC 20234

Issued January 1982

Final Report

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director

TABLE OF CONTENTS

Page

ABSTRACT i i i

1. INTRODUCTION 1

2. SUMMARY OF ALGORITHMS 4

2.1 Traditional Approaches 4

2.1.1 Birge Ratio Algorithm 4

2.1.2 Two-Stage Birge Ratio Algorithm 5

2.2 The VNIIM Algorithm 6

2.3 Generalizations of the VNIIM Algorithm 9

2.3.1 Inverse Algorithm 10

2.3.2 Natural Log Algorithm 11

2.3.3 Geometrical Mean Algorithm 11

• 2.3.4- Simple Mean Algorithm 12

2.4 Algorithms in Which Random and Systematic Uncertainties

Are Separated 13

2.4.1 Internal Birge Ratio Algorithm 14

2.4.2 VNIIM Algorithm for Systematic Uncertainties .... 15

2.4.3 VNIIM Algorithm with Weights 17

2.5 The Extended Least- Squares Algorithm 18

3. NUMERICAL RESULTS AND DISCUSSION 23

3.1 Summary of Input Data 23

3.1.1 1973 Adjustment Data 23

3.1.2 1969 Adjustment Data 26

3.1.3 1963 Adjustment Data 26

3.2 Results and Discussion 27

3.2.1 Table 5 28

3.2.2 Table 6 30

i
TABLES OF CONTENTS (Continued)

Page

3.2.3 Table 7 33

3.2.4 Table 8 35

3.2.5 Tables 9 and 10 35

3.2.6 Tables 11 and 12 37

3.2.7 Table 13 38

4. CONCLUSIONS 40

5. ACKNOWLEDGEMENTS 42

6. FOOTNOTES 43

7. REFERENCES 46

3. TABLES 49

9. FIGURE CAPTIONS AND FIGURES 69

1 i
NUMERICAL COMPARISONS OF SEVERAL ALGORITHMS FOR TREATING

INCONSISTENT DATA IN A LEAST- SQUARES ADJUSTMENT

OF THE FUNDAMENTAL CONSTANTS

B. N. Taylor

Electrical Measurements and Standards Division

Center for Absolute Physical Quantities

National Bureau of Standards

Washington, D. C. 20234

ABSTRACT

A number of recently proposed algorithms for treating inconsistent or

discrepant data in a least-squares adjustment of the fundamental physical

constants, along with several new but related algorithms, are compared in

detail. The comparisons are first made by means of the numerical results

the algorithms yield when applied to the same data considered by Cohen

and Taylor in their 1973 adjustment which led to the recommended set of constants adopted by CODATA and in current use. A selected number of the

algorithms are then further compared through the numerical results they

yield when applied to the data considered by Taylor, Parker, and Langenberg

in their 1969 adjustment and by Cohen and DuMond in their 1963 adjustment.

The principal conclusion of this paper is that the actual algorithm used to

carry out an adjustment is much less important than the data finally selected

for inclusion in the adjustment.

Key Words: Data analysis; discrepant data; fundamental constants; inconsistent

data; least-squares adjustments; physical constants.

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1 . INTRODUCTION

In the three most recent least-squares adjustments of the fundamental

physical constants, those carried out by Cohen and DuMond in 1963 [1],* Taylor,

Parker, and Langenberg in 1969 [2], and Cohen and Taylor in 1973 [3], the input

data were considerably censored because of large discrepancies among the

data. For example, in the 1963 adjustment the value of the fine-structure

constant a, determined from the hydrogen ground-state hyperfine splitting,

was discarded; in the 1969 adjustment two "high values" of the magnetic

moment of the proton in H2 O in units of the nuclear magneton were

eliminated; and in the 1973 adjustment two values of the Faraday constant

F, were censored. While in each case there was some additional supporting

evidence for discarding the suspect data, the principal justification was

their significant disagreement with the remaining data.

Not surprisingly, "Murphy's Law" was apparently operating in each of

these three adjustments — subsequent measurements indicated that it was

highly likely that the wrong data had been discarded. For example, the value

of Of derived from Parker, Taylor, and Langenberg' s ac Josephson effect determination of 2e/h in 1967 showed that the hydrogen hyperfine splitting value

of a discarded by Cohen and DuMond in their 1963 adjustment was more nearly

correct than the value of of they retained which had been derived from the

deuterium fine-structure measurements of Lamb and coworkers. The two values

of Pp'/pj^ discarded by Taylor and colleagues in their 1969 adjustment were

subsequently shown to be more nearly correct than the three values they

retained by the highly consistent, sub-part-per-mil 1 ion (ppm) determinations

of Pet ley and Morris and of Mamyrin and coworkers. And the two values of F

eliminated by Cohen and Taylor in their 1973 adjustment have received signif-

*Footnotes begin on p. 43 and literature references begin on p. 46.
leant support from the recent high precision, coulometric redetermination of

F by Bower and Davis; and from Kibble and Hunt's measurement of the proton

gyromagnetic ratio in H2 O by the so-called high field method. (See Refs. [1-6]

for further details.)

Censoring the wrong data in the 1963 and 1969 adjustments has led to large

changes (in comparison with their assigned uncertainties) in the recommended

values of several constants from one adjustment to another. For example, the

1963 recommended value of a ^ was 137.0388(6) (4.6 ppm) while the 1969 recommended value was 137.03602(21) (1.5 ppm), a decrease of 20 ppm or over four

combined standard deviations;^ the 1969 recommended value of u '/u., was

p N

2.792709(17) (6.2 ppm) while the 1973 (and present) recommended value is

2.7927740(11) (0.38 ppm), an increase of 23 ppm or nearly four combined

standard deviations; and because of the situation with the Faraday constant

discussed above, similar significant changes in the 1973 recommended values

of several constants are likely to result with the completion of the 1982

adjustment which is currently being carried out by Cohen and Taylor under

the auspices of the CODATA Task Group on Fundamental Constants.

The author has argued [7] that such changes are really not as serious

as they might first appear and that the insight gained during the course of

the critical review which necessarily accompanies a least-squares adjustment

of the constants is much more important than the recommended values themselves.

However, others, in particular a group at the Mendeleyev Institute of Metrology

(VNIIM) U.S.S.R. , are very much disturbed by such large changes in the recommended values. Moreover, this group feels that such large variations can be

significarrtly reduced and values closer to the true (but unknown) values obtained by including all of the data available at a given epoch in the adjustment, even if some are discrepant. To this end, the VNIIM group [8-12] has
3

developed a specific algorithm for including discrepant data in a least squares

adjustment and has applied it to modified versions of the 1963, 1969, and

1973 data as well as to more recent data. They conclude that in general, the

recommended values of the constants from one adjustment to another are in better

agreement if all of the data available at the time of each adjustment are

included using the new algorithm, and that the values are closer to the true

values.

The theoretical basis for the VNIIM algorithm is not well founded. In

an attempt to place the handling of discrepant data in a least-squares adjustment on firmer theoretical ground, E.R. Cohen [13-17] has examined the proposed algorithm, has developed a number of alternatives, and has compared

several of them with the VNIIM algorithm by applying them to a modification

of the 1973 data.

Although the papers of both the VNIIM group and Cohen have included

numerical examples, these have not been given in any great detail nor have

they been particularly extensive. Since it is likely that the 1982 adjustment will be carried out using one of the proposed algorithms or an appropriate modification thereof, the author feels it is important that they

become more widely known and that the detailed results of their application

to the uncensored and unmodified data of past adjustments be made available

for critical review. Thus, it is the purpose of this paper first to compare

these various algorithms, as well as several new but closely related algorithms

developed by the author, through the detailed numerical results they yield

when applied to the same data considered by Cohen and Taylor in their 1973

adjustment; and then to compare further a selected number of the algorithms

through the numerical results they yield when applied to the data considered

by Taylor, Parker, and Langenberg in 1969 and by Cohen and OuMond in 1963.
2. SUMMARY OF ALGORITHMS

We summarize here the various algorithms for treating discrepant data

which will be compared in detail in the present paper. ^ However, it should be

borne in mind throughout this paper that the underlying reason for trying to

devise such algorithms is the belief that if discrepant data are included in

an adjustment, and it turns out at a later date that these data are in fact

reliable, then the changes in the recommended values of the constants from one

adjustment to the next will be less than if the discrepant data had been discarded. More importantly, the recommended values should be closer to the true

values. Although the discussion in Section 1 implies that this might possibly

have been the case with the last three adjustments, it would definitely not

have been the case if the unreliable nature of the discarded data had in fact

been confirmed by subsequent experiments. If this had occurred, then including

the discrepant data would also have led to disturbing variations in the constants from one adjustment to the next and shifts from their true values. In

other words, including discrepant data will reduce such variations and shifts

if it later turns out that the originally discrepant data are reliable, but

will contribute to such variations and shifts if it later turns out that the

discrepant data are in fact unreliable. (Guess right and you win, guess wrong

and you loose!

)

2.1. Traditional Approaches

2.1.1. Birqe Ratio Algorithm . One of the simplest ways of objectively treating discrepant data in a least-squares adjustment, and one which sees con2 1 /2

siderable use in data analysis, involves the Birge ratio Rg = (x /F) , where

is the familiar statistic chi-squared for the adjustment and F is the number

of degrees of freedom: F = N - M where N is the number of items of stochastic
- 5

-

input data and M the number of unknowns or adjustable constants. In this

approach, one first carries out an adjustment using all of the available data

2 with their a priori assigned uncertainties a^. (weights w. = ^/G. ), computes

Rg, and then carries out a second adjustment with new uncertainties a.' = RgO^.

2 2 (weights w.‘ = 1 /Rq cr. ). Since this leads to a value of chi-squared for the

I D I

' 2 second adjustment equal to its statistically expected value x = F, it may be

viewed as being statistically consistent.**’®

While the adjusted values of the unknowns are the same in the two adjustments because each a priori uncertainty in the second adjustment is changed by

the same multiplicative factor Rg, the uncertainties of the adjusted values of

the unknowns in the two adjustments differ by this multiplicative factor. For

the discrepant data case (Rg > 1), the increase in uncertainty of the adjusted

values is a direct result of the inconsistencies among the data with the larger

uncertainties better reflecting the true reliability of the adjusted values of

the unknowns. (This approach may also be described as calculating the uncertainties on the basis of external consistency; see Refs. [1-3].)

Although the Birge ratio method of handling discrepant data is simple and

objective, it inappropriately treats the uncertainties assigned to all of the

2 stochastic input data as equally reliable and all values of (a. Vo^ within

some range as equally probable. This leads to the undesirable result of having

the a priori uncertainty of a highly consistent input datum (residual much

less than unity — see Footnote 1) with possibly a well defined uncertainty

increased by the same factor as a highly discrepant datum (residual much greater

than unity) with perhaps a poorly defined uncertainty.

2.1.2. Two-Stage Birge Ratio Algorithm. There is an obvious generalization of

the above approach which was used in part in the 1973 adjustment and which
partially responds to this last difficulty. It involves using the Birge ratio

in two distinct stages. First, the weighted average of each subgroup of

stochastic input data of the same kind (e.g. , all the values of a) is computed

and the uncertainty of each weighted average expanded by its Birge ratio if

greater than unity. Then, these individual weighted averages with their uncertainties (expanded or not as the case may be) are used as input data in an

adjustment to obtain the adjusted values of the unknowns. Rg this

second adjustment is greater than unity, then it is repeated with the uncertainties of the individual weighted averages expanded by this Birge ratio. Thus,

each data subset is first made internally consistent, and then the weighted

averages of the different subsets are made consistent with each other.

While this approach is also simple and objective, and to some extent

reduces the influence of a discrepant input datum on the bulk of the more reliable data, it does have the drawback of possibly giving greater weight in

the adjustment to the indirect values of a particular quantity than to the

directly measured values. That is, the first step of the procedure decreases

the weight that the directly measured values would have had in determining the

adjusted values of the unknowns (assuming Rg for the weighted average is

greater than unity), thus allowing the indirect values to have a greater

influence in determining the unknowns. (See Refs. [1-3] for a further

discussion of direct and indirect values.)

2.2. The VNIIM Algorithm

The VNIIM group [8-12] suggests altering the usual least-squares procedure

by changing the a priori uncertainties a. assigned each input datum to new

values a.' such that the sum
7

S = I (R,^ - 1)2 (1)

1=1

2 2

is minimized, where R. = (cr^. Va^ . This minimization is to be carried out

' 2 subject to the constraint that the value of the statistic x calculated for

the adjustment employing the new a.' is equal to its expected value F = N - M,

the degrees of freedom of the adjustment. The adjusted values of the

unknowns X^‘ (of = 1...M), are those which minimize the usual least-squares

expression

Q' = X [A. - S,(X„)]2/a,'2 = £ [r,'(X)]2 . (2)

i=l

1 1 1 1 Of

where A. is the numerical value of the i^*^ input datum and A.(X ) is A.

expressed in terms of the unknowns or adjustable constants X^. The sum of the

squared normalized residuals r.' evaluated with X = X

' is the statistic chisquared for the adjustment and thus we have

= F =

NI

i=l

[r.'(X ')]'

L

^ 'Of (3)

The values of X^' and a.' are found, therefore, by simultaneously minimizing

Eqs. (1) and (2) subject to Q'(min) = F, which is Eq. (3).

The simultaneous minimization of Eqs. (1) and (2) may be carried out by

combining them into a single relation through a Lagrange multiplier p, and

minimizing

NI

i=l

Q'* = m(r7 - 1)2] (4)
Since the term multiplying p in Eq. (4) is independent of the the leastsquares solution of Eq. (4) obtained by setting 3Q'*/9X^ = 0 is the same

as the usual least-squares solution but carried out using the revised uncertainties a.'. The latter may be obtained by setting 8Q'*/3ct.

= 0 and using

Eq. (3) to eliminate p. The result is

N

4 2 2 ' 2 2 2

R. (R/ ’ 1) = R/(r. VF) I r/(r/ - IT 1)

j=l J J (5)

In practice, Eq. (5) may be solved iteratively by, for example, first

carrying out an adjustment using the original a. and taking as the first

iterate R.loo = R-,, where Rq is the Birge ratio. These values of R. are then

1 substituted into the right-hand side of Eq. (5) with r

.

' replaced by r . , and

the new values of R. to be used in the next iteration determined by solving

Eq. (5) with the R^. on the left-hand side taken as unknowns.® The procedure

is then continued until

j x

- F is sufficiently small, say 10 , and/or the

R. and X^‘ change by negligible amounts from one iteration to the next.

As noted earlier, a firm theoretical basis for Eq. (1) is yet to be

given. However, the VNIIM group has justified it on the following grounds:

2

One may introduce a probability density function <|).(R. ) on the assumption

2

that the R. are random quantities each with average value equal to unity and

with the same variance a

2

. This means there is a certain... probability that the

a priori assigned uncertainty a. of each stochastic input datum will differ

from its true value and that all of the a priori assigned uncertainties are

equally reliable. The values of the R. which give a consistent adjustment

can then be found from the equation

^ 2 n (l).(R. ) = maximum

1=1 '

' ( 6 )
- 9

-

' 2 with the requirement that x = F. The choice of the functions (}). in

the form of a uniform distribution leads to the solution R. = R„ for all i

1 D (the Birge ratio algorithm of Section 2.1.1), while the choice of the (t>. in

the form of a normal distribution leads to Eqs. (1) and (5). The uniform

2 distribution corresponds to assuming that all values of R. (within a specified

range) are equally probable, while the normal distribution corresponds to

2 assuming that the more R. differs from unity, the less probable is that value

of R.^.’’

1 In summary, the algorithm proposed by the VNIIM group replaces the

somewhat subjective approach of completely discarding discrepant data with

an objective technique which retains all of the data but alters each of

the a priori assigned uncertainties in accordance with what Cohen [13] terms

a "cost function", that is, Eq. (1). This cost function, in combination with

' 2 the requirement x = leads to larger values of cr^-'/a. for the discrepant

data than for the consistent data, thereby reducing but not entirely eliminating

their influence on the adjusted values of the unknowns. However, the VNIIM

algorithm still suffers from the improbable assumption that all of the a priori

assigned uncertainties are equally reliable. One well knows that the uncertainties in some experiments are much easier to evaluate than in others, that

some workers carry out their experiments with greater care than others, etc.

2-3. Generalizations of the VNIIM Algorithm

In order to better understand Eq. (1) Cohen [13] has generalized it to

S = I W. f.(w.,w. ') = minimum , (7)

i

where the a priori weights w. and new weights w^. ' are related to the uncer2 ' 2

tainties and ' in the usual way, that is, w^. = 1/a. and w. ' = 1/a^. ;
10

and W. is a weighting factor which may be different for each cost function

component f , but which we presently assume to be unity because a completely

objective way of arriving at such weights is not at all cl ear. ^ The only

restrictions needed on the f. are that they be a positive definite function of

w

.

' such that f^. - 0 for w. ' = w. and f^. > 0 for w. ' ^ w..

In terms of weights, the VNIIM group's proposed cost function, Eg. (1),

may be written as

S = Z

N

(Aw./w.')^=

2^22

I (r/ - 1)^ , (8)

i=l

^

^ i=l

^

2 ^ where Aw. = w. - w.' and as before R. Ill =

1 ^ 1 K

2-3.1. Inverse Algorithm. Cohen notes [13] that an expression similar to

Eg. (8) and perhaps egually valid is

N

2

^ 2 2 S= I (Aw./w.)^= I (1 /r/ - 1)^ , (9)

i=l

^

^ i=l

^

which we term the "inverse" of the VNIIM algorithm because of the way

2

R. enters. Eguations (8) and (9) are approximately egual if w. ' and w.

are approximately egual, but differ significantly if w. ' and w. differ

' 2 significantly. Proceeding as before with the constraint x = F. one can

show that the expression corresponding to Eg. (5) for this cost function is

1 - 1/R-^ = R.^(r.'^/F) I (VR.^ - . (10)

1 1 T j=l J J

A major difficulty with Eg. (9) is that it leads to meaningless negative

values of w.' for highly discrepant items of stochastic input data because

it is finite for w. ' =0 (R- = “). For such data, one must impose the
11 -

2 additional constraint w.' > 0 (0 < R. < »). The end result is that this

1 - 1 -

algorithm essentially discards highly discrepant items of input data by

2 diminishing their weights to negligible values (R. » 1). Since these items

are not available to influence the adjusted values of the unknowns, some of

the latter can differ markedly from the values which result from the VNIIM

algorithm and other algorithms which do not require the additional constraint

w^. ' > 0. (The numerical examples in Section 3 will clearly demonstrate this.)

