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NBS TECHNICAL NOTE 952

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Designs for the Calibration

of Standards of Mass

52
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and

technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to pro- mote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Institute for Computer Sciences and Technology, the Office for Information Programs, and the

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THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consist- ent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of the Office of Measurement Services, and the following center and divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center for Radiation Research — Laboratory Astrophysics 2 — Cryogenics 2 — Electromagnetics 2 — Time and Frequency*.

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THE OFFICE OF EXPERIMENTAL TECHNOLOGY INCENTIVES PROGRAM seeks to affect public policy and process

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1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

2 Located at Boulder, Colorado 80302.
Designs for the Calibration of

Standards of Mass

Technical ^6V<

J. M. Cameron, M. C. Croarkin,

and R. C. Raybold

-AilONAi BUREAU

MG 2 3 1978

n 6+- qlqlq

uj"7S3

ooqsrX

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

asft of

^fAU 0?

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

U»S. NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued June 1977
National Bureau of Standards Technical Note 952

Nat. Bur. Stand. (U.S.), Tech. Note 952, 64 pages (June 1977) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1977

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13.46:952). Stock No. 003-003- Price $2.10 (Add 25 percent additional for other than U.S. mailing).
DESIGNS FOR THE CALIBRATION OF STANDARDS OF MASS

by

J. M. Cameron, M. C. Croarkin, R. C. Raybold

Office of Measurement Services

Institute for Basic Standards

National Bureau of Standards

This report presents a collection of designs for the intercomparison

of sets of weights for use in precision calibration of standards of mass. These include a number of previously unpublished designs which

have an additional weight in each set to serve as the check standard for

monitoring the performance of the weighing process. Also included are

the classical designs of Benoit and Hayford. The complete least squares

analysis is presented in integer form (i.e., with a common division) for

the most widely used designs; and for the others, the standard deviations

are given for various weight combinations when used as an ascending or

as a descending series. Designs for sets of nominally equal objects,

the 2 2 . . . 1 1 . . . series, the binary sequences, the 5 2 2 11

series, and the 5 3 2 11 and some miscellaneous series are given.

Key Words: Design of experiments, least squares, mass calibration,

statistical design, weighing design

INTRODUCTION

Calibration of a set of weights consists of assigning values for

the unknown weights in terms of the known mass of one or more standards,

For high precision work, this involves the use of the balance as a

comparator which measures the difference between two objects (or two

groups of objects) which must have nominally the same mass because of

the small "on-scale" range of the comparator. In deriving units which

are subdivisions of the basic unit or multiples thereof, a variety of

different weighing sets have been used because of convenience or other

practical considerations. A typical set is the 5 3 2 1 series which

bridges the range from 10 to 1. In this paper, designs are presented
for sets of weights of the same nominal size, for the most common

subdivisions currently in use, and for a miscellaneous group included

for completeness. In most cases, the designs provide for a check

standard, treated as an additional unknown weight, to be used for

monitoring the performance of the measuring process.

Precision weighing is usually done by some form of transposition

weighing on a two-pan balance and by substitution methods on a one-pan

balance. Matters relating to weighing procedures are discussed in

[8, 9] . For the purposes of this report, it will be assumed that a

well behaved comparator is available and that measurements of differences

in the mass of two objects or groups of objects are corrected for air

buoyancy effects and other environmental or procedural factors. It is

further assumed that the measurements are uncorrelated in the statistical

sense and all are of equal precision. (These latter two assumptions

are non-trivial and special care has to be taken to insure their

validity so that the random error component of the uncertainty is

properly evaluated.)

NEED FOR A CHECK STANDARD

In a calibration laboratory, it is necessary to have checks on the

measurement process to provide assurance that the process measures what

it was intended to measure and that it does so with a nearly constant

precision [10] . A direct check on the limiting mean of the measurement

process is provided if a known weight is calibrated regularly as if it

were an unknown test weight. If the value obtained for the weight

differs from its accepted value by an amount larger than can be

accounted for by the imprecision of measurement, then the process

2
would be regarded as being out of control. One is saying that if he

cannot calibrate his own weight correctly, he can have little confidence

in the values for the calibration of unknown weights derived from the

same data.

There is another equally important reason for routine calibration

of the same weight—the results on it provide the true measure of the

variability of the process. In the course of a year the weighings would

have been done under diverse weighing conditions and, hence, the sequence

of values would reflect the actual variability of the process—variability

which may not be reflected in the internal agreement of one series of

measurements

.