Thus, the inverse algorithm would not seem to completely satisfy the underlying

reason for devising such algorithms as was discussed at the start of Section 2.

2.3.2. Natural Log Algorithm . Cohen also suggests a cost function intermediate

between Eqs. (8) and (9):

N . N

^ ^

S = Z 2n (w./w.')= Z R. , (11)

• i=l

^

^ i=l

^

which leads to

R.^ 2n R.111 = R.^(r.'^/F) Z £nR. . (12)

j=l ^

He points out that this expression has several advantages. First, it is invariant as to whether S is considered to be a function of the weights w. or

uncertainties a^. . Second, the penalty is exactly the same whether R^. = p

or R^. = 1/p, for example, whether ct. is doubled or cut in half. This desirable property is approximately true for Eqs. (7) and (8) only when R. is very

near unity.

2.3.3. Geometrical Mean Algorithm . It occurred to us that the geometrical

1 /2 mean (w.w^.') could have been taken as the "normalization" of Aw^. in Eqs.
(8) and (9) (i.e. , the denominators in the expression for S) in place of

either w^. ' or w. which lead to the VNIIM and inverse algorithms, respectively. The resulting cost function would be

S =

N

I

i=l

(Awp Vw.w.' =

1 1

N

I

i=l

(R. - dVrT (13)

with

R* - 1 = rT(p 'V) I (R.'^ - 1)/r/ . (U)

j=l J J

Comparing Eqs. (13) and (14) with Eqs. (8) and (5) leads to the conclusion

that the geometrical mean and VNIIM algorithms should yield similar results.

2.3.4. Simple Mean Algorithm . If the normalization of Aw^. is taken to be the

simple mean (w. + w^. ')/2, instead of the geometrical mean, then one obtains

the cost function

S = 4

N

1

i=l

(Aw^‘ 7(w. w.

1 )2 =

N

4 1

i=l

(R7 - i)^/(R^.^ l)‘ (15)

with

r^crT - i)/(rT + 1)^ = RT(r/^/F)

N

I

j=l

R,^(R,^ - i)/(r/ + 1)^. OS)

J J W

Comparing Eqs. (15) and (16) with Eqs. (9) and (10) leads to the conclusion

that the simple mean and inverse algorithms should also yield similar results.

In particular, the same additional constraint imposed for the inverse algorithm,

namely w. ' >0, must be imposed in this case as well since Eq. (15) is finite

for w. ' =0 (R.^ = o»).

1 '
13 -

2.4. Algorithms in Which Random and Systematic

Uncertainties Are Separated

It is common practice to report the total uncertainty a of an experimental result as the square root of the sum of the squares of the random

uncertainty component and the systematic uncertainty component

In Ref. [13] Cohen suggests that in many fundamental constant determinations

the random uncertainty is much better defined than the assigned systematic

uncertainty. The reason is that is usually obtained in a well prescribed,

statistically sound way from repeated observations. In contrast, is often

based on rough estimates using approaches which differ widely from one experi

menter to another. Moreover, in some cases important sources of systematic

error may be entirely overlooked. He therefore suggests that in treating

discrepant data, it may not be unreasonable to require that remain fixed

and to vary a solely by varying cr^.

On the other hand, one could argue that Eq. (17) is not statistically

well justified and that in complex experiments requiring several ancillary

measurements, each with its own random and systematic undertainty components,

the random and systematic uncertainty components of the final result cannot

really be separated in a meaningful way. Furthermore, the uncertainty

(variance) in a„ is given approximately by [18]

where n is the number of repeated measurements. In many experiments n is

so small that is quite uncertain (e.g. , for n = 10 the one-standarddeviation uncertainty in is 0.22 CTj^). As we shall see in Section 3, even

for a set of input data with some highly discrepant items the majority of the

(17)

(18)
14 -

I 2 values of a- ‘/a. required to achieve x - using one of the more reasonable

of the proposed algorithms is 1.5 or less, with only a few as large as 2.0.

Since such changes in cr. are not at all inconsistent with the typical uncertainty in there would seem to be no compelling reason to allow only to

vary. Nevertheless, for the purpose of better understanding the various

proposed algorithms as well as for completeness, we shall examine this

possibility.

The idea of only allowing to vary can in principle be applied to all

of the algorithms so far presented. However, in light of the above discussion, but also because it would lead to a large increase in the number of

algorithms we would have to consider, only two representative examples will be

examined. Since the natural log and geometrical mean algorithms yield results

similar to the VNIIM algorithm, we shall not explore them here. Similarly,

because the inverse and simple mean algorithms are alike and because they lead

to values of cr.j'/a^. for highly discrepant data which are so large that these

data are essentially completely discarded from the adjustment, we shall disregard them as well. We shall also disregard the two-stage Birge ratio algorithm

since it is not meaningful to separate the uncertainties of the weighted

averages obtained in the first stage into random and systematic components.

Thus, only the Birge ratio and VNIIM algorithms will be considered.

2.4.1. Internal Birge Ratio Algorithm . Assuming the random uncertainty to be

fixed, Cohen [13] has shown that the Birge ratio algorithm of Section 2.1.1

' 2 defined by a.' = Rpa. with x - ^ becomes

CT.

1 (19a)

or equivalently.
15 -

> - rr 2 ^

' 2 ,/, 2 ^ 2 . -. 1/2 ^ '=^51 ^

‘"ST ^i

= [(T.^ +

^1 ’

(19b)

(19c)

with once again x = P-^ Here as before a. and a.' are, respectively, the

a priori and altered total uncertainties of the i^*^ input datum; is the

datum's a priori and assumed fixed random uncertainty; a^-' its altered

systematic uncertainty; a^- the datum's a priori assigned -systematic uncertainty; T. = (cr^./a^.) ; and Rtd an "internal Birge ratio" such that when

1 Id

all of the CTg^- are increased by the same factor R^g (i.e. , = Rjg ^si^’

' 2

the resulting a^. ' are such that the adjustment is consistent, that is, x =

Equation (19a) is, of course, what one might intuitively expect. An iterative

procedure may be used to calculate R^g with Rg as the first iterate. Since the

(jg^. are in general finite, R^g will in general exceed Rg. Unfortunately, the

disadvantages of the Birge ratio algorithm discussed in Section 2.1.1 are still

prevalent in this modification.

2.4.2. VNIIM Algorithm for Systematic Uncertainties . Application of the VNIIM

algorithm, Eq. (1), only to the systematic uncertainty components yields as a

cost function

N N

2 2 2 2

5 = I - 1]^ = 2 (Rr/ - Tr , i=1 i=l( 20 )

2 2

where R^^. a '/a^.) . As above, Eq. (17) is used to relate a. to and

' 2

a<-.; and a.' to cjn- and a-.'. With the usual constraint x ~ Py "the expression

j 1 1 K1 o 1 analogous to Eq. (5), obtained by differentiating Eq. (4) with respect to a^-'

2 2

rather than a-' (and with R. replaced by R<-. ), is

1 1 0

1

(T,2 . R3//(R3,2 - 1) = (t/ . R3/)(r,'2/F)| (t.2 . R3/)(R3/ -1), (21)
lb

2 2 where, as before, T. = ((t^./g-.) . Equation (21) may be treated in a manner

1 K I 5

1

similar to Eq. (5).

Since it is assumed that the random uncertainties remain unchanged,

the cost of changing ct. to a.' is borne solely by the systematic uncertainties. However, the form of Eq. (20) is such that the cost of this change

is the same whether is a major or minor contributor to the total a priori

assigned uncertainty a. — in other words, whether the change in leads

to a large or small change in cr. . Intuitively, one would expect that the

penalty should be less for changes in which do not lead to large changes

in a^.

.

At first glance, one might consider addressing this last criticism by

altering Eq. (20) to the following:

S = = MR3.2 - 1)2/(t/ . 1)2, (22)

2 2 2 2 . where, as before, T,. a ^^ 5 -} = desirable feature

of this expression is that the cost of changing a particular a priori systematic

uncertainty depends on the size of the corresponding random uncertainty. That

is, for a given change in the cost is greater for erg,! larger than

Oo- (T. < 1) than for smaller than a„. (T. >1). If T. >> 1, then large

changes in still cost relatively little. This is as one might expect

since it is difficult to investigate potential systematic errors in an experiment dominated by a large random uncertainty. Thus, changing the a priori

assigned systematic uncertainty by comparatively large amounts for such experiments is not particularly disturbing and should not result in a large penalty.

Surprisingly, however, it is readily shown that Eq. (22) is identical to the

original VNIIM algorithm and therefore offers nothing new.
- 17 -

2.4.3. VNIIM Algorithm with Weights . As pointed out in Footnote 7 in connection with the discussion of Eq. (7), the weights W^. may be used to take into

account the relative reliability of the a priori assigned uncertainties .

One possible choice for such weights is

(23)

2 2 where, as before, T^. = (Cp^./cr^p . This weighting factor behaves as seems

appropriate: W^. * is near unity for experiments dominated by systematic

uncertainty, thus leading to a large contribution to the cost function S from

the cost function component f . ; and is significantly diminished for experiments

dominated by random uncertainty, leading to a small contribution to S from f^.

.

For example, W.* = 0.9 for T^. = 1/3 while W.* = 0.1 for T. = 3. Although this

again assumes that experiments in which is dominant are inherently more

reliable than experiments in which is dominant, the difficulties discussed

earlier (p. 13, second paragraph) regarding the validity of allowing only

to vary are not prevalent in the use of Eq. (23) for the weights W^. since there

is no special assumption that cTj^. is fixed.

From Eq. (7) the generalization of the VNIIM algorithm to the case where

the weights are constants (i.e. , independent of a.' and X^) such as are given

by Eq. (23) yields the cost function

S = I W.*(R.2 - 1)2 . i=1 .(24)

The expression analogous to Eq. (5) is then

R.^(R.2 - 1) = R.2(r.'2/w.*F) I W.*R.2(R ^ - 1)

T 1 11 1 j_i J J J

( 25 )
18 -

While we could explore the effect of including weights for the other

algorithms we have discussed, as well as a weighting factor other than

Eq. (23), we choose not to for essentially the same reasons given at the end

of Section 2.4. The VNIIM algorithm should be sufficiently representative of

the method. We do note that all of the previously given equations analogous to

Eqs. (1) and (5) can be “converted" to their proper equivalents of Eqs. (24)

and (25) by following the exact form of the latter with regard to the position

of the factor

1

2.5. The Extended Least-Squares Algorithm

In Refs. [14,15,17], Cohen has attempted to develop an algorithm for treating discrepant data which is more firmly based on sound statistical principles

than is the VNIIM algorithm and its various derivatives, and which also does

not have the limitations of the Birge ratio algorithms. The basic idea behind

his approach is to recognize that the a priori uncertainty assigned each

stochastic input datura is itself uncertain, and to assume that this uncertainty

may be characterized by a “confidence parameter" v^. . He defines £. as the true

th 2

(but unknown) error in the i input datum, cr^. as the true (but unknown)

2 variance of s^. , and s. as the a priori estimate of this true variance, that

is, s. is the a priori assigned uncertainty of the i^^ input datum. (Note that

this notation differs from what we have been using up to this point.) Cohen

further assumes that the probability distribution for the errors £^. is approximately Gaussian, and makes other reasonable postulates about the quantities ,

(j. , and s.. He also introduces a factor of 2 into his definition

< s.^ s .^ > = 0.^(1 + 26../v.) , (26)

1 j 1 J ij 1

where 5.^ is the Kronecker delta, so that v^. can be identified with the number
19 -

of independent observations or degrees of freedom characteristic of the experiment which determined the i^ datum. That is, Eq. (26) implies that the uncerth tainty of the a priori assigned uncertainty of the i datum (variance of its

variance) is

var(s.^) = 2ct^.Vv^. , (27a)

or

var(s.) = a^.^/2v^. . (27b)

The identification of v^. in Eq. (27b) with n in Eq. (18) is obvious. Cohen

emphasizes, however, that it is unnecessary to give v. such a strict interpretation. One may identify it with the "effective" number of independent observations which defines the variance of the a priori assigned variance, or as

stated. above, as simply a parameter which measures the reliability of the

a priori uncertainty assigned the i input datum.

^2 Cohen goes on to define a. and X as the minimum variance, unbiased ^ 1 a

2 estimators of a. and the adjustable constants or unknowns, respectively.

For he takes a linear combination of the stochastic input data, and

2 2 for a. a linear combination of s. and the differences between all of the

1 1 input data and their adjusted values. His approach to altering the a priori

2 assigned uncertainties s. is thus embodied in his expression for a. . The

requirement that the estimators be both unbiased and minimum variance

2 allows him to solve for both a. and X with the result

1 a

a.2 = (v.s,2 H. * F) , (28)

where as before F = N - M is the number of degrees of freedom for the adjustment

2 ^ 2

and X is the statistic chi-squared. The latter depends implicitly on the a.

since the X^ are obtained in the usual way using as the uncertainty to be
20 -

associated with the i^^ input datum (i.e. , the weight of the i^^ input datura

2

is w- - 1 /ct. ). But as Cohen points out, Eq. (28) is hardly a solution

^2 2 since it expresses o’. , the estimate for the unknown variance cr^.

, in terms

of the unknown variance itself. If it were known, then there would be no

need to estimate it.*

Cohen notes though that it is possible to obtain a meaningful solution

2 by assuming a. and var(s^ to be independently estimated, resulting in the

2 2 replacement of in Eq. (28) by its estimate a. and of v. by its estimate

2/2var(s.) as obtained from Eq. (27b). If then var(s.) is estimated by

s^.^/2v^. [see Eq. (18)], Eq. (28) finally becomes

= [1 * (x^ - F)/v,] S.2 , (29)

or in terms of the notation used with the other algorithms we have previously

discussed,

= [T + (x'^ - o. . (30)

Equation (30) gives the new, expanded uncertainty cr. ‘ for the i*"" stochastic

input datum in terms of its known a priori assigned uncertainty a.; the known

confidence parameter for a., v.; the known degrees of freedom F; and the value

' 2 of X calculated for the adjustment using the a.'. Of course, the relation is

‘ 2

not as straightforward as it appears since x depends implicitly on all of the

a.'. Consequently, as for several of the other algorithms previously discussed,

some sort of iterative procedure must be used until a sufficiently self

consistent result is obtained. Cohen gives one such approach in Ref. [17].

There are four especially interesting features of this algorithm as it is

represented in Eqs. (29) or (30). First, items of input data with the same confidence parameter v. will have their a priori uncertainties a,, expanded by the
21 -

same factor. This means that if v. is the same for all of the data, the adjusted values of the unknowns will be the same as for the Birge ratio algorithm

but with smaller uncertainties. Second, the more reliable the input datum (i.e.,

the larger its confidence parameter), the less its a priori uncertainty will be

altered (as v^. -»• », a.' /a. ^1). Third, since the a priori uncertainties a^. are

' 2 positive and since the ' must also be real and positive, x > F - v^. (min).

' 2 That IS, X Tor the adjustment must be greater than the number of degrees of

freedom for the adjustment minus the smallest confidence parameter. Fourth,

' 2 the algorithm will yield the expected value x = F, in only two cases; if

V. = 0 for each datum (it then reduces to the Birge ratio algorithm); and if a^.

for each datum just happens to be equal to ' . In general, however, the

' 2 2 algorithm will yield a value of x between F and the value of x obtained

using the a priori uncertainties cr. . If most of the v^. are large, implying

1 2 2 that the a. are reliable (small uncertainty), then x will be closer to x

than to F. On the other hand, if the v. are generally small (e.g,

,

near unity),

' 2 implying that the a. are rather uncertain, then x will be closer to F.

The principal difficulty with this algorithm is arriving at meaningful

confidence parameters v^. for each datum. In Ref. [17] Cohen applies the

algorithm to the spectroscopic data bearing on a

^ considered in the 1973

adjustment of Cohen and Taylor [3] and attempts to derive a value of v^. in

an objective way for each datum by separating its total uncertainty into

2

two components: Component A, characterized by variance cr^^. and degrees

of freedom v^^. , is that portion of the total uncertainty or variance which

is based on a statistical analysis of repeated measurements (essentially,

the random component of Section 2.4). Component B, characterized by

variance 2 and degrees of freedom Vg^. , is

. then the remaining ...variance

(essentially the systematic component cr^ of Section 2.4). These quantities
22 -

are related by

4

cr.

1V

i

4 4

9 (31a)

with

a.

1 2

(31b)

In calculating v. for the data in question, Cohen assumed Vg. =1, He also

decided to restrict v.. to 10 or less because the actual values of v. . were

Ai Ai so large for a number of experiments (due to a large number of separate

line-center measurements) that they implied a reliability for these experiments which could not be justified by past experience- Thus, a considerable

degree of subjectivity had to be injected into the calculation.

Because of the difficulty in obtaining meaningful values of v^. , we shall

only investigate the extended least-squares algorithm in this paper by applying it to the 1973 data. To obtain values of v. for these data, we use the

relation

V. = l/2x.^ , (32)

where x-ct. is the assumed one-standard-deviation uncertainty of the a priori

assigned uncertainty a.. Simply stated, if ct^. is assumed to be uncertain by

10%, X. = 0.1 and v. = 50; if the assumed uncertainty in a. is 50%, x. = 0.5

and v^. = 2; etc. [Eq. (32) follows from Eqs. (18) and (27b)]. A brief discussion of how values for the x. were arrived at for the 1973 data will be

given in Section 3.
23 -

3. NUMERICAL RESULTS AND DISCUSSION

3.1. Summary of Input Data

Here we present and discuss the numerical results which the algorithms of

Section 2 yield when applied to the data considered for use in the 1973, 1969,

and 1963 least-squares adjustments. These algorithms are succinctly summarized

in Table 1 for easy reference,* while the relevant data are summarized in Tables 2,

2a, 3, 3a, 4, and 4a. The following comments apply to these data.

3.1.1. 1973 Adjustment Data . Table 2 is essentially Cohen and Taylor's Table 27.1

[3] and includes all of the items of input data they considered for use in their

1973 adjustment. No data discussed by these authors in Ref. [3] were excluded

from their Table 27.1 solely because they were discrepant. The several items

they discussed but did not include in the table were discarded on the basis of

a very large uncertainty compared with the uncertainties of other quantities of

the same type, or because the experiment was incomplete of preliminary. However,

there was no discussion in Ref. [3] of the deuterium fine-structure measurement

of Of

^ by Lamb and coworkers or the high field determination of Yp' by Thomas

and colleagues. The accuracy of both of these experiments was still competitive

in 1973 and there were no compelling experimental or theoretical reasons for

disbelieving them. Cohen and Taylor's principal basis for excluding these

experiments was their historical gross disagreement with a large amount of

other data thought to be reliable. Thus, if one were strictly to adhere to

the philosophy discussed at the start of Section 2, one could argue that

these data should be included in Table 2. We choose not to, however, because

the evidence against the reliability of these experiments seems incontrovertible.

The point is that one should not blindly include a datum so inconsistent with

all of the remaining data that it is clearly wrong. While there obviously has

to be some subjectivity associated with such decisions, and each situation has

*The tables begin on p. 49.
24 -

to be handled separately, as a rule of thumb one might arbitrarily take something like a five standard deviation discrepancy, or more precisely a normalized residual of five, as an indication that a particular datum should at

least be considered for exclusion.

In this regard, we note that item 10.4 of Table 2, the value of a

^ obtained from the Kaufman, Lamb ^

al_. determination of the AE splitting in

hydrogen, is discrepant by over seven standard deviations (normalized residual

of 7.9 — see Table 5, algorithm 2). One might, therefore, consider discarding

this item. Indeed, the VNII.M group did just that in applying their algorithm

to the 1973 data [8]. However, we shall initially retain it in order to investigate how the various algorithms handle highly discrepant items.

Our decomposition of the a priori assigned uncertainties ct. into their

random and systematic components is similar to that given by Cohen in Ref. [13]

and is based on the detailed discussion contained in Refs. [2] and [3]- and the

original papers. However, this decomposition was done with only limited

accuracy because we did not feel that the time and effort required to improve

it significantly was warranted in light of the historical nature of much of the

data and the limited purpose of the present work — our aim here is not to

derive a complete set of constants of unquestioned reliability but rather to

examine the various algorithms and their limitations. Furthermore, for many

experiments it is extremely difficult and often meaningless to try and separate

unequivocally the random and systematic components of uncertainty.

Our estimate of the value of each x . , and thus the confidence parameter

V. through the relation v. = l/2x. , is based on personal knowledge of

the experiments in question, their historical difficulty, the degree of conservatism exhibited by the experimenters in assigning their final uncertainties, and other similar factors. Although they are a rather subjective set
25 -

of numbers which could differ significantly from a set derived by another

author on the same basis or by other means, the extended least-squares

algorithm is relatively insensitive to the values of v^. assumed and our set of

X. should suffice to demonstrate the basic features of the algorithm. It

is clear from Table 2 that we feel only a few a priori assigned uncertainties

are highly reliable.

The auxiliary constants employed with the 1973 data are those used by

Cohen and Taylor (Table 11.1 of Ref. [3]) rather than the present best values.

Similarly, the auxiliary constants employed with the 1969 and 1963 data are

those used at the time (Table XI of Ref. [2] and Table III of Ref. [1],

respectively). We choose this approach in order to minimize any effect of

hindsight. That is, one of our goals is to answer the following question: If

these algorithms had been available at the time of the 1963, 1969 and 1973

least-squares adjustments, what would the result have been, all other things

being equal?

Table 2a (and its counterparts for the 1969 and 1963 data. Tables 3a

and 4a) gives the input data to be used in the application of the two-stage

Birge ratio algorithm. These data follow directly from Table 2 (Tables 3

and 4 for the 1969 and 1963 data, respectively) and are simply the weighted

averages of the various groups of like data using their assigned a priori

uncertainties cr^. . The uncertainties given in Table 2a (Tables 3a and 4a for

the 1969 and 1963 data, respectively) are those resulting from the weighted

averages but increased by the Birge ratios for the weighted averages when

greater than unity.

The unknowns or adjustable constants used in the application of the

various algorithms to the 1973 data are those used in the 1973 adjustment:

“ ’ ^BI69'^^’ ^BI69^^’
26 -

3.1.2. 1969 Adjustment Data . Table 3 is essentially Taylor, Parker, and

Langenberg's Table XVI [2] and includes all of the data they considered for use

in their 1969 adjustment. As for the 1973 data just discussed, the only items

initially discarded by Taylor et al. on the basis of inconsistency were the

data they discussed but did not include for consideration were discarded on the

basis of a large uncertainty compared with the uncertainties of similar quantities, an incomplete or preliminary experiment, or insufficient information. We

use item 22a rather than 22b of Table XVI, Ref. [2], because even in 1969 the

theoretical uncertainties in the expression for AE^ were negligible. This

places the full burden of any inconsistencies on the experimental measurements

or 1963 data the algorithms requiring separation of random and systematic

uncertainties because of the difficulty in- separating the uncertainties in this

way for historical data and because of the limited justification for the approach.

The application of these algorithms to the 1973 data should suffice to demonstrate their properties. Similarly, because of the difficulty in reliably

estimating the confidence parcimeter v. , the extended least-squares algorithm

is applied only to the 1973 data.

The adjustable constants used to analyze the 1969 data are a \ e,

N^, and A.

3.1.3. 1963 Adjustment Data . Table 4 is taken from Cohen and DuMond's Table X

[1] with two exceptions. First, so that the results using the 1963 data may be

readily compared with the results using the 1969 and 1973 data, the proton

moment in nuclear magnetons and the proton gyromagnetic ratio do not include

the 26.0 ppm correcton for diamagnetism. Second, the a priori assigned

uncertainties of the two values of N^A are taken from Table V of Ref. [1]

As noted earlier, we choose not to apply to either the 1969
27 -

rather than from Table X because in the latter these uncertainties were

arbitrarily expanded by a factor of three to compensate for the inconsistency

of the two values. All of the other data listed in Table X of Ref. [1] were

excluded for consideration by Cohen and DuMond on the basis of a comparatively

large uncertainty, an incomplete or preliminary experiment or theoretical

calculation, non-availability of the result at the time the 1963 adjustment

was actually carried out, lack of sufficient information to evaluate the

reliability of the experiment, insurmountable difficulties in assuring that

the results of certain X-ray experiments were expressed on the same kxu

scale, etc. It should be especially noted that in contrast to the 1969 and

1973 adjustments, the Lamb et al . value of or

^ (item number 6.2) and the Thomas

et al. value of (item number 2.3) had been included for consideration.

Also, because Cohen and DuMond took the value of the ratio of the as-maintained to absolute ampere as an exactly known auxiliary constant, it does not

appear as an item of stochastic input data. This also has the effect of

eliminating the distinction between low and high field values of the proton

gyromagnetic ratio — all three values of v

‘ 'in Table 4 are expressed in SI

P

units and should in principle be identical to each other.

The adjustable constants used to analyze the 1963 data are of \ e,

and A.

3.2. Results and Discussion

The results of our calculations are given in Tables 5 through 13 and in

Figs. 1 through 11. In performing the numerical work, a sufficient number of

v-5 iterations were carried out where appropriate to ensure that 2 t: X

- F < 10

and that a.* /a. showed essentially no change in the third decimal place and

the adjusted values in the tenth. We have attempted to make each table and
28 -

figure as self contained and understandable as possible so as not to require

detailed explanation. However, a number of clarifying remarks, comments, and

observations are in order.

3.2.1. Table 5 . The data used to obtain the 1973 recommended values were

strongly censored (items 3.1, 3.2, 9.1, and 10.4 of Table 2 were deleted — see

Alg. 1, Table 6). Hence the adjusted values for a number of constants resulting

from the application of most of the algorithms of Table 1 to the uncensored

1973 data differ significantly from the corresponding recommended values in

comparison with the uncertainties of the latter (compare for example Alg. 2

with Alg. 1). This was alluded to in Section 1.

Algorithm 3 differs from Alg. 2 only in that the uncertainties of the

output values of Alg. 3 are 2.18 times the uncertainties of the corresponding

output values of Alg. 2. The factor 2.18 is, of- course, the Birge ratio of

Alg. 2.

The 1973 data exhibit not only inconsistencies among quantities of the

same kind (see Table 2a), but major inconsistencies between the different data

subgroups. This is evidenced by a value of 2.21 for the second Birge ratio of

Alg. 4, that is, the Birge ratio characterizing the adjustment involving the

12 weighted averages of Table 2a.

‘ 2

The internal Birge ratio required to make x - ^ Alg. 10 is 2.54,

which may be compared with the value Rg = 2.18 for Alg. 2. As pointed out in

Section 2.4.1, since the are in general finite, R^g will always be larger

than R

B' 2 _

For the extended least-squares algorithm, Alg. 13, x =29.5. It may be

compared with x^ = 119.1 for Alg. 2 and with F = 25. As noted in Section 2.5,

'22

this algorithm will always yield F < x 1 X . the actual value depending on

the v^. . For example, Alg. 13 was repeated with each x. of Table 2 reduced
- 29 -

by a factor of 2 and thus each v. increased by a factor of 4. The adjusted

values of the various constants changed by only 0 to 0.3 ppm compared with

' 2

the values given in Alg. 13, Table 5, but x =36.7 (Rg‘ = 1.21). The

uncertainties of the adjustable constants were reduced by roughly 10%. (Note

that for larger values of v., the cr

.

' are generally smaller and thus the

uncertainties of the output values are smaller). The fact that this algorithm

is more dependent on the relative values of the v. than on the actual values

themselves should simplify the assignment of a meaningful set of confidence

parameters to a given set of data.

The most interesting point to note in Table 5 is that with the exception

of Alg. 1 for which the data were highly censored, the changes which occur

in the adjusted values of the constants (in comparison with their uncertainties)

from one algorithm to another are surprisingly small. This feature, quite

astonishing when one considers that the* input data include the extremely discrepant item 10.4 and Alg. 6 and Alg. 9 actually discard it (see Table 6),

becomes quite clear when the results in Table 5 for a particular constant

are graphically compared’* as in Figs. 1 through 8. With the possible exception of the Of ^ plot, which can be explained by the direct dependence of of

^ on

item 10.4, a major portion of the points in each figure are contained within

each other's error bars. Table 5 and its counterparts. Table 7 (1973 data with

item 10.4 deleted). Table 9 (1969 data), and Table 11 (1963 data), lead us to

the principle conclusion of the present work: The actual algorithm used to

carry out a least-squares adjustment of the constants is much less important

than the particular items of stochastic input data selected for inclusion in

the adjustment. Indeed, Alg. 2 of Table 5, for which all of the 1973 data were

included without any alterations in their a priori uncertainties, does not differ

in a major way from most of the other algorithms, even with regard to the

*The figure captions and figures begin on p. 69.
30

uncertainties of the resulting adjusted values. This emphasizes the need in

any adjustment to evaluate the reliability of the available experimental data

as carefully as possible.

3.2.2. Table 6 . The two-stage Birge ratio algorithm has not been included

because it is the 12 weighted averages of the 12 different data subgroups rather

than the 31 separate items of input data which enter the final adjustment.

The censored and altered nature of the data used to obtain the 1973

recommended values is evident in Alg. 1.

As first noted in Section 3.1.1, the highly discrepant nature of item 10.4

is apparent from its normalized residual r^^ ^

= 7.91 in Alg. 2. By comparison,

all of the other items are significantly less discrepant. In particular, one

might wonder why items 3.1 and 3.2 were deleted from Alg. 1.

In Alg. 3, all of the a priori uncertainties have been expanded by the

factor Rg = 2.18 of Alg. 2 and therefore each r. in Alg. 3 is reduced by this

factor in comparison with its corresponding value in Alg. 2.

The functioning of the VNIIM algorithm is clearly seen in Alg. 5. The

large increase in the a priori uncertainties of the two most discrepant items,

(or'/CT)^Q ^ = 2.87 and (a'/a)g

^ = 2.10, is apparent. These expansions of a.

are in marked contrast to the negligible expansions of a. for the highly consistent items 1.1, 4.2, and 6.1. Because item 10.4 is significantly smaller

than the other values of a

^ in Table 2, and its weight is significantly

reduced in Alg. 5, the adjusted value of a ^ resulting from Alg. 5 is rather

larger than the value resulting from Alg. 2 (see Table 5 and Fig. 1).

As pointed out in Section 2.3.1, the inverse algorithm tends to discard

highly discrepant items and this is apparent in Alg. 6 where we see that

(CT'/a)^Q ^ = (a‘/a)g

^ The absence of these items severely reduces the

overall “strain" [1] in the adjustment and leads to much smaller increases in
31 -

the remaining a^. than for^ the corresponding a. in the VNIIM algorithm (compare

a particular a.'/a. for Alg. 6 with the same a.' /a. for Alg. 5). The fact that

Alg. 6 completely discards item 10.4 explains why its resulting value of of

^

differs so markedly from the value of Alg. 2 (see Table 5 and Fig. 1).

The natural log algorithm, Alg. 7, is similar to the VNIIM algorithm

but tends to expand the a. of highly discrepant items by a larger factor. For

example, Alg. 7 gives (aVCT).jQ ^ = 4.22 compared with 2.87 for Alg. 5. As a

result, the values of a .' for the remaining items of input data are generally

rather less than the corresponding values for Alg. 5.

The geometrical mean algorithm yields results between the VNIIM and natural

log algorithms as can be seen by comparing Alg. 8 with Algs. 7 and 5.

As pointed out in Section 2.3.4, the simple mean algorithm is similar to

the inverse algorithm. However, it is less severe in its treatment of highly

discrepant items and as can be seen from Alg. 9, censors only item 10.4 while

Alg. 6 censors both 10.4 and 9.1. As a result, the remaining values of cr.j'/a.

for Alg. 9 are somewhat larger than the corresponding values for Alg. 6 because

the overall strain in the adjustment is not reduced to the same extent.

For the internal Birge ratio algorithm, Rjg = 2.54 is the expansion

factor by which each a priori systematic uncertainty is increased to

achieve x

'2 = F = 25. Since ct '2222 = + R^g a^. , the expansion factor for

each a. depends on the ratio T. = — the smaller T. , the more nearly

1 1 K1 o 1 I o. '/a. will equal for large values of T^. , a.' /a. will be closer to

unity. This is apparent in Alg. 10 where we see that (a'/CT).|

^ =2.54 and

(aVcj)g

^ = 1.19. This feature of the algorithm can have an important effect

if a particular datum is somewhat discrepant but has a relatively large value

of T^. . For example, (ct'/ct)^

^ = 1.19 means that this particular input value

for the Faraday constant F carries a relatively large weight in the adjustment
32 -

and determines the final output value of F to a greater extent than it does in

some of the other algorithms where (CT'/a)^ -| is as high as 1.75. This explains

the relatively large difference between the value of F resulting from Alg. 10

and from some of the other algorithms (see Table 5 and Fig. 8). A similar but

opposite effect also explains why of

^ is so different for Alg. 10 than for the

other algorithms. Item 4.2, a measurement of the proton gyromagnetic ratio,

determines a highly accurate, indirect value of cf

^ which is numerically larger

by about 13 ppm than item 10.4. Because T^

^ is relatively small, the uncertainty of item 4.2 is expanded by a relatively large factor: (a'/cr)^

^ “ 2.43.

The uncertainty of the indirect value of a

^ is similarly increased with the

net result that it carries much less weight in Alg. 10 relative to item 10.4

than it does for the other algorithms.

The general comments made regarding Alg. 10 apply to Alg. 11 although

the corresponding values of and a.' /a. are radically different between

the two.

The VNIIM with weights algorithm, Alg. 12, yields normalized residuals and

values of which are quite similar to those of the VNIIM algorithm as one

2 -1

might expect since the weights W^. * = (T. + 1) are rather similar for most of

the data. (If W.* was the same for each datum, the two algorithms would be

identical.) However, for those items with unusually large values of T^. (small

W.*) such as items 3.1 and 10.6, the differences in cr^. '/a^. for the two algorithms

are rather large as can be seen by comparing Alg. 12 with Alg. 5 for these

two items. As discussed in Section 2.4.3, the reason is that for small W.*,

the "cost" of expanding a^. by a large factor is significantly reduced.

As pointed out in Section 2.5, for the extended least-squares algorithm,

Alg. 13, those items of input data with identical confidence parameters v^.

end up with identical values of o. Va. . It is also clear from Alg. 13 that
33 -

for those items of input data with reliable uncertainties or large v^. , cr^. ' /cr^.

is near unity. For example, (cr'/a)g

^ “ 1-04. It is of interest

to note that if all of the v. were identical and equal to 2 (x^. = 0.5), then

(j.'/o. = a'/a = 1.97 and x

^ = 30.8; if all the v. = 10 (x. = 0.22), then

' 2

(j'/cr = 1.67 and x - 42.9. These numbers should be compared with Rg = 2.18

and x^ = 119.1 for Alg. 2.^3

3.2.3. Table 7 . Table 7 is identical to Table 5 except that the highly discrepant item 10.4 has been deleted from the 1973 data. Many of the more general

comments and observations made with regard to Table 5 are therefore applicable

to Table 7.