If an unknown test weight was repeatedly measured, one would expect

variability similar to that shown by the check standard. If one has a

single measurement on an unknown, it would be like a random selection

from the sequence. From the sequence of values on the check standard,

one can establish limits to the variability of the process and, because

of the equivalence in the method of measurements for both the standard

and the unknown, one can legitimately transfer the properties of this

sequence to the unknowns.

To establish that the measurement process is in control requires also

that the measurements be internally consistent within the limits of

random error. If more weighings are made than there are unknowns, then

there will be a "closure" error because the values for the observations

calculated from "best" values for the weights will differ from that

actually observed. The standard deviation computed from these deviations

can be tested against the long-run value of this process parameter.

3
In summary then, the schedule of measurements for calibration

should include provision for a check standard and also for within-run

redundancy. The decision as to which one of a number of possible

schedules or designs to use for intercomparison of a set of weights

depends on items such as the variance associated with individual weights

or combinations thereof. The least squares analysis from which the

values for the weights and their variances are calculated is presented

in the next section.
LEAST SQUARES ANALYSIS

We begin then with a set of n observations, y-]_, y„,

involving k objects whose values, 3., 3„, . . . ^k are to be determined,

The set of observations can be represented by the equations for their

expected values, E(y.),

E(y

i

} =XlA + X

12

3

2

• • • xiA

E(y

2

) = x

2131 + x

22 6

2

. . . x2k3k

(1)

E(y ) = x 6: + x _3~ . n nl 1 n2 2 nk k or in matrix form E(y) = X3 where the element f x. ., of the X matrix

is if the weight is absent, and 1 or -1 depending on the direction

of the comparison. In this note we shall adopt the convention of

using just the signs so that, for example, all possible comparisons

(ignoring direction) of 4 nominally equal objects will have the

representation

3 e 1 2

B. 3.

x = 1 -1

1 -1

1 -1

1 -1

1 -1

1 -1
In the least squares analysis one forms the normal equations

X'X3 = X'y

where the entries in X'X are merely the sums of squares and sums of

cross products of the columns of X. In the above case, one gets

3 -1 -1 -1 3 =

" Y

l + Y

2 +y3~

-1 3 -1 -1 "y

i + ^4 + ^5

-1 -1 3 -1 ~y 2 ~ ^4 + ^6

-1 -1 -1 3 _-y 3 - y 5 " y 6 _ where 3 is the column vector with elements 3, # 3~, 3_/ 3,» the caret

being used to denote the fact that the values are functions of the

observations, and not the sought-after values, 3.

It can easily be verified in this case that the system of

equations is not of full rank (e.g., the column totals are zero) and

this is a property of all designs where only differences are measured.

In mass calibration, one has one or more standards whose value can be

taken as known and these provide the restraint on the system needed to

give a unique set of answers. Usually these involve a starting kilogram

or a unique summation such as 5 + 3 + 2 which has been determined in a

previous series or is the initial unit value for an ascending series

such as the 1, 2, 3, 5 series. One can write the restraint* in the form

r

l

B l + r

2

B

2

. . + rk k m (2)

and use the method of Lagrangian multipliers (with multipliers 2A.) to

minimize the function

*In all cases treated here a single restraint is sufficient. See

Zelen [12] and Goldman and Zelen [6] for a discussion of the general

case.
$ = E (deviations) 2 + 2\(r, $, + . . . r,6 - m) 11 k k

(3)

The normal equations now contain an additional "unknown," namely X and

written out in full are as follows:

+ Zx n xjL • • • ^xxj. + r A = Exv (4)

2 1 2

K

2

Zx

2 x

1

3

1 + Zx

2 B

2

1 k"k

' * ' ZX

2 XA + r

2 X = ZX

2 Y

where

Zx

kxl

3

1 + Exkx

2 3

2

. . . Zx

k k

+ r

k

X = Zx

k

y

r

i

B

l + r2B

2 '

' ' rA

Ex.x. = I x..x.,

i 3 k=l ik jk

Ex . y = Z x y

i k=l ik k

or in matrix notation

X'X X'y

m

The solution may be written out formally as follows:

X'y

m

CX'

h'X' m

(5)