Deleting item 10.4 reduces the Birge ratio from 2.18 to 1.39 for Alg. 2 and

thus the uncertainties of the output values of Alg. 3, Table 7, are 1.39 times

larger than the uncertainties of the corresponding output values of Alg. 2,

same table.

The inconsistencies among the various values of a

^ are significantly

reduced with the deletion of item 10.4 (compare Rg = 2.90 for the weighted

average 10a in Table 2 with Rg = 0.95 for the weighted average 10b). The

inconsistencies between the different data subgroups is also reduced: The

value for the second Birge ratio of Alg. 4, Table 7, is 2.13 compared with

2.21 for Alg. 4, Table 5.

The value of the internal Birge ratio R^g for Alg. 10, Table 7, is 1.63

compared with 2.54 for Alg. 10, Table 5.

' 2

For Alg. 13, Table 7, the extended least squares algorithm, x =25.3

with F = 24 compared to x

^ = 29.5 with F = 25 for Alg. 13, Table 5. Thus,

eliminating the discrepant item 10.4 leads to a value rather closer to F.

For the case where each v^. is increased by a factor of 4 (see Section 3.2.1),
34 -

X

^ = 27.9 with F = 24 for Alg. 13, Table 7, compared to x

^ = 36.7 with

F = 25 for Alg. 13, Table 5.

With the removal of item 10.4 the changes which occur in the adjusted values

of the constants from one algorithm to another are even smaller than those in

Table 5. This is clear from Fig. 9 where we graphically compare the values of

-1

Of resulting from the 13 algorithms of Table 7. In comparing Fig. 9 to its

counterpart from Table 5, Fig. 1, we see that the variations are signficantly

reduced. This is as expected since or

^ is directly dependent on item 10.4.

The reduction in the variations of the other constants is not so dramatic

because they depend less critically on item 10.4, but of course, their variations were less in the first place. Table 7 reinforces the principal conclusion of this paper: The algorithm used to carry out an adjustment is much less

important than the data selected for inclusion in the adjustment.

It is of interest to consider how "robust" the various algorithms are .

with respect to the deletion of a highly discrepant datum. In comparing the

adjusted values of like constants resulting from the same algorithms in Tables 7

and 5, we find surprisingly small changes. For example, the value of resulting from Alg. 5, Table 7, exceeds the value of resulting from Alg. 5, Table 5,

by only 0.7 ppm. (The uncertainties of these respective values of are 5.0 and

6.0 ppm.) Figures 10 and 11 graphically compare in this way Table 7 with Table 5

for selected constants. As can be seen from the figures, Alg. 2 (and the similar

Alg. 3) is the algorithm most sensitive to the deletion of the discrepant item

10.4, while Alg. 9 is the least sensitive. This is not surprising since Alg. 2

does not expand the a priori uncertainties of discrepant data in any way and

thus they carry full weight. In contrast, Alg. 9 expands the uncertainties of

highly discrepant items to such an extent that item 10.4 was mathematically

discarded from Alg. 9, Table 5 (see Table 6 and Section 3.2.2). The actual
35 -

data were therefore essentially the same for Alg. 9 in both Tables 5 and 7.

(Although the inverse algorithm treats discrepant data in a similar way, it

shows greater variations in Fig. 10 because of the way it treats item 3,1 —

compare the change in (aVa)^ -| between Tables 6 and 3 for Alg, 6 with the

change for Alg. 9.

)

3.2.4. Table 8 . Table 8 is again identical to its counterpart, Table 6,

with the exception that the highly discrepant item 10.4 has been deleted. Thus,

many of the general remarks made regarding Table 6 apply to it as well.

With the deletion of item 10.4, the most discrepant datum is item 9.1

as can be seen in Alg. 2 where r^ .j = -3.26. While this large a residual is

not necessarily unacceptable, the residuals for the remaining data are significantly less and quite acceptable. Thus Alg. 3, in which each cr. has been

expanded by the factor Rg = 1 , 39 of Alg. 2, might have been quite a reasonable

treatment of the 1973 data in the context of the viewpoint of the VNIIM group.

Indeed, if one strictly adheres to their viewpoint, there is no justification

for discarding any input datum except item 10.4 from the 1973 data (see also

the discussion of Section 3.1.1).

With the removal of the highly discrepant item 10.4, the overall strain

in each adjustment is significantly reduced and the individual values of

a.'/a. for 11 of the 13 algorithms in Table 8 are generally much smaller than

the corresponding values in Table 6. The two exceptions are Alg. 6 and Alg, 9.

Since these algorithms had already mathematically discarded item 10.4 in Table 6,

the changes in cr^'/a. between Tables 6 and 8 for the two algorithms are much

smaller.

3.2.5. Tables 9 and 10. Since the general features of the application of the

various algorithms to the 1969 (and 1963) data are the same as for the 1973
36

data, we shall severely limit our comments concerning Tables 9 and 10 (and

Tables 11 and 12).

The two main points to note concerning Table 9 are first that because the

data used to obtain the 1969 recommended values were highly censored (items 6.3,

6.4, 7.1, 8.1, 10.1, 10.2, 11.2 and 11.3 of Table 3 were discarded — see

Alg, 1, Table 10), the adjusted values for several constants resulting from

Algs. 2 through 9 differ significantly from the corresponding recommended values.

These differences are mainly due to the deletion of items 6.3 and 6.4 from the

1969 data, the so-called "high values" of Pp'/pj^ (see Section 1).

Second, as we now expect, the changes which occur in the adjusted values

of the constants from one algorithm to another are relatively small with perhaps

the exception of the inverse algorithm, Alg. 6. As can be seen from Table 10,

this algorithm mathematically discards the discrepant item 6.4 and thus this

datum has no influence whatsoever on the adjusted values resulting from Alg. 6.

In contrast, all of the remaining algorithms, including the somewhat similar

Alg. 9, retain it but with an expanded uncertainty. This again points up the

fact that Alg. 9 is less severe in its treatment of discrepant data than Alg. 6.

However, a close examination of Table 10 shows that it is more severe in its

treatment of such data than the VNIIM or other VNIIM related algorithms and

thus might provide a good compromise between discarding discrepant items entirely, and giving them excessive weight. Indeed, we recall that Fig. 10 showed

Alg. 9 to be the least sensitive of the algorithms to the deletion of the

highly discrepant item 10.4 from the 1973 data.

With reference to Table 10, it is evident from Alg. 2 that in contrast

to the 1973 data, the 1969 data contain no single item so discrepant that one

could justify discarding it if one were strictly adhering to the viewpoint

of the VNIIM group.
37 -

3.2.6. Tables 11 and 12 . The 1963 recommended values given in Table 11 are

those actually calculated from the input data of Table 4 and differ slightly

from the values given in Ref. [1]. That is, we were not able to duplicate

exactly the recommended values given in Ref. [1] from the given input data.

However, the differences are relatively minor and have little bearing on the

conclusions to be drawn. It should also be noted that a recommended value for

A was never actually given in Ref. [1] so we have chosen a plausible value

resulting from one of the exploratory adjustments carried out in Ref. [1].

Since it serves primarily as a fiducial point for comparing the results of

the various algorithms, its value is not critical.

We make three observations regarding Tables 11 and 12. First, in contrast

to the situation which prevails with the 1973 and 1969 data, there are only

small differences (in comparison with their uncertainties) between the 1963

recommended values of the constants and the values which result when all of the

1963 data are included (compare Alg. 2 with Alg. 1, Table 11). This is somewhat

surprising in view of the degree to which the 1963 data were censored in order

to obtain the 1963 recommended values — items 2.3, 3.1, 4.1, 4.2, 4.3, 5.1, 5.2,

and 6.2 of Table 4 were deleted (see Alg. 1, Table 12). Although this might

lead one to conclude that the 1963 data were in relatively good agreement, the

value Rg = 1.92 for Alg. 2 and the several large residuals for Alg. 2 in Table 12

show that the 1963 data were in fact highly inconsistent. But it does point

up a peculiarity which can occur in an adjustment — including discrepant

data does not always have as large an impact as one might expect.

Second, there are interesting variations in the way Algs. 5 through 9 treat

the discrepant items 2.9 and 5.1. For example, referring to Table 12, Alg. 6

mathematically discards item 5.1 and retains item 2.3 even though item 2.3 is

the more discrepant datum (i.e. , in Alg. 2, 3 ~ "3.40 while r^ .j = -2.51).
38 -

Algorithms 7, 8, and 9 also expand the uncertainty of item 5.1 by a larger

factor than for item 2.3, although for Alg. 8 the difference is entirely

negligible. On the other hand, for Alg. 5, the VNIIM algorithm, item 2.3

is expanded by a larger factor than for item 5.1. Nevertheless, in spite

of these differences, the variations in the adjusted values of the constants

(in comparison with their uncertainties) from one algorithm to another are

surprisingly small as can be seen from Table 11.

Third, referring to Table 12, Alg. 2, one sees that as for the 1969 data,

there is no single input datum so discrepant among the 1963 data that one could

justify discarding it if one were strictly adhering to the viewpoint of the

VNIIM group.

3.2.7. Table 13 . The aim of Table 13 is to give some indication of which if

any of several selected algorithms of Table 1 is "best", that is, would have

led to the smallest changes in the recommended values of the constants from

one adjustment to the next (i.e., from 1963 to 1969 to 1973), and would have

given values closest to the true values. (Note however, that these are not

necessarily compatible requirements!)

Table 13 follows directly from Tables 7, 9, and 11 and in general is to

be read horizontally; reading the three "data" columns vertically and comparing

like constants is equivalent to reading Tables 7, 9, and 11 horizontally.

As a further aid to reading the table, we point out the following: the

1969 recommended value for N^ has an uncertainty of 6.6 ppm and exceeds the

1973 recommended value by 20.6 ppm; the 1963 recommended value for N^ has an

uncertainty of 15 ppm and exceeds the 1973 recommended value by 82 ppm. The

1963 value thus exceeds the 1969 value by 61 ppm.

Continuing in the same vein, for the application of the VNIIM algorithm,

Alg. 5, to the 1973 data (but with item 10.4 deleted because of its highly
39 -

discrepant nature), one finds that the value of has an uncertainty of 5.0 ppm

and exceeds the 1973 recommended value by 11.0 ppm; that the value of

resulting from the application of Alg. 5 to the 1969 data has an uncertainty

of 7.0 ppm and exceeds the value of resulting from the application of Alg. 5

to the 1973 data by 0.1 ppm; and that the value of resulting from the application of Alg. 5 to the 1963 data has an uncertainty of 15 ppm and exceeds the

value of resulting from the application of Alg. 5 to the 1973 data by 69 ppm.

Thus, the 1963 Alg. 5 value exceeds the 1969 Alg, 5 value by 69 ppm.

We see then that the results from the application of a particular algorithm

to the 1969 and 1963 data are given relative to the results for the application

of the same algorithm to the 1973 data (with item 10.4 deleted), while the

results for the application of the algorithm to the 1973 data are given relative

to the 1973 recommended values. This means that the algorithm for which the

1969 and 1963 results for a particular constant as given in the table are closest

in value to each other, and for which the magnitudes of the values are closest

to zero, is the "best" in the sense discussed above. This assumes that the

application of the particular algorithm in question to the 1973 data (with

item 10.4 deleted) results in values of the constants which are "correct," that

is, the true values. With the completion of the 1982 and subsequent future

adjustments, each of which will presumably be closer to the truth than its

predecessor, we will be able to extend Table 13 and test more critically for

the best algorithms. For the moment, however, we must content ourselves with

looking only at the 1963 and 1969 results for a particular algorithm relative

to each other and to the result of applying that algorithm to the 1973 data --

until the gathering and analysis of the 1982 data is completed, there will be

nothing to which the latter may be compared.
40

A detailed examination of Table 13 leads us to conclude that there is no

clearly best algorithm. The relative changes in the recommended values between

1963 and 1969 and 1969 and 1973, as well as the magnitude of the changes, are

not all that different for any of the algorithms. For example, in comparing

the VNIIM algorithm, Alg. 5, to the recommended values algorithm, Alg. 1,

we see that while the numbers in the table for the 1963 data are generally

smaller for Alg. 5 than for Alg. 1, the reverse is true for the 1969 data.

The changes in the constants from 1963 to 1969 for the two algorithms are also

similar. Although we leave it to the reader to draw his or her own conclusions

from Table 13, we believe that it does not in any way demonstrate the supposed

benefit of routinely including discrepant data in a least-squares adjustment of

the constants.

4. CONCLUSIONS

It is the author's opinion that the development of a purely objective means

based on sound statistical principles for optimally incorporating discrepant

data in a least-squares adjustment of the constants is a goal yet to be achieved.

The various Birge ratio approaches, the VNIIM algorithm and its variations, as

well as the extended least-squares algorithm of E. R. Cohen, all have their

limitations as discussed in this paper.

It is the author's view that a computer program cannot and should not

replace the sound judgement of the conscientious, thoughtful reviewer who,

through his personal knowledge of the experiments and experimenters, is best

able to judge the reliability of a measurement. In retrospect, perhaps more

faith should have been placed in this judgement during the course of past

adjustments than in statistical measures of inconsistency. That is, there

was perhaps too great a tendency to discard only mildly discrepant items of

J
41 -

input data rather than to retain them and adjust for the inconsistencies via,

for example, the Birge ratio. For as we believe we have clearly demonstrated in

this paper, the actual algorithm one uses to carry out an adjustment is not

nearly as critical as the actual items of input data one selects for inclusion

in the adjustment.

Of course, there is still the basic question which in the author's

opinion is yet to be clearly resolved: Does including what appears to be

discrepant data in an adjustment really lead both to smaller changes in the

recommended values of various constants from one adjustment to the next and

to values which are closer to the true values? Although discarding some

apparently discrepant items in the last three adjustments has led to undesirable

consequences, if subsequent experiments had in fact substantiated the unreliable

nature of the discarded data, then the decisions to discard would have gone

unnoticed. Indeed, the reader should not forget that the decision to

discard the highly discrepant value of a

^ of Kaufman, Lamb ^ (item 10.4

of Table 2) from both the 1969 and 1973 adjustments has been well supported

by recent experiments [5,6] and that if this datum had been included, it would

have led to highly erroneous recommended values for the fine-structure constant.

Thus, in contrast to the VNIIM Group's point of view, the author feels that we

must wait for the completion of several more adjustments before we can say

with any degree of confidence that indiscriminantly including all but the most

highly discrepant data is really advantageous.

Finally, although it is premature to say just what algorithm will be

used to carry out the 1982 adjustment presently underway, it is certainly

our intention to explore the implications of at least several of the algorithms

discussed in this paper. This should lead to a good appreciation of the inherent

reliability of the final recommended values resulting from that adjustment and

mark yet another milestone on a long journey begun in 1929 by R. T. Birge.
- 42 -

5. ACKNOWLEDGEMENTS

I should 1ika to thank S. A. Taylor for her invaluable and highly competent

assistance with the computer calculations, E. R. Cohen for several helpful

discussions, and the Office of Standard Reference Data, National Bureau of

Standards, for partial support.
- 43 -

6. FOOTNOTES

^The inconsistency of the i^*^ stochastic input datum in an adjustment is manifested by the excessive contribution of its normalized residual r. to the

2 2 2 statistic X Tor the adjustment, where x = 2r- . The normalized residual of

an input datum is its deviation from its least-squares adjusted value divided

by (or normalized to) its a priori uncertainty. In a typical adjustment, r^.

should be unity or less for a consistent datum. A value of r. significantly

greater than unity implies that the datum is discrepant. (See Refs. [1,2,3,16]

for a detailed discussion of the least-squares technique as applied to the

fundamental constants.)

^Uncertainties are given as one standard deviation estimates throughout this

paper.

^Much of the discussion in Section 2 follows Refs, [13-17],

2 1/2

‘‘The standard deviation of x is (2F) [1] and thus taking a.' /a. in the interval

Rn/7l+(2/F)^'^^ < a.'/cr. < Rq/-/i-(2/F)^^^ so that x

^ lies in the interval ~ 1 1 — D

1 /? '9 1/9

F - (2F) 1 X < F (2F) would not be unreasonable. Taking ' /a^. = Rg

' 2 so that X = F, its expected value, may be viewed as a good compromise

between excessive pessimism and excessive optimism concerning the overall

reliability of the input data.

^Throughout, the prime means that the so indicated quantities are based on

the adjustment carried out with the altered uncertainties , Unprimed

quantities are based on the adjustment carried out with the a priori uncertainties a..
44 -

^Equation (5) and its counterparts throughout this paper are written so as to

minimize the explicit dependence of the right-hand side on cr

.

' ; the quantity

2 ' 2 . R. r^. is only implicitly dependent on a.' through the values of the adjustable

constants or unknowns.

0 0 o "^For the uniform distribution, <j)^.(R. ) = 0 for R. < 1 - A and R. > 1 + A, = 1/2A for 1 “ A < R^.^ < 1

-t- A, with A = crV3 ; for the normal

distribution, (j>^.(R.^) = (1 /ctV2;i) exp[-(R.^ - l)^/2a^]. However, see Ref. [16],

pp. 591-593, for a detailed discussion of the problem of justifying the VNIIM

algorithm on the basis of a normalizable probability distribution.

^Taking W. 5^ 1 is one approach to overcoming the assumption that all a priori

uncertainties are equally reliable. The question is how to arrive at such

weights with a minimum of subjectivity. One particular approach will be

discussed in Section 2.4.3.

^It is pointed out in Ref. [13] that the condition x = F - 1 is perhaps more

reasonable since the additional variable Rjg has been introduced into the

problem and the number of degrees of freedom has been reduced by one. However,

we shall use F since it is consistent with what was done for the previous

algorithms. The difference between the two choices is not significant for the

typical adjustment with a large number of items of input data (i.e.

,

large N).

See also Footnote 4.

^°In this regard, it is interesting to note that the equivalent of Eq. (21) for

the VNIIM algorithm in the form represented by Eq. (22) may be derived from

Eq. (21) by introducing the weighting factor W^. ^ = [1/(T.^ + 1)]^ in the latter.
- 45 -

2

(7. and V. had been considered to be independently estimated instead of

2 2 2

(j. and var(sp, cr. in Eq. (28) would have been replaced by a^. to yield,

'2 “ 1/2

irr analogy with Eq. (30), ct. ' = [1 + (F - x )/v^.] . However, this

' 2 expression requires the physically unreal constraint that x < F + v^. (min).

' 2 That IS, X would be determined by the input datum with the least reliable

uncertainty or smallest confidence parameter, independent of the size of the

uncertainty itself and the datum's level of agreement with the other experiments. The constraint imposed by Eq. (30) is much more physically real since

it implies that there is a minimum overall level of consistency as character-

' 2 Tied by the value of x which can be achieved for the adjustment and that this

level is dependent upon the input datum with the least reliable uncertainty.

^2piots are not given for their obvious negligible

variations (mainly due to their small a priori uncertainties), nor for or

m^ because their variations are nearly identical to those of F and \

respectively.

^^It can be shown that if all of the v. are identical and equal to v, then

a.Va. H a' /a = (V2/2)*c7(F/v - 1)^ + 2 FRq^/v - (F/v - 1)]^'"^ 11 and x'^ = X^(cr‘/cr)^,

D 2 where Rg and x are, respectively, the Birge ratio and chi -squared for the

adjustment carried out using the <j^. (i.e. , Alg. 2). It also follows that

Rg' = Rg/(aVa), where Rg' is the Birge ratio which would result from the

adjustment carried out using the a.'.
- 46 -

7. REFERENCES

[1] Cohen, E.R. , and DuMond, J.W.M, , Our Knowledge of the Fundamental

Constants of Physics and Chemistry in 1965, Rev. Mod. Phys. 537

(1965).

[2] Taylor, 3.N. , Parker, W.H., and Langenberg, O.N. , Determination of

2e/h, Using Macroscopic Quantum Phase Coherence in Superconductors:

Implications for Quantum Electrodynamics and the Fundamental Physical

Constants, Rev. Mod. Phys. 4^, 375 (1969).

[3] Cohen, E.R. , and Taylor, B.N. , The 1973 Least-Squares Adjustment of the

Fundamental Constants, J. Phys. Chem. Ref. Data 2, 663 (1973).

[4] Taylor, B.N. , Is the Present Realization of the Absolute Ampere in

Error?, Metrologia 1_2, 81 (1976).

[5] Taylor, B.N. , and Cohen, E.R. , Present Status of the Fundamental

Constants, in Atomic Masses and ‘Fundamehtal Constants 5 , Ed. by

J.H. Sanders and A.H. Wapstra (Plenum Press, New York, 1976), p. 663.

[6] Cohen, E.R. , Status of the Fundamental Constants, in Atomic Masses and

Fundamental Constants 6 , Ed. by J.A. Nolen, Jr. and W. Benenson (Plenum

Press, New York, 1980), p. 525.

[7] Taylor, B.N. , Comments on Least-Squares Adjustments of the Constants,

in Precision Measurement and Fundamental Constants . Ed. by D.N. Langenberg

and B.N. Taylor, Natl. Bur. Stand. (U.S.), Spec. Publ. 343 (Aug. 1971),

p. 495.

[8] Tooninsky, V.S. , and Kholin, S.V. , Concerning the Changes in the Methods

for Adjustment of the Physical Constants, report from the D.I. Mendeleyev

Research Institute of Metrology (VNIIM), Leningrad, U.S.S.R. , 1975.

This paper has appeared in Russian in the U.S.S.R. journal Metrologiya

(Metrology), issue No. 8, p. 3 (1975).
47 -

[9] Mamyrin, B.A. , The Principles of Adjustment of the Fundamental Constants,

in Proceedings of the Fifth Biennial International CODATA Conference , Ed.

by B. Dreyfus (Pergamon Press, New York, 1977), p. 25.

[10] Gorbatsevich, S.V., Krasnov, K.A. , and Tooninsky, V.S., VNIIM Procedures

for Effecting an Adjustment of the Fundamental Physical Constants,

report from the O.I. Mendeleyev Institute of Metrology (VNIIM),

Leningrad, U.S.S.R.

[11] Tooninsky, V.S., and Krasnov, K.A. , On Criteria for Determining the

Best Values of the Fundamental Physical Constants, report 01.0607 TK-2

of the D.I. Mendeleyev Institute of Metrology (VNIIM), Leningrad,

U.S.S.R. , June 1978.

[12] Oleynik, B.N. , Gorbatzevitch, S.V. , Tooninsky, V.S., and Kholin, V.M.

,

Improvement of Treatment of Data of Fundamental Physical Quantities,

report of the D.I. Mendeleyev -Institute of Metrology (VNIIM), Leningrad,

U.S.S.R.

[13] Cohen, E.R. , "Extended" Least- Squares , Report SCTR-76-1, Science Center,

Rockwell International, January 1976.

[14] Cohen, E. R. , Maximum-Likelihood Criterion for Adjusting Weights in a

Least-Squares Analysis, Report SCTR-78-9, Science Center, Rockwell

International, August 1978.

[15] Cohen, E.R. , An Extended Least-Squares Algorithm for Treating Inconsistent Data, Report SCTR-78-11, Science Center, Rockwell International,

December 1978.

[16] Cohen, E.R. , Determining the Best Numerical Values of the Fundamental

Physical Constants, in Metrology and Fundamental Constants , Proceedings

of the International School of Physics "Enrico Fermi", Course LXVIII,

Ed. by A. Ferro Mi lone, P. Giacomo, and S. Leschiutta (North-Hol land

Publ. Co., Amsterdam, 1980), p. 581.
48 -

[17] Cohen, E.R. , An Extended Least-Squares Algorithm for Treating Inconsistent Data, in Precision Measurement and Fundamental Constants II , Ed. by

B. N. Taylor and W. D. Phillips, Natl. Bur. Stand. (U.S.), Spec. Publ.

617, in press.

[18] A. Hald, Statistical Theory with Engineering Applications (John Wiley &

Sons, New York, 1965), p. 300.
- 49 -

8. TABLES

Table 1. Summary of algorithms to be considered in the present work.

Algorithm^

Identifying . characteristic^Section where

di scussed

1. 1973, 1969, or 1963

recommended values

Varies from one adjustment

to another

Refs. [3,2,1]

2. A priori assigned

uncertainties and all

available data used

•

3. Birge ratio =

'’b‘^1 2.1.1

4. Two-stage Birge ratio = RqCT. (twice)

D 1 2.1.2

5. VNIIM S =

N

2 (R/ - 1)^, X = F i=l

^ 2.2, 2.3

6. Inverse S =

N

2 '2

I (R. - 1)^ X

^ = F i=l

^ 2.3.

1

7. Natural log s =

N

2 2 '2

I 2n^R/, X = F i=l

^ 2.3.2

8. Geometrical mean s =

N

2 2 2*2

£ (R/ - dvr/, X

^ = F

i=l

^ ^ 2.3.3

9. Simple mean s = 4 Z (R,2 - 1)^/(R^^ + 1)2, =

1=1 ^ ’ F 2.3.4

10. Internal Birge ratio

1 ^i

^ =

®R1^

" ''ibSi^- f 2.4.1

11. VNIIM- systematic

uncertainties

s = ^Rs,T- 1)2, x' 2 = F

i=l

2.4.2

12. VNIIM with weights s =

N' 2 2 ' 2

I w.*(R/ - D^. X = f

i=l

^ ^ 2.4.3

13. Extended leastsquares = [1 + (x'^ - F)/v.]'^2 2.5

The algorithm number will be referenced in the tables, figures, and text.

R. = (<T, Vcr.)^; ^
Table

2.

Summary

of

data

considered

for

use

in

the

1973

adjustment

of

Cohen

and

Taylor

^3].

- 50 -

a> -o CM ro >— C\i CM CO 0“ in 00 CO iT) i- & o fvj cnJ CM ro CO O' un in in O LU z

o CO CO m CM o CM m o o o o CM CM CM o p- iT> Ln o o CM CO o o o o a o o

CM CO CO in bO CM «si CM

in CM CM CM oIf) i— --

o o o o o o o m o a o o o o o o o m nr CO CO m CO o tn m bO o o o o o o o o o' d d d d d d d o

CM bO or* 00 <n oo nr VO o 00 vn CO o <n O r^- CM CO LT> CO CM CM CM <T> bn CO o — CO d d d d d d d d d d >— d

(TV 00 oo CM m (TV o (Tt 00

CO a CO o CO — d CO — CO in CM d d 26 vd 32

o tn — m CO a bO o CM O

oCM O'

bn o d bO d cvl d — <M d d CM o

<3» in oo m o o o o

CO CMOO 00 O d in n d d CM d VO d d 33 d 33

— CM CO CM >— CM CO «r- 1— CM — CM — CM CO _ CM CM CM CO CO nr O' d n d d

Short

wavelength

limit

16.7
N.A^

6.059768

16

2.8

15

0.18

0.70

1.02

XRCD -

silicon

50

Ttiese

are

the

identification

numbers

for

the

indicated

items

used

in

Ref.

[3].
-50- i;': -

*4,' , ^:.'.^tAS^. — -“

1^ -• ir- w P. P ,,s^ " "

f»^"S

-i. -

I WSr Is-—'•”

i-. 'v’-Cft

' ^'-- “ •fi: . ,;. 4S -'

^

r- 8:; I 1 t S^: ? I

* «'1.^’'S

I S I f I* I I 1.* “* ‘f £ Vi.«.

I S -*: * S "®f .-\f. s*Sr g / t'’S •i c-^S, a s^s-tx -

* •s s': a

« ^

I, I li g I'^l.i r i

it £ t- .|:4 |a|

" i-.f f f n ii i i':i'^r.-

- "I f | :f

__ . i-"& » R -"§

si|® »r ,E .# ,^8 , f , .s

^ ^ a ; ? a Cl '^1

-ssi liJ

iO ' -© $ s

.

fel ,

: f Q %0> O- j «# ; »J

>4nw« -

2i ^ » li V'- K?a- f4 ^

r % ; .,T' f i- . T vi’gT- ^

J-i®

fi &•'•

!

s-**sJ t i a^T^

g:'-a « tt v-sa^^s-s #-..a"#^s,,s

• «.

‘ if.*sit

- 1-* , ^e*. ® fefi © ; »^ k», -' S'

(S- •»

.

S^' <*«: l*.i’ -'Hif

«e'

' *~ . ©* ifl '% . . V

'^ *^‘ ,;V^T& ‘ - ^

^- *^- '

i<i . if* M ^ ,.^‘

- s|-rfc .#]ij’ l;

#1^ $ir

'Wg •* rS ll -t! I ??» 5

I

^ ,-|g "Ai" -<-i

V: ®. -,m

, .„ Kf' fSjj i» , ©f

v> •'

. -fc:

p^ : a% 1 ^ ^ It C-k- B^

^ -_ ^

-i J # 1 §;is ^

®“

, fe i? ;i: %fi 4 4-

-w¥M^ ^

-

. I- 5 . „ -r-' - " ^ :f^»' igi*: to*j ‘vMr';i*~

- - '- - - p,^"' >• i%i»-p«'^ - -w •% — _j«, "KV ti- sg ~J* 'j^ "Lft"' *' •

nfe-
- 51 -

Table 2a. Input data derived from Table 2 to be used in the application of the

two-stage Birge ratio algorithm to the 1973 data; see text.

Value^

Bi rge

ratio of

Quantity

(weighted

average)

weighted

average

Uncertai nty^

(ppm)

Degrees

^ of freedom^

1 ^8169'^^ 1 - 0.54 X 10-« - 0.19 -

2 ^BI69^^ 1 0.0 X lo"^ 0.17 4.1 2

3 F 9. 648679 0.21 5.6 1

4 Yp' (low) 2.6751290 1.43 2.3 3

5 Yp'(high) 2.675121 1.16 7.8 1

6 2. 7927740 0.39 0.38 1

7 A 1 . 0020609 0.89 9.

1

2

8 v' 6.059730 0.82 14 1

9 24.21398 1.15 16 1

10a 137.03516 2.90 2.5 5

10b

«'

137.03571 0.95 1.1 4

11 3. 1833479 0.24 2.2 2

12 v(Mhfs) 4463303.8 - 2.0 -

^These identification numbers correspond to those given in Table 2 Item 10a

is the weighted average of all of items 10 in Table 2; 10b is the weighted

average of items 10 with 10.4 deleted.

be-Since there is only one value of type 1 and 12 data in Table 2, a weighted

average cannot be carried out; the values remain unchanged. The units for

F are '^8169*^ •mol ^

; for yp'(low), 10® s*'' *^BI69 ’ Yq' (high),

for 10 mol ^

; for

-3

, 10 kxu; and for v(Mhfs)

,

kHz.

This column gives the total ppm uncertainty of the weighted average, multiplied by the corresponding Birge ratio if the latter is greater than unity.

^The degrees of freedom for each weighted average is the number of items of

like data minus one.
52

Table 3. Summary of data considered for use in the 1969 adjustment of Taylor,

Parker and Langenberg [2].

Item No.^ Quantity Value^Uncertainty

(ppm) Description

Original

No.*^

1.1 2e/h 4.335976 2.4 Josephson effect 1

2.1 1 + 10.2 X lO'® 9.7 NBS Pellat balance 2

2.2

^NBS^

^ 1 + 9.2 X 10‘® 7.7 NBS current balance 3

2.3 1 + 8.0 X O

1

o NPL current balance 4

3.

1

F 9.648570 6.8 Silver coulometer 5

4.

1

V

' (low) 2.6751525 3.7 NBS 6

4.2 O

CL 2.675144 5.8 NPL 7

5.1 Yp'(high) 2.675105 7.4 KhGIMIP (Kharkov) 8

6.

1

2.792690 11 NBS Omegatron 9

6.2 2.792701 26 Inverse cyclotron 10

6.3 ‘'p'^t'N 2.792832 20 Cyclotron 11

6.4 2. 792794 6.2 Mass spectrometer 12

6. 5 I'p'S 2.792746 ’19 NPL Omegatron 13

7.

1

hc/e 12373.15 33 Short-wavelength

1 imit

14

8.

1

A 1.002030 38 Ruled gratings 17

9.

1

6.05972 37 XRCO-calcite 15

9.2 6.059768 16 XRCD-si 1 icon 16

10.1 24.21263 38 Annihilation-H20 18

10.2 24.21421 15 Annihilationtantalum

19

11.1 137.03591 2.6 Hydrogen hyperfine

splitting

20

11.2

-1

at 137.03545 4.3 . Hydrogen finestructure

21

11.3

-1

at 137.03505 2.4 Hydrogen finestructure

22a

^This is the item identification number to be used in the tables and discussion

of the present work.

*^The units for 2e/h are 10^^ ^

’ Yp'(low),

10 ® s’''-T^g5’''; for Yp'(high), 10 ® A^g5•s•kg^ for hc/e, V^g^-kxu; for

N^A®, 10^® mol and for 10 ® kxu.

'’These are the identification numbers for the indicated items used in Ref. [2].
- 53 -

Table 3a. Input data derived from Table 3 to be used in the application of the

two- stage Birge ratio algorithm to the 1969 data; see text.

Item

No.^ Quanti ty

Value^

(weighted

average)

Birge

ratio of

weighted

average

• c Uncertainty

(ppm)

Degrees . of freedom*^

1 2e/h 4.835976 - 2.4 -

2 1 + 8.8 X lO'® 0.14' 4.3 2

3 F 9.648570 - 6.8 -

4 Yp' (low) 2.6751501 0.46 3.1 1

5 Yp'(high) 2.675105 - 7.4 -

6 2.792768 1.68 8.2 4

7 hc/e 12373.15 - 33 -

8 A 1.002030 - 38 2

9 v' 6.059751 0.2Q 14 1

10 24.21399 1.59 22 1

11 a-' 137.0354 1.28 2.1 O

^These identification numbers correspond to those given in Table 3.

^Since there is only one value of type 1, 3, 5, 7, and 8 data in Table 3, a

weighted average cannot be carried out; the values remain unchanged. The

units for 2e/h are 10^^ ^ Yp'C^o'^).

10® s’^-T|^g2 Yp'(high), 10® for hc/e, Vj^g^-kxu;

for 10^® mol and for 10 ® kxu.

'"This column gives the ppm uncertainty of the weighted average, multiplied

by the corresponding Birge ratio if the latter is greater than unity.

'^The degrees of freedom for each weighted average is the number of items of

like data minus one.
- 54 -

Table 4. Summary of data considered for use in the 1963 adjustment of Cohen

and DuMond [1].

Item

No.^ Quantity Val ue^

Uncertai nty

(ppm) Description

Original

No.^

1.1 F 9.648682 6.8 NBS Silver coulometer 6

2.1 "p' 2.6751224 3.0 NBS low field 7

2.2 ^p' 2.6751184 3.0 NPL low field 8

2.3 ^p' 2.6752104 9.3 NBS high field 9

3.

1

2.792833 20 Cyclotron 3

3.2 2.792684 9.0 NBS Omegatron 4

3.3 2.792697 25 Inverse cyclotron 5

4. 1 A 1.002020 35 Ruled grating, CuKa.j 10.

4.2 A 1.'0021 10 ' 75 Ruled grating, CuKa.j 11

4.3 A 1.002011 33 Ruled grating, AlKa.|a

2

12

5.

1

6.06018 18 XRCD, CuKa.j 13

5.2 6.05972 17 XRCD, MoKa.| 14

6.

1

“1 a 137.0352 12 Hydrogen hyperfine splitting 1

6.2

-1 a 137.0388 4.6 Hydrogen fine-structure 2

^This is the item identification number to be used in the tables and discussion

of the present work.