(6)

where r' = (r,r . . . r, ) . 12 k

To facilitate computation it is convenient to have the values,

8, written out as linear functions of the y's and m, i.e., 8 = [CX 1 , h]|y

This leads to a set of multipliers of the observations of the form
3

1 = gilY

l + g

l2y

2

* ' ' g inY

n

+ V (7)

3, = g, ,y, + g, n y~ . . . g. y + h. m

k kl 1 k2 2 kn n k

These multipliers, g . and n ., are given in Appendix B in transposed

form for some of the designs. The matrix C is important because the

variances and covariances of the estimates are given by

S\ *N Variance ("3.)=C..a , Covariance (3., 3.) = C. .0 (8) i n i j i]

The quantity, 2

, is the variance (square of the long run value of the

standard deviation) associated with the process. In a set of n observations on k items and r = 1 restraints one has n-k+r=n-k+l

degrees of freedom for a standard deviation, s, formed by

? 1 2 s = : E (deviations) . (9) n - k + 1 l

(deviation). = y. - (x. $ + x.„3 . . • x $ ) l l ll 1 i2 2 lk k

One can write these deviations as a function of the observations by

noting that the predicted values are just x3 and the deviations are

thus

dev = y - X3 = y - X[CX' ,h] Y = y - [XCX',0]

= [i-xcx'ly

(10)

which can be written as deV

l = dllYl + d

12Y

2

* • ' d

lnYn

(11)

n nl 1 nZ 2 nn n
The array of coefficients, d. ., is given in Appendix B for some of the designs,

Weights are often used in combination and one needs to know the

standard deviation for the various sums. For a sum of two items, 3.

and 3 . , one has

D Var(3. + 3.) = Var(3.) + Var(8.) + 2Cov(3., 3.)

i : i j id

and for a linear combination

L = £ § + U, . . . *>.%. (12) 11 2 2 k k

Variance (L) = i'Clo2

where £' = (£ , £ . . . £, ) , C comes from the inverse of

the matrix of normal equations [see equation (6) ] . in Appendix A each

design has a list of the factors D. for computing the standard deviations for all usual weight combinations, L. where Variance (L.) = D

2

<J 2

.

DESIGNS FOR WEIGHING

The criteria for good weighing designs depend to some extent on

the use intended for the resulting values. For example, if the

weights are to be used independently of each other, then one would

want the standard deviation [oVc . . from formula (8) ] for the value for

11

each unknown weight to be the minimum possible. If the weights are to

be used in combination, then one wants the variance of all appropriate

linear functions to be as small as possible.

Further, the desirability of a design depends somewhat on the

restraint being used. In some cases, one's judgment of a design

changes depending on whether one starts with a summation as known (e.g.,

5+3+2) and works down, or with a unit as known and works up (e.g.,

by use of a 1, 2, 3, 5 series) . For a given number of measurements

9
only a finite set of possible designs exist for a series and only

occasionally is one of these designs uniformly and undeniably "best".

The designs are grouped into categories in Appendix A: A. Designs

for Nominally Equal Groups, B. Designs for the 2 2 . . . 1 1 . . . Series,

C. The 5, 3, 2, 1 and 5, 2, 2, 1 Series, D. Binary and Miscellaneous Series,

and E. Designs for Direct Reading. The two most widely known collections

of weighing designs are those of Hayford [7] and Benoit [l] which, although

they do not make provision for a check standard, are listed in this

appendix. In Appendix B the complete analysis is given for five of the

most commonly used designs. For the others, the complete analysis is on

file with the authors, and the factors for computing the standard

deviations for different weight combinations are given in Appendix A.

A. DESIGNS FOR NOMINALLY EQUAL GROUPS

A.l All distinct intercomparisons . If k weights are to be intercompared by measuring the difference between weights of the k (k - l)/2

distinct pairings, then a general analysis can be written out as a function

of the number of weights that are regarded as known and used as the

restraint. The inverse of the normal equations with the sum of the

first m of the k weights taken as known is as follows:

kl-J -J 1

-J kl-J

±'

-1-1 _1_

mk ml-J kl

ml+J k±

kl' kl'

:i3)

where 1' = (1, 1, . . . 1) and J is a matrix of all ones and the

matrices on the diagonal are of dimension mxm, (k-m)x(k-m), and lxl.