*^The units for F are

4 -1

10 C-mo1 ; for Y ‘

,

10^ and for N^A^ 10^^ mol'\

^These are the identification numbers for the indicated items used in Ref. [1].
55

Table 4a. Input data derived from Table 4 to be used in the application of the

two- stage Birge ratio algorithm to the 1963 data; see text.

Item

No.^ Quantity

Val ue^

(weighted

average)

Bi rge

ratio of

weighted

average

Uncertai nty

(ppm)

r Degrees

^ of freedom^

1 F 9.648682 - 6.8 -

2 ^p' 2.6751248 2.50 5.

1

2

3 2.792708 1.72 13 2

4 A 1.002024 0.86 23 2

5 6.05993 3.09 38 1

6

-1

Of 137.0384 2.00 8.5 1

^These identification numbers correspond to those given in Table 4.

^Since there is only one value of type 1 data in Table 4, a weighted average

cannot be carried out; the value remains unchanged. The units for F are

10^ C*mol for Yp'

,

s ^-T and for N^A^, 10^^ mol \

This column gives the ppm uncertainty of the weighted average, multiplied

by the corresponding Birge ratio if the latter is greater than unity.

^The degrees of freedom for each weighted average is the number of items of

like data minus one.
New

adjusted

value

(expressed

asa

ppm

change

relative

to

the

corresponding

1973

recommended

value)

and

its

ppm

uncertainty

for

the

indicated

algorithm - 56 -

uO

ro <T^ CO c

*.0 tn uO

n

n «T CO

u

0)

dd CM F— CM CM CM o . « > FV c 1 CM o F-F

1—• o CO M3 o n o m <Ti GO o os O o d CM d d d rr d 1 1 1 1 1 oo CM o

s: FF- o O 00 tn CM CM o F— (— d o in CM d lO d m CO

d z>

00 o o

o> o cn CM ro C*3 •sO

n o w

lT) o d

dd

d

d

ddd

d

d . m 1 « 1 1 1 3) • « o CO

O! •fK CM 1 w 00 00 <o o <T» 03 00 03 03 00

O • d d d d n d d o ! O 0> 01 —

«— iQ s u o CM in 00 O

CQ

dd ro

d lO

dd CM

d

d lOd o 1 1 ' f 1 1 1 <4*

(/) CM CO

Of CM CM CM 00 in

n 00 CM CO 00 o rr

•Fd

d •0"d

m

ddd CO1 03 fO o u o <sO 1—

1—• o 00 1— CM CM

n 00

n o ro

F CM o d «T CM CM n d d d d z 1 1 i 1 uo

<4_

J . • F- >» c^

•M o 00 o C uO I— 00 o n o 00 o o CO 03 u d d ro t—•

CMm CM CM

d 1 a M o

S. < 0) t3

n

u

O 00 CM CM CO

n

n o 0/ CM 3 CM d d «T d CM CM d d d d d 1 1 1 1 1 1 1 t ow

Fff CM 00 1 03 o lO m m o> »—•

03 00 CO /“o d d d CM tn CM CM d d CM CM d S £ 5S CO

Q.C o CM 1 o

*o <T3 o 1 C 4^ FW

O CM fF»s

n F» in »:r «sO 01

0>X CO s^

m CM u. i 0> u Sw< X CM CM CM >*1^ *— o fO H- 3 c o m lT) f^ o 03 n lO o «3“ • FiV 3 o in «T pN. 00 m in CO <c 1“

ro

OO cn

n F— z 1 > O o CM d CM m CM n 03 00 n CM Lrt CM O 00 o CM o 03 u O + o n vn n o CO

ro o d FM FW ro FF» d 03 d CM CM

(Ji w

fO

—

>v c a.

4-) < 3 — Z 4-> <T» o. z OJ C O lO z z o

fO 1— 3 1 O < CO z 01 Q. a. O' 25 d z < < z E u. >• z - ^ CM

II II CM CM X X

03O*

II CO

CM

it

II

vj

X

00

CMc

II

CD

•o

O CO c

LU OS

and

the

second

Birge

ratio -

2.21;

for

adjustments 5

through

12,
57

CVJ

<u cn O 0» CM cn <n c ja 0) 0) i- cn 00 o 00 05 CM CM 00 in [ o n) •o B • • « • • • • • . . • • 1 1— C 3 O o m cn 00 CM cn m in cn cn o ! 05 O' f (tj c 0) s- (51 u ^ o X 1 4>> <4-> UJ -(-> o in cn C 1 (n Cvj o cn cn cn 05 05 cn B* o 1 a. 0) 1 o >, . (T3 • • t • . . « • • t « a. > w -l-> cn 05 1— o f— m o o cn 00 px. d <TJ •f— c r— 1 f—

1 1 1 1 1 05 o> 1 ! (U > (0 i sz CO •<-

>-» ITJ •fj L. O 0) z i £ •M — u j o C 0) c CM 05

1. 0) S- 3 2 CM 1— 00 B"

m B" CM CM CM B"

<4- £ • « • • -M 0) £ £ £ 4^ <3 d d cn in CM cn d <n m cn o 05 CO cn O.J3 H-4 ^ C 3 c 1—4 O) 1 •f— fO ‘f* z •<- •O X: 1/5 L. > 05 O o cn

(i— (T3

U -MO 2 00 o 00 in r«-

m px. CM 00 o 3 •f- O) . • « « • • • • U) on £ -- <N o d cn d CM m r>. in . in d 0) Q.-0 fB f—• 1 1 1 1 1 1 1 s- 05 1 O. 3 1

(T3 -O

<0 0)

w 1) /-s ^ , c 1/) 0) (13 CO

(T5 IB 3 U 05 o o <-> pM> «fi» U •»- 00 CM o m 00 P4- 05 in o fx«. px. BCO Q (fl "O Z 4-1 • . • c o 0) > c •-4 4^ C d d in CM in CM CM d d CM CM O o CO •*— u fl 1/) "O Z £ B 1 •1^ 0> 0) :> 05 4^ -O (TJ J- -O 4-> u 05 1^ n 0) o a c • CO 01 m o CM cn m »» r**. 05 1— in o X 0) 1— > u • • • . » • « • u <1) 0) £ -— (51 C d d 00 O B* o 05 05 o (U g 3 1 1 1 1 1 l-«* 1 o 1 0) >3 U

to 4i. 3 0< o 1— u B o J > m »— 1 r- o 00 00 05

CO

O 1 a 05

—

O 1— CM CM >n rv. px. 05 CM «T 00

<u 0» 1— 03 -f- • « • • • « « • 3 Q *-) C 4^ o d cn 00 cn cn d <n m cn o to 05 i. B

fO 3 C 05 i- > F— 4-> 3 O -O C 05 00

01

BC 1-4 O) o 00

m cn 00 cn px. o 05 1 <U o 2 O. • CM d cn m d CM CM d cn d o cO 05 151 O CQ 1 pan 1 1 1 1 1 1 h- z 0> 1 u a Vi» i-

(B o ou

CO£

4-> CM 05 00

00

O <n cn ro 05 00 00 cn o t. o O 05 o' d d CM in CM CM m d CM CM d c O) O o c s

CO to 05 a. >1

m £ 0.4^ 1 o &. <u £ C o 1 /4—S

(TJ -C

O T3 •— o B" CM

m

N in

O.U

C

B r-*

m in pxx. px- 5— £ 05 B 4-> X cn cn CM O L. ;- X CM CM CM o U o 05 05 o cn m r*. o 05 n n o B cn 3 u <3 in B" p>i P*4 00 fx< cn in cn rx» 1— c cn • O • o cn r« in pa» (X^ 05 B 3 o o CM o CM cn CM n 05 00 in CM r— > • CM o 00 o CM o fx- 05 o r*. 1 O o m n n n • • cn . 4-> r— <n pmm cn d 05 oi CM CM cou j >. 1 Lf) 4-» c; < ] Z 1 Oi 4-5 05 05 a. 3. 1 c o <n z \ B r-« H-H — —

fO 3 1 OQ < CO z 05 a. a. ! 1— O' a c; z <c < 3. 05 £ U- ?• 3. i '3* 10

CVJ

11 ca

as. a
Normalized

residuals

and

a.'/u.

for

indicated

algorithm

58

CO

a. II

,1 3

•a

-ac

fO

a-> *3

Oic>

O)

uO o

3}CV

o3a.6ou< uo

35

<0

II n

01 o to CM 05 00 05 ^ p^ ro f) CM 05

O CM cn ps. CM ps. o 05 o 00 05 3 o o o O o P*5 CM ro O CM CM CM CO O o CM o CM o — o o O O O CM s

01

i-0 E

fn 05 05

_ vO if) VO 05 f) PN. 05 VO cn 03 1— 05 o to o o CM ^ _,

O 3 o o cn CM CM VO o cn o <n VO 00 — f) CM 00 05 o «T o ps 3 3 3 o d d d d d d d d » d 1 dt d d d d d1 d1 di 1 C o CM Ps. p"^ vO 00 CM vO o p*». cn to <n o *T CM 05 o 05 tO 05 ps o «n 05 u fO o o 05 t3 o If) f) VO o o rr CM 'O o O 3 CM vO CM 3 01 22 s o .1 01 m 05 o r»N m m rr p^ «T 05 CM 05 PN» CM 1— a to CM ps iw CM CM 05 3 3 il o u o cn o o o o 00 o cn 05 f) ps. o f) CM VO '

Ps f)

— ps uf) 'T 3

00 o d

dd • d

d

d

d

d

ddd 1 dd

d1 d CMddt d1 d o 05 05 p^ <o m uf) ro P««. o f) vO o o CM o cn CM f) o CM 05 f) n* ps 3 CM

03 o o o 05 o 'T

m f) o o cn CM f) o ' i 

O — o CM CM f) CM — o 3 1 3

05O Z rr CM CM o 05 03 CM p^ 3 f) 00 3 ps. 00 uf) f5 CM cn n* ps Ps CO o <n cn ps o O <r> o CM O f*5 3 O 05 o cn 05 f) ps. o f) CM VO o ps '

CM 3 3 n- 3

PN.

dddd T*dd

d

d

d

dO

dd

d

dd CMdd1 dd1

01 o CM

m CM if) cn f) o o vO <o Ps

O >o o 00 VO CM CM cn 3 tA o O o o CM CM o cn o o a o 3 o o o o o o o 3 U a a

01>c

ro o tf) o if5 CM p^ 05 05 cn Ps cn f) 05

O VO 00 05

O CM 3 CM CM <r

a o m ps.

m P*5 00 CM o cn o m CO 05 o CM o cn 00 CM 05

O

— ps 3 3 3 o o d d d d '• d d d d d d d d d d d d d d “

CM 05 05 uf) GO

m o f5 VO CM CM <o ps o 05 VO vO o ps. uf) 3 3 3 o m m r*» 00 o P^ P»i» VO GO o CM <o cn ps. o O cn 00 Ps cn cn ps z> VO If) o ro 05 GO o cn 3 P**- VO P»«- p^ <n cn O f) CM 05 to 05 CM O cn cn CM CM o <Ti Lf> o» CO o 05 05 3 o o cn 03 VO o o m ’

CM if)

O o VO 3 3 3 ‘

iri o d

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dd “ d ci

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d

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d

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d cn

—

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00 oo 00 00 00 00 03 00 3 3 00 3 3 00 03 00 CO CO 00 00 00 00 CO 00 GO CO 3 3 3 3 3 1

3>05 o CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM CM i u

a-> CO m vO

O Lf) CO o P»** rr VO <n 05 o VO 05O VO cnO VO CM ps Ps ps 3 im <“ CM VO vO U5 o 05 05 O VO CM Ps O o CM to VO ps cn 3 3 ! d d d d d d '

d d d d d d d d d d d d d d d cn d d d1 d d d 1 > o o o o o o O o o o o o o o o o o o o o o o o o o o o o o o 3 i u o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 3 1 o C u fO a 4-1U to o CM 05 CO o vO CM p** f) f) CM CM CM ps 05 CM <NI f) 05 05 PS VO 3 ps 3 CM 3 < 01 m o c*5 t3 m vO CM o VO o o <r m f) CO S.. o o CM 05 f) 05 cn 3 ps — 3 c o d d d CM CM CM F- CM d d d d d cn d d d d Ps d d /«• 1 fvi 3 » 1 » 1 t ' 1 '

*o o o o o m

m PO rn o o o 3 00 CO CO CO 00 00 o o o o o o o o o

01 o o o o rr o o o O CM CM CM CM CM CM n* n* 'T ro •o *3 *3 “3 3

pN. c V

V Si 03 05 0) a-> 4-> 4^ V 0) <u 'T CM p^ VO P^ f) 05 <n VO f) cn ps 05 f) 05 cn 00 05 cn ps 3 3 o o CM 01 V o VO 3 05 o CM cn 05 00 05 01 cn <o CM Ps 11 3 3 cn u *3 •3

0)u

dddd

dddd1 dd1 dddI d

d1 ddd

ddj d1 d1 O 1 > j

•— c s o a.

a.

UW 05 cn CM & 0) p^ p^ If) CO VO o o o o 3 00 o 05 uf) cn cn 3 3 ps 3 1 CM < o o d 1 05 d d 05 CM d p^ VO d O cn 05 cn to 3 f) cn cn cn cn CM CM 05 3 cn cn —•

CM cn s CM m CM CM m CM CM CM cn CM CM CM cn If) 3 — CM cn 1

4iJ o F— csi CM CM rnd

d

dd ps. ps. ps 00

d 05 05ddd

d

dd .

r— CM t z 1
Normdlizecl

residuals,

a.

Vo.,

and

a,.'/Or.

for

indicated

algorithms

- 59 -

> ^

HA

a —

V ^ ^ a. a a. s ^

X

-• o>

w C Z € > 01 «-»

. 01 — > u

— o

(9 M- C 4-»

OJ

s» O CO

^ •Si ^ CVi CNJ CSJ csj CSJ CM fsi CSJ CSJ CM CM CM «M ?M «M CM CMCMCMCMCM — ^CMCM

^ CM — m in CM — o ^ o —

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d d ^

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r*» m la o d o

CM —

d d

in in cn m to in 0*1

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in — cn

rn CM c**^ in in CM CM r*^ CM 00

CM in ^ o — CM cn 00 CM nO

CM ^ CM CM ?M I— —

'

*— CM — i

I o

CM in o>

CM «— CM CM CM

vO 00 9^ — — — oo ^

vo — in n ooo»— o — oo o o o o o o o o o

inininin«ninininininininininininininininininininin

cmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcmcm

vO cn CM ^

CM CM

00 o ^ s\

CM CM

vO ^ 9> CM CM CM CM n — 00 CM ^ in oo o — CM CM CM

in in n in

CM CM Cvi CM n o ^ n

CM CM

CM in d d

^ CM O

d d

o> oo — o *— d d 0 0 — 00

o o CM cn d d

n o

d —

in n o o

cMcocninin — cMCMvn — CMCMCM

o o o o ^ en cn r*. d d d d — o n

d d d d d

o o r**. n*-* d d d

aa

a —

CM n —

— n CM o — — —

— COOOCMOf**-

d ^ ^ d — . ob

O O >— O —

i0 9>0 0 —

CO in r*» ® —

cn 00 d d — ^ CM —

O CM O’! CM — —

^ r^. d d f*^ o

CM O

d d vo QO n (VI M CM — cn — VO d — <— m CM CM o

— — CMCn —

d CM CM CM d

CM — CM <n ^ CM —

in in VO

CM — CM C»^ »— CM —

00 O 9>

CM —

d d

vo — cMcn —

Q ^ ^ ^ csi

Thus,

only

the

value

fur
New

adjusted

value

(expressed

asa

ppm

change

relative

to

the

corresponding

1973

recommended

value)

and

its

ppm

uncertainty

for

the

indicated

algorithm - 60 V

2.£ C

roV o

0)

—

CJ eu

00

dd od

GOd

05o d3

rodd o

CS4 o d

lD

CVJ od

CSJ O

Ln

00 ^

CM ^ <o d ^ ^ CM

0%— CM

lD

tT)

(T5 d

d

CM

CM CM

GO

rodod

CO

rod

CM

<od

00

roOod

00d

<n

<o ood odO

CM CM CM

—

OO

CM

'!

’I

® I o I ^2 3O CM

•3

OO

d

d 3 O — 3 od od 3

II

o» o

OJ*'-

<y)

M f Od

d

lD <j> *—

d

urt 3d 33 3 3 d d 3

CM

CMd 33 3

CM 3CM

4-> c

ro

a o

CM 33 33 3 3

ro ^

roO 3d 3 CM

CM 3 ^ ^ d3CM 3CM 3rod o33

ro3O

II 3 3

CM 33 3d 3— 3 CM

tl

O)

fO

cn

•*- 3 U fO a. ro 5o o 3 3 C S > 0) 3.W £ Q. a O 3 U C 4-> 0) ^

s- 0) 0} u

ro 3 C ^ 3 >

CM a>

CO ^ 33 3

CM 3rod 3 3

ro

— 33 3CM3

rr 3 ro 3 33 3 3 3

0}

&.

—

fO I c3

3^ 3 3 a

For

this

adjustment x=

^6.6. F=

24,

and=

1.39,

For

adjustment

4,

and

the

second

Birge

ratio =

2.13;

for

adjustments 5

through

12,

x'^=F
61

01 o 05 f—• c CM 05 U 05 1- PN. CM VO

m o PO VO 05 VO Vf1 o z o T3 03 . • • • 05 -*-> *- C 3 o d CM VO CM CM CM CM

-•-i 05 O'

03 i3 o > -»-5 to O (0 P -M X 1 c UJ -4-» 05 o 05 ‘f- 01 CM o lO VO Lf3 CM

m CT> lT3 CM CM

o. c > (T3 . 03 • . « a. •»* •f- +J m 05 o’ o f— vn VO d lD

ddd fO i. 1 1^ 1 1 1 i c <0 05 1 1 01 05 U > 05 C w U 3

03 £ 05 £ +j O (/) 05 Q. •r® 03 05 u 03 C Q, 3 to f—

VO 05 05 00 PO

w 03 01 . 4-» Z to s -> o d Lfi CM f— CM lO

d CM CM

05 c U -i-> £ t-i z 05 •1- Z *—1 03 £ £ -<-> z •- a,-o •>- > 05 00 o in Q. C i- 3 o o o O o m 00 05 VO VO 3 3 (0 o

KTi 03 03 CM

dd 05 vOd 00 00

V T3 r— 1 1 1 1 1 1 j. 03 ^ OJ 03 3

on >— "O o 05 05 05 > -M 0)

(T3 (— to 03 05 00 05

to -o

U (J •— VO f—> m cn CM CM VO 05 VO

05 05 05 •<- S •-- *5 . • . * . C t. "O "O t—1 <-> C d d CM vf3 CM CM 'T CM CM o a. c c fO u 1 X 0) •<- Z £ <Q 1 05 £ > 05 -i-> o '^^ £ 05 •M S- lT3 o 05 i O J= . 01 0) o m CM VO 05 05 00

•fJ <0 I 05

U -l-> >-->10 • • • • . u t 3 05 o d cn vO 00 d id P— m r>* r>^ 05 03 r- i. 3 1 p* I 1 1 !• p» o 03 1 1 05 > cn

05 T3 05

05 o W

to 03 o r>*

O 3 C

— o 00 PO 00 uf3 05 vO 05 cn 00 VO

05 03 •>- 3 5— o -O C -I-)

d

d CM in CM CM

d CM CM

5— I—

d 03C t.

03 (T3 05

O 05 i. >• M 3 CS. W O 05 <D C 05 05 CM

•c

W 5— Z 05 H-4 03 VO o m vfl VO <T3 cn 00

05 05 1 s- i- • . t • w o I i. o d cn VO 00 d id P» cn >s.

«y> o c CO 1 f— 1 1 1 1 f— i»> 3 05 u 1 1 o j3 o fO (T3 h05 £ d o z CM 05

l/J £ 00 o VO cn

m 05 00 00 o £ 05 -o • . •M 05 o d VfJ CM n CM CM un VO cvi CM c •M o o c £ Z 05 O. > VO O 4-J £ 0.-M VO

03 e C o 1 P^ P^

03 #-• 3 O -O •»- r—

p— ^ o CM VO p^ P^ p^ lD 3. <T3 U C 03 r— p-» Lf3 vr VO r*. r*^ e w a; rg w X (T) 'SaiO 5W^ PO CM o 3 Nmi/ X CM CM CM s.^ 5—

o O 05 05 o ro lD O 05 VO VO O m 3 u VO LT3 1^ 00 rv. pp) lO m r«» r— c . o o m lT3 f—

05 03 3 o O CM

d CM ro CM VO 05 00 Vf3 /-> ^ > CM O 00 O CM O •g* T3 O + o P* VO VO r— VO VO m • • • • 4-> VO cn P- <o 05 05 CM ou

>>

a c 'v.

05

w 05 05 a.

•M* c VO VO 3.

03

—

<t3 3 1 CO *c OQ 3. 05

C

1— ar 3 a z < < a cu £ LL. >•

>~D *TO01

ro

rno

<Ti

<T)

<~o OOI

03

iT) OI

00

POo

r*.

PVJ 03

«\1

3.

O.

m

VO

II o

Q£

T3
Normalized

residuals

and

o.'/o.

for

indicated

algorithm

62

•!'

if

.iI 3

*3

O)

01

aj

<3> o 3 CSJ 3 ro CM O’ 3 3 P-H CM CM 3 3 3 3 3 3 3 O CM CM 3 3 3 3 3 O' 3 CM o 3 3 3 CO CO 3 CM CM CM O O 3 CM 3 CM 3 3 3 3 «T CM 3 3 3 CM

Q. 3 S

01 on is OJ

<*> r*» r*«. CM 3 CO 3 3 o 3 CM pK 3 3 Px 3 3 3 3 3 O 3 3 3 3 3 o on IT) m CM CM 3 3 3 CO 3 3 3 3 3 li* O CM 3 3 V 3 3 px 3 3 3 cri 3 o d1 d dt 3 d d d d 3 d 3 d d d d d 1 1 o 3 rr m m 3 3 3 3 CM 3 3 3 o 3 3 3 3 3 O CM 3 3 3 3 3 CM 3 3 3 u rt3 o 3 3 CO 3 3 CO CO CM O 3 3 CM 3 3 3 3 3 3 O' CM 3 CM 1 w Oi 3 OJ

fiS o 3J 0> <T3 csj 3 3 p^ 3 ps» 3 r*.. 3 O 3 3 CM 3 3 3 CM 3 3 3 3 3 Px 3 U o lO 3 r». CM CM 3 3 3 CO 3 3 3 3 Px CM 3 Px CM fXN. 3 U 3 3 3 O’ 3 O , 3 d d d d <• d d d d d d d d d d 3 d d d 30 o 3 m CM O' 3 3 3 P«- 3 3 3 px^ px o 3 3 CM o 3 3 3 CM 3 3 . 3 fO o 3 3 CO m 3 CM CO CM O 3 3 CM 3 3 O 3 3 3 O' CM 3 3 CM 3 3 z !TS 3 OJ z 3 3 3 3 CM 3 3 CM O 3 PN. 3 3 px 3 3 3 px 3 px uO 3 3 3 3 3 !l o iCi 3 CM I— CM 3 3 3 CO 3 3 3 3 CM 3 CM Px 3 U 3 3 3 3 3 o i 3 . .1 d d d d d d d d d d “ d d d d 3 d d d

01 o C\J 3 CM 3 CM CM 3 CO 3 PN. 3 3 Px. Px Px 3 3 3 3 3 CM <r CM 3 o un o 3 3 3 3 <o CM 3 CM 3 3 3 3 3 3 3 3 3 3 3 3 3 I u a

0)> 4,^ c OJ 3 3 3 CO P^ 3 o 3 CM 3 3 3 3 3 3 3 3 3 px 3 3 3 px CM 3 CM CM 3 o 3 CSJ 3 CO <o CM 3 CM 3 CO 3 3 CM 3 3 3 CM 3 3 3 CM 3 01 3 3 3 3 3 d d d d d d d d d d d d d d d d d d d (1j o -o 3 3 3 3 3 o o 3 o o 3 px* CM 3 3 3 o 3 3 3 O’ CM o 3 CM CO 3 CO CO ro o 3 3 3 CM O 3 3 3 CM 3 CM CM 3 Z 3

xJ z U > ?n CO 3 3 3 3 p^ o Px, Po* 3 o 3 O 3 O CM CM CM px 3 3 3 3 ! O tn <r Ps* 3 CM 3 3 3 3 CO 3 3 3 3 .px CM 3 3 CM p„ 3 OJ 3 3 3 O' 3 3 3

irs

ddd

ddddd

d

d

d

d

dddddd

a^ 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 m 3 1 ro 3 m CO CO CO m CO CO CO m 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 0)z O • (Bi 4-) OJ il CO fO jn 3 O' 3 3 3 3 o CO 3 Px CM 3 3 O Px o O 3 3 3 3 PsJ 1 u 3 3 3 3 3 3 3 3 o 3 3 3 3 3 CM 3 3 3 OJ 3 3 O' 3 3 d d d d d d d d d d d d d CM d d d d d d d d d ’1 > o 3 3 3 3 3 O 3 3 3 3 3 3 3 3- 3 3 3 3 3 3 3 3 3 3 3 3 3 3 ii o :1 u o 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 i o c 3 ua OJ &. r»*. 3 3 3 3 3 3 3 3 3 3 3 Px 3 o 3 O 3 3 3 3 3 3 3 o < Oi O r«* 3 3 3 CO 3 3 ro O 3 3 3 O 3 3 3 Px 3 CM CM 3 3 OJ 3 3 3 Px 1 u 3 c o d d d d d d d d d 3 d d d CM d d d

<N 3 » i

*3 3 3 3 3 3 CO CO CO 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 o o o 3 3 3 3 CM CM CM CM CM CM o O o O' O' o O O 1 m *3 3 3 3 3

r**C OJ 01 OJ OJ 0^ 0) w F OJ OJ OJ OJ CM P»* 3 p'*. 3 3 3 3 3 px» 3 O Px 3 3 3 3 3 3 3 px 3 3 3 , o 3 CM 0» OJ 3 3 3 3 3 CM 3 3 3 3 u 3 3 CM px OJ O' 3 3 3 O

(J 3 3 3 3

0»u

dd

ddddd

d

ddddd

ddddd

d

d1 dd» d

>> •— c U -KF o &

a.

uu 3 3 CM

CL 0^ r>* r>* 3 3 3 3 3 3 3 o O 3 3 —* 3 3 uO 3 3 3 3 px 3 < C 0 d 3 p"-* 3 3 3 csj O 3 p««. 3 d d 3 3 3 3 3 3 3 ro 3 3 CM CM 3 O CM ! 3 3 3 CM 3

il S „» CM m CM CM CO o CM CM r- CM 3 CM — CM CM 3 O' 3 3 CM 3 _ 1 ^ il o CM CM CM CO CO O’ O o o 3 3 3 d Px, px px 3 3 3 3 d d d 3 d d — 1— OJ z u
Tdble 8

(cont'd). A

complement

to

Table 7

(cont'd)

giving

the

individual

normalised

residualb

and

altered

uncertainties

resulting

from - 63 -

'iit

•3

b CO 9 9 imm 9 9 CO 9 9 9 9 _ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 V «VJ 9 CM 9 ro CM CM CM O 9 9 9 9 9 9 9 9 9 9 9 9 ro 9 *3 3 9 C 3

01 CT

'TJ »/> 07

>/»

X 1 s UJ kT) CO CM 9 9 9 9 f>i. 9 9 Oo. 9 9 9 CO 9 9 O* 9 9 <b 9 9 9 9 £ O un CO 9 ro 9 CM 9 9 9 ro 9 ro 9 9 9 9 9 ro 9 9 • ro <b d d d d d d d d 9 d 9 d CM 9 9 9 d d d d 9 3 O 1!1 T3 Z 01

a-> • w o 9 9 CM ro 9 ro

VO CM 9 «TO 9 9 9 9 9 CM 9 9 — CM 9 2 o 9 CM 9 fO ro CM CM CM 9 9 o CM 9 9 9 CM 9 ro 9 9 U 9 Z •o £ c 35 b Z> 0» CNJ 9 9 9 9 9 CM 9 9 O' 9 9 9 ro 9 9 9 9 b •O CM 9 r>» 9 im 2 o *T 9 9 CM — 9 9 CM 9 ro 9 9 9 CM 9 9 CM CM 9 ro 9 9 9 9 d d d d 7 — d — - d — d d - d d 7 d rsj d d d 9 - 9 9 d 9 9

’i V) 3

go o 9 9 9 9 9 9 9 9 9 9 9 9 O' 9 CM 9 CM 9 o- 9 — 9 3 o <M •“ <Tt CM 9 9 9 9 9 9 9 CO 9 9 9 9 CM 9 CM 9 CM ro CM 9

•o v/> u 0) z w 9 i fO c o LfJ CM 9 V 9 9 9 9 CM ro rs. ro on 9 ro 9 9 9 ro 1 z R o 9 9 9 ro 9 9 9 9 9 9 O o ro 9 9 9 9 9 9 CM 9 CM 9 'O ,j 0 > <u flj \ u 01 >» 0) 9 3 </> uc CM 9 9 ro 9 ro CM CO r«*. ro 9 9 «r ro 9 9 9 9 CM 9 9 9 9 ro 3 o fN. 9 9 CM 9 CM 9 9 — 9 ro 9 ro 9 9 9 — CO 9 9 O' 9 1 d d d d d - d - - - - d d d d d 7 d 7 d d 9 d - d d 9 d - 3o

(b 1 rO m cn ro ro ro ro ro ro fO ro CO CO ro CO ro CO ro CO ro CO <o ro ro ro CO CO ro CO 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

01 o c wf9 9 •V 9 f««» 9 9 ro 9 9 9 9 9 CM CM CM 9 9 CO 9 9 9 CO 9 0) 9 9 CM «r 9 3 CM CM 9 9 9 9 9 9 9 ro 9 9 cO 9 9 <o CO <o 9 w CM 9 ro 9 ! (» c 01 o 35 z b9 O CD

CO 9 9 9 9 9 <o tr «T 9 9 9 CM <o ro CM 9 CO 1 o 9 9 CM r*. CM 9 9 9 9 CM 9 CM 9 CM CM 9 9 r««. 9 9 9 9 9 . . d d d d .d d d d d d 9 d d d CM d d d 9 9 d 9 9 o o 9 9 CM 9 CM 9 9 9 9 9 9 CM CM CM CM CM CM CM ro CM CM CM CM CM ro 9 CM CM i o 9 9 9 9 CM 9 ro 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ““ ro 9 ® 1 **

osj

d <n 9 9 CM CM 9 CM CM CM

d CM 9

.i o 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 o 1 uD rn ro 9 ro 9 9 9 9 rs. r** C'* r»* r*ik O' 9 1 d d d d d d d d d d d d d d d d d d d d d 9 d d d d d d d 9 d

CM 9 9 9 9 9 9 9 00 9 ro >«. 9 9 9 9 9 CM 9 9 9 9 CM 9 1 9 CM 9 CO CM CM CM 9 9 ro 9 9 9 9 9 9 9 wO 9 9 CM d d d d d d d d d d 9 ** d d *“ d d d d d CM d d ® 1 ; Sa 3> 9 a 9 00 CM 9 r** 9 9 9 9 ro 9 ro ro 9 9 CM 9 9 ro o- 9 9 1 go 3 d V ro 00 <d CO 9 fN» CM d d 9 9 CM 9 9 CM CM d CM 9 ro CM > CM ro CM 3 •wC 3 n 9 ro CM

•M 9 9 9 o 9 9 9 9 9 9 9 9 CO ro 9 9 9 1 < <b O d 9 r»>» 9 9 9 d CM 9 (>• 9 d d ro 9 ro 9 9 9 ro ro ro CM CM 9 CM 1 uc3

ro ro CM ro 3 CM ro CM CM CO CM CM CM ro CM CM CM ro 9 9 CM CO b d w z CM CM CM rd ro O* 9 9 9 9 0-. r** 9 9 9 9 d 9 d d d d CM ! <9

For

agorithms

12

and

13,
New

adjusted

value

(expressed

asa

ppm

change

relative

to

the

corresponding

1969

recommended

value)

and

its - 64 -

O)

*3

*3

03O0^

0^

llII o

31

rs

0)

*3C

0^o

a.

a.

0^

n CM ro 01 CM o ,M|,* a. c £ 3 (-M O CM n fM* n n CO n O5

00 o*

n CO o oi 00

<jy 1 1 d p— CM

d1 COd

s.

C3 Pm ro CM

n PM> ro p~ 1 OJ OJ £ (—• vO CM n PM* vd n ro n o

Of

CJ3 3 CO PM* o ro o MM CO

(J 00 MM

dd CO rM O" P«M 00 » 1 1 ; 3 in PM. ro CM o ro piM PM. 1 3 31 CM n PM* vb n ro n 4-) O 3 r“ Z 00 n o ro 00 GO GO CM oi 1 cb r-M d d CM d d 1 r>-. 1 1 1

<u

iJl rn *:r

n MM

O PMM oi o o u

p—•

v3 CM

n PM.

nn ro

d >

CM a^

— ,M*» 00 CM fM “31 CM CO d . ro n pmI d CM CM 1 1 z lT) o 00 PM GP» n n CO oi

MMmM Pm CM

n P^ vb

n COn z> MD o> ro P*M n CM o CM

lO CM 31

d ,M.

d rb 00

n rb 1 (

Oj o ' i 3 •I— 1 3 4-) O O fM* MM oo CM CM n CM «T ! W 3

un CM oi CO PM*

n

d 30 PM.

d ! O 0) i s 31 CM. 00 n rM CM o VO rM 1 CM cb MM d MM CO PM, 3" p» 1 03 1 1 1

01 00 lD

n

n CM CO i a O

p— 00 CO pM*.

n

d

d VO lO

•#M 4-i CO 3U oi lD in ro o n

n lO

—

rn CM CM CM CM

ddd I 1 1 1 1 u

>,

4.^ CM 03 lO 30 oo fMn CM 30 ro o c

in CM ro

d

n «r CM 3- 3 a 4-»U oi in in ro O n n VO < 0)

CJ CM !-• CM CM CM 1—

ddd c 1 MM 1 1 1 1 MM

CM 3

*3

0> tn

n ron o n p— CM

-3C £ > 1— in CM

n p^ vbn ro

d OJ O.W £ Q. C•FM n o *3 3 1 yM* u C MM o ro o y>Ms /-Mk yM* CM yM.

01 3 U CM o (—

n o 00 pM* s. <v NMtP> Mm*' n n n >w^ *M1 o; u CM X pM* s.^ o

o> 3 c O 31n p—n 00 o PM* 01 3 o n in o> n PM* CM o ’

<T» 3 CO M* o l-M p^

n VO pM* > o CM d CM CM n oi 00 n CM

CM

OO CM o rM 01 ' o 4- o n VO p— lO n p^

ro

n M—

ndd CM CM 3

>» < z 4J o*» ? L c n \ 1 33 1 <c CO 0) 3. 1 O' 3 z <c < 0^ n £ 1.^ >• 3. 01 oo

Oj ^I 03 1/ < 00 z o uoc3 o

il CO

QSi *35

00

<sD m

CNJ

“3

rn

II

00

QC

11

CO ac35

II

H

II II

Csj CM X

X

*3

03 m

II CMX

t/)

*33o

LU U

;l I

and

the

second

Birge

ratio =

1.40;

for

all

other

adjustments

in

Table
Table

10. A

complement

to

Table 9

giving

the

individual

normalized

residuals

and

altered

uncertainties

resulting

from

the

application

of

algorithms

of

Table 1

to

all

of

the

data

of

the

1969

adjustment

as

given

in

Table

3. - 65 - O o CO CO CO 00 o o 3 3 CO CO CM CO CM o o o o o o o O CM o CM 00 o >— 3 o 3 3 O 3 <r

a. c s•• OJs o un 05 vn cn P^ VO

A 00 PS, fS^ CM CO 3 A 3 CMA

cvi o uO 00 vA CM 00 p^ F— 3 3 3 A CO pSB,

AA j oi d d1 o d1 d1 d d d d I 1 d1 d1 d1 d d d O 1 c CM o vA 05 VO VO A o A CO 3 P'S, CM A A o o o o a o fM o CO o CO o p—•

CM 3 3 3 A o 3 A

tu 0) F* so

0) f8 uO rn 00 00 F»' <—•

CO 00 o CO o PS^ 3 CM o 3 PSb

a (J tT5 CM CM o lO vA 00 >.o CM uA CM p^ 3 3 3 A CO 05

A O’ j o d d d d d d d F~> d d d d d d o d F-> si 1 1 1 ' 1 1 1 1 1

es o VO vO CO CM lA CO p^ o P* o A CM 3 A

A

A } o a o o a o o CO a CO p^ o CM o O 3 05 3 3 'T ! u

O 3 MB fBM p^ f* p— P“ p— ! 05 W 05

03 o V z VD rN, CM fN, lO CM lA o A 00 CO CM CO

A o 3 o P—A

CM CM o lT5 vn S VO CM vA CM 3 O o A CO 3 A O’

TJV o d

dddd

d

d »•

d F»dd i«—d

dddA p—* i 4-) 1 1 1 1 1 1 1 1 1 Uoc V C\J o o o o CO o CM o A CM o F— o Ps F— A o o o o o a o o o o F» o 3 3 o 3 3 A 3 3 <— 1 S. a

<a> o >c

CO o o CO LA oo CM CM o A /s%

A

A

A CO

A F» 1 O CM 05 CM 00 CM 00 o A 3 o A CO 3 A

dd1 d1 dadddd

d 1 ddd1 —d1 dd CM

O

d

— 1 o'”

•oc un iT5 cr» 00 VO

a vA 05 CM o 3 A CM CM o A o o o CM CM CO kO o CM CO 3 P-» 3 PS* ^ ! z fJi NN 1^ p** p““ 1 MB

fO Z ! 3 > 00 tr> CM CM o oo 00 CO CO CO 1— 3 o CM PSb CO 3 3 CM CM VO 00 VO CM lA p's. o PSB 05 A CO 3 O’ ^ i d d d d d d d d d p. d d d d d d CM A d F— 1 1 1 u t

*a

0)N VO VO VO VO

A VO vO 3 3 A

A

A

A 3 AA

«r

0105 o s U 1 O ca iT5 VO 00 00 o VO 05 VO oo o 05 o — A 05 A 3 p— 1 z CM CM VT) 00 <T O' A A 3 CM A CM CM A CM CM 1 ro d1 d d d d d d d d d d d d d d d CM d d

>. ! w o o o o o a o o O a o O o 3 o o 3 3 O o 3 3 o c o o o a o a o o o o o O o 3 3 3 o 3 A A 3 A

fO

a •4-» 1 im < 0) CO lT5 o 00 CO CO CM F* A A rs F* A CM ps. CM 3 ps% u 00 m CO CM 00 a CO 3 CO CM CM A CO ps. 1 c

CNJ 3 dJ

dd

d

d CM

d CMddd

d COAd

a o o o o o o o o o o 3 3 O 3 1i V o o o o o o o o o a o 3 3 A 1 05 •a *o a 3 3 3 3 3 3 c r* F* FM 41 4) p"^ 4J 41 — 4J QJ p-» 41 ^ i 05 01 W 4f> 4.^ 4iJ 4iJ 4-^ ' e 4^ 4) 4J 4^ a 41 41 g rr <T» 1— CO F» FW 1^ 1— CO 3 F— F» F» p—» o o F«- VO 41 4) PSs 4> 4) o 4J 4) CO 41 4) :! u o •a 3 3 3 A A "O , a; d d d d d d d d d o d d d d ! u 1 1 1 | >

•—

C

U s O fo w a. j u u

O. 0) o 00 00 CM

A CO O’

(J *« c 0 cvi 05 vd VO* CO lA VO o vd 05 fO 3 PSp

A 3 A CM CM CM CM CO CO CO CO ' TM s CM m CM CM CO A F-» CM CM CM CO

0) o CM CM CM cn VO

d VO vd vd PSb 3 05 05

dd p~ »• 1 z * '* ll
1963

adjustment

as

given

in

Table - 66 -

II

(Ti u

a.

a.

A3

OJc3

</) 01o

tj

Oi wu0)

0)

to

rr

01

fT3

01 o 9 a

a. 9" 9* rs.

a CSJ 9 a’ CM

a £ C CSJ 9

t/) OiE ps.

a CSJ PN. CM

a

a CM

(ji

d csi r— csi

dd 1 to

4->

T3c •i» (9 L. c tO 9*

a p**.

W 9

Oi Oi

a 9 a 9 a CM

a 0) s £ CSJ 3 o

r— Oi l-w 9 a 9 tn

a

a 9 9 aa > U • d — a d d d d

0) ‘ ' ’

oi ' ES CSJ

a

a o 9 u U 9* a a a 9 a csj a Oj 3 CM u •N> 3J 9 O m Z un 9- o a a p^ a CM

<T> S d

a iiv l-K

dd

d r*>* 1 1 1 O)-.- c ^

•— o

X3 Oi c — Oi

m

a

a 9 O 9 to o. . w 9- rs. a CM 9 a csi a lO *3 Oi CM

0) 0) >

1.

w

W 9 no o CM

aaaaa p^ o u U a csi a a a d o to 1 i 0) c

01

O £ a

a

a o 01 u z > o 9- a a 9 a a a CM a CM z 9 >, > a a 9 a o a a 0^ c u in d 9 o d d . - 9 1 t 1 1 a ^ W

O) ^ C <ii ,iw 9 U a ^ c 01 O 1 U 3 c o a CM o 9 S E W 9 d a a a P^ a p^ a d a, Q. to W CSJ CSI 9 a

CL Q. Oi O Oi s 9 i 3J a a a a a a a a L. u

•f— >—» 9 a

da 1— csi

a o 9 a 1—• 1 t 4- 1 'a . -

o» to

toto

a 0)

L. 1—

a 9 a Oi a X a o d CSJ a <— a o a o 1 Qi u CSJ •— CSJ 9 CM O 4-1 a 9

01U

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a 9 1 1 t 1 > o

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01 < Oi

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a p^ • 1 z u Oi to c 9» d a 9 csi a <N4 3 1 1 1 1 u a o o 4m' * 1 Q.

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a s C 5s CM

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<v 9 Ua CSJ

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m 3 c a

a

a

aa

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Ol 9 O CSJ CSI CSJaa

a

a CM Z o > CSJ o o CM o 9 p*^ a s

r*^ o o a

a 1—

aa p^ 1 i.

m o a r* a a a a csj csi w 9 to O 9 4-) l-M 5S ‘1—

4-1 c — z 3 u. a Oi u 9 rv — o 3 1 < Oi a. Q. a tw or 3 z < Oi z S u. >• 3. ! iQ

0)

01

u)

OJ

as, cc

a;>

5)

0)

ulo

>>

0)E3CV

toor

cnmo

II

*ocm

II o

ti

CSJ

X

«vj a>

II (a

0£

T3c

<9 Oo

ro

II

fSi

Xo H j / 7 ii A

and

the

second

Birge

ratio =

1.14;

for

all

other

adjustments

in

Table

10,x=F=

10

and

R,
Table

12. A

complement

to

Table

11

giviny

the

individual

normalized

residuals

and

altered

uncertainties

resulting

from

the

application

of - 67 -

4» o o fSJ rN oo o 00 ro 3 o o o O lO 3 o o <— o eg 3 o 3 o

a.

C

IP eg (**» 3

4» to a

lO to to 00 3 m CO 3 m ro o 00 3 o y* o ro *—

w

Odd

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4) P tn ro to eg to to 00 00 eg r— 3 y-* o U 1— to eg 3 3 3 3 3 3 d d d d d d d d o 00 ' 1 s

1— (— to eg 3 3 o 3 3 p^ 3 3

(TS o o o 3 O 3 ro y— 3 3 e.. 3 u o 3

O) 4-» 3)

<0

O z (— 3 «r 00 e*. tO 3 (-• 3 p—

CM 3 CO 3 3 3 'T iw t>mm *o

4)

dd

d y-*

d

d

d

ddd F*

d1 u

<r j 13C 4» o o o 3 eg p^ 3 3 3 OJ Ji o o o tr o 3 o 3 3 o eg a ! U 3 ^ 1 s» 4» tO o > ^ ! C 3 to m e^ 3 3 00 3 c 1 o m 3 00 lO O 3 eg o eg 3 eg 1 o

to

ddddd

dd

d )iMdd

d

41 •«» > 1 o

cn 1 c 3 o» to eg o 3 3 00

(A 1 <Q o o eg o> o tO <n 3 3 03 o z (/) w t m c Z V 1 3 > o <r o ro eg to o 3 eg e : *3 o 3 eg 3 P^ 3 00 W

U) 1 </) in

ddd 1— IMdd

dd

dd F-»d 3 ' 4) 1 1' *3

41 1 IN fSI CM eg eg eg eg eg eg eg eg eg eg eg eg 3 3 3 3 3 3 3 3 3 3 3 3 3 4> ( a o e u

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** c d

dd

m eg

d

dd egd cnI 3 1 1 1 o

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(/) s > u ••- C o U e

O)O ro Q.

•*- -W

fTJUU & 4J 00 o o er> 3 3 U < c o to ro rn d 3 d to tO 3 ro 3 P^ eg 3 eg eg m en e eg m eg eo _ eg cn eg eg

4) 6 1—

CSJ eg <N4

md ro «r 3 3 3 3 z
68 "

Table 13. Comparison of adjusted values of selected constants resulting from the application of

selected algorithms of Table 1 to the 1973 data with item 10.4 deleted, and to the

1969 and 1963 data.®

Value*^ and ppm uncertainty 1963 - 1969

differences

A1 gorithm Quanti ty 1973 data 1969 data 1963 data in ppm

-1 a 137.03604(11) 0.82 -0. 12 1.5 20 4.6 20 6.022045(31) 5.0 20.6 6.6 82 15 61

1 . Recomtiended

e 1.6021892(46) 2.9 1.5 4.4 -57 14 -58

values

h 6.626176(36) 5.4 3.

1

7.6 -93 23 -96

m

e

9. 109534(47) 5.1 2.6 6.0 -53 14 -56

-1

ct -0.26 0.81 -2.7 1.8 19 3.1 21 \

12.0 5.3 -2.8 8.5 65 27 68

3. Birge ratio e -5.6 3.0 9.

1

5.5 -48 25 -57

h -11.4 5.7 15.6 9.9 -77 41 -93

'”e -11.9 5.3 9.9 8.

1

-40 25 -50

-0.54 1.46 -1.7 2.0 22 9.2 23 12.9 8.4 -1.0 9.0 74 29 75

4. Two-stage e -5.6 4.9 8.8 5.8 -57 28 -66 Birge ratio h -11.8 9.1 16.

1

10.2 -92 47 -108

m

e -12.9 3.4 .12.5 8.2 -49 30 -61

-0.31 0.69 -1.9 1.5 20 4.6 22 11.0 5.0 0.1 7.0 69 15 69

5. VNIIM e -5.0 2.7 7.9 4.4 -51 14 -59

h -10.3 5.2 14.

1

7.9 -82 23 -96

m

e -11.0 4.9 10.0 6.5 -42 15 -52

-0. 10 0.63 -2.2 1.3 19 4.3 21 7.9 4.6 8.8 6.4 69 14 60

6. Inverse e -3.3 2.5 9.

1

4.0 -49 13 -59

h -7.7 4.3 16.2 7.

1

-80 22 -96

m

e -7.9 4.6 11.6 5.9 -42 14 -54

-1 d -0.25 0.66 -1.7 1.4 20 4.4 21 \

10.0 4.7 3.1 6.4 70 14 67

9. Simple mean e -4.6 2.6 7.7 4.

1

-51 13 -58

h -9.4 5.0 13.9 7.3 -81 22 -95

m -9.9 4.7 10.3 5.9 -42 14 -52

®Item 10.4 has been deleted from the 1973 data as given in Table 2 because of its extremely

discrepant nature as discussed in the text.

*^The values for the 1969 and 1963 data are given in ppm relative to the corresponding values for

the 1973 data, while the values for the 1973 data are given relative to the 1973 recommended

values — see text. The units for e are 10 C; for h, 10 J-s; for m^, 10 kg; and for

N^, 10^^ mol'\
69 -

9. FIGURE CAPTIONS AND FIGURES

Figure 1: Graphical comparison of the values of a ^ resulting from the application of the 13 algorithms of Table 1 to the 1973 data (i.e., a plot

of the results for a ^ given in Table 5). Each value is expressed

as a ppm change relative to the 1973 recommended value.

Figure 2: As in Fig. 1 but for N^.

Figure 3: As in Fig. 1 but for ^gjgg/A.

Figure 4: As in Fig. 1 but for A,

Figure 5: As in Fig. 1 but for u /u . M P

Figure 6: As in Fig. 1 but for e.

Figure 7: As in Fig. 1 but for h.

Figure 8: As in Fig. 1 but for F.

Figure 9; As in Fig. 1 but with item 10.4 deleted (i.e., a plot of the results

for a

^ given in Table 7).
70 -

Figure 10: Graphical representation of the changes in the adjusted values of

selected constants resulting from the application of the indicated

algorithms of Table 1 to the 1973 data, and to the 1973 data with

item 10.4 deleted (i.e.

,

a comparison of Tables 7 and 5). The

points are the ppm differences between the values given in Table 7

and those given in Table 5. Each division on the vertical scale

is one ppm.

Figure 11. As in Fig. 10.
71

rO

O

05

CO

CO

CVJ

(Ludd) |_o Nl 39NVH0

FIG. I

ALGORITHM

NO.
72

rO

CVJ

O

(Ji

CO

1^

CD

lO

rO

CVJ

<o ^ CO ^

(ujdd) fgi 30NVH0

CVJ O

CVJ I FIG. 2

ALGORITHM

NO.
73

ALGORITHM

NO.
74

(Uidd) y Nl 39NVH0

rO

O

Gi

GO

r^

FIG. 4

ALGORITHM

NO.
1 • I 1

1

1

. 1 r~

1

• 1

• I

1

1 « I

I

1

1 •

• I

^

I

1 •

1 —

I

I

1

H

I

1

1

• I

I

1

i. • I 1

1——

1

r ..1

f

^

^

1

1 1 1 ^ cvj O ^ to (ujdd)

fji 39fgvH0

fO

CJ

g

Oi

00

N

m

po

CVJ

FIG. 5

ALGORITHM

NO.
76

rO

g

CD

00

lO

CM

CMOC^'I'COCOO^

(ujdd ) a Ml 30NVH0 FIG.6

ALGORITHM

NO.
77

ALGORITHM

NO.
78

CD^OJOCOCO^OJOOJ^

O

<Ji

00

lO

rO

<M

(UJdd ) j Ml 39NVH0 FIG. 8

ALGORITHM

NO.
79

CM O CM

I I (ujdd) |_D Nl 39NVH0

ro

I F1G.9

ALGORITHM

NO.
ALGORITHM

- 80 -

Oi

d

z

00

d

d

lO

dS

CM

COO

FIG.IO

ONE

ppm

/DIVISION
ALGORITHM

81

rod

d

o

d

FIG.II

ONE

ppm

/DIVISION

NBS»n4A ^EV. 2»eC) J.S. DEPT. OP" COMM.

4. TITLE AND SUBTITLE

Numerical Comparisons of Several Algorithms for Treating Inconsistent Data in a Least-Squares Adjustment of the Fundamental Constants

5. AUTHOR(S)

B. N. Taylor

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. NATIONAL BUREAU OF STANOAROS

DEPARTMENT OF COMMERCE 8. Type of Report i Period Coverea WASHINGTON, D.C. 20234

BIBLIOGRAPHIC DATA

SHEET I See instructions)

1. PUBLICATION OR

REPORT NO.

NB SIR -2426

2. Performing Organ. Report No. 3. Publication Date

January 1982

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

National Bureau of Standards

Electrical Measurements and Standards Division, Bldg. 220, Room B258

Office of Standard Reference Data, Bldg. 221, Room A321

10.

supplementary notes

Document describes a computer program; SF-I8S, FiPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

A number of recently proposed algorithms for treating inconsistent or discrepant

data in a least-squares adjustment of the fundamental physical constants, along with

several new but related algorithms, are compared in detail. The comparisons are first

made by means of the numerical' results the algorithms yield when applied to the same data considered by Cohen and Taylor in their 1973 adjustment which led to the

recommended set of constants adopted by CODATA and in current use. A selected number | of the algorithms are then further compared through the numerical results they yield j when applied to the data considered by Taylor, Parker and Langenberg in their 1969 j adjustment and by Cohen and DuMond in their 1963 adjustment. The principal conclusion!

of this paper is that the actual algorithm used to carry out an adjustment is much I less important than the data finally selected for inclusion in the adjustment.

12.

KEY WORDS (Six to twelve entries; alphaoetical order; capi tali ze only proper names; and separate Key words bv semicolon s) Data analysis; discrepant data; fundamental constants; inconsistent data; least-squares

adjustments; physical constants.

13. AVAILABILITY

X i Unlimited

^ For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Xi Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES ;

15. Price

310.50

jSCOmM' 5043.030
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