Thus the standard deviation of the value for weights within the

restraint is o/^-r- and for the unknowns, a/~-. The standard

mk

deviation of a sum of h unknowns

_jnk

, /h(h + m)

is 0/ : . mk

10
The 3 . are given by

m

t. It r\ i 1 i K

3. = -: —r~ + — for weights within restraint

k (14)

T. Z T. 3 i , m+l i , K

p. = -r~ + ;— + ~~ for unknown weights

l k mk m

3

where T. is the sum of the y values invlving 3. in the positive sense

minus the sum of y values involving 3. in the negative sense (e.g., if

E(y) = 3

i - 3 • , then y would be added to T. but subtracted from T . )

, and

m

K is the value of the restraint (13 = K)

.

1

The standard deviation s is given by

S ' =

(k-1) (k-2) {Zdev2}

ET 2 {Ey 2 - _i} (15)

(k-1) (k-2) *i k

Designs for which a linear drift with time is balanced out are also

included (see [4] for details of the analysis)

.

A. 2 Subsets of all distinct intercomparisons . For large k (say

k >_ 6) the number of possible pairings becomes large enough that the

time involved in completing the measurements leads to a degradation of

the precision as environmental changes, operator, fatigue, etc., become

important. For that reason, subsets of the k(k-l)/2 pairings are used

to form the design.

In some of the designs the sum of all weights is taken as the

restraint. This is appropriate when the design is used to monitor

within group behavior. In others, there is an implied grouping into

two classes the sum for one of which is taken as the restraint.

11
A. 3 Designs involving grouping of weights . When differences

between groups of two or more weights are measured, a reduction in

the variance of the values can be achieved in comparison with an equal

number of differences between single weights. However, for large k

the problems of identifying and handling the groupings may outweigh

the possible gain in efficiency. Bose and Cameron [2, 3] have tabulated

all designs up to k = 13 and give methods of construction for k <_ 50,

for the special case of designs balanced so that all weights appear

equally often with each other on the same pan and a similar property

holds for their occurrence in opposite pans. Partially balanced

designs have been developed by Suryanarayana and Chakravarti [5, 11],

B. DESIGNS FOR THE 2 2 . . . 1 1 1 . . . SERIES

When weights of nominal size 1 and 2 are involved, the construction

and analysis of the designs does not possess the simplicity and

symmetry of the case of all equal weights. The designs given under

Section B in Appendix A are a listing of those in common use. It is

not known how close they are to an "optimum" design.

C. THE 5,3,2,1 AND 5,2,2,1 SERIES

The most commonly used sequences for weight sets are the 5,3,2,1

and the 5,2,2,1. These series permit one to achieve nominal values from

one to ten and are the basic series for calibrating weight sets from

1 kilogram down to one milligram. The designs with the check standard

[C2, CIO] are the ones used for all the calibrations of such series

at the National Bureau of Standards, and the complete analysis for

each of these two designs is given in Appendix B. A complete example

from mass calibration at NBS is given for the 5,3,2,1,1,1 series on page 14.

12
D. BINARY AND MISCELLANEOUS SERIES

Binary series used for pound sets and some other miscellaneous

series used at NBS are included for completeness.

E. DESIGNS FOR DIRECT READING WITH CONSTANT LOAD

When a single pan balance is used as a direct reading device, the

time required to complete a set of measurements can be greatly reduced,

often with a significant reduction in the variance associated with the

measurements. To eliminate the effects of a linear drift in the

measurements, the set of weighings are repeated in reverse order, e.g.,

A,B,C,D,D,C,B,A for the four nominally equal weights A, B, C, and D.

The designs are, of course, applicable for groups of weights which are

nominally equal as shown in the tables for the 5,2,2,1 series.

An example is given of the use of these direct reading designs in

mass calibration on page 17.

TREND ELIMINATION

When responses are time dependent due to temperature and atmospheric

changes, proper ordering can make the values for the weights independent

of any drift effect. The designs A. 1.3, A. 2. 5, A. 2. 2, A. 2. 4, E.l, and

E.2 have this property. If one wishes to use the trend eliminating

property of the design (they are valid as given, of course) , then one

can account for a drift effect of the form . . . -3A, -2A, -A, 0, A,

2A, 3A . . . if n is odd or by . . . -5A, -3A, -A, A, 3A, 5A . . . if

n is even. This will not change the computations for the weights or

their variances. However, the degrees of freedom are reduced by one

and the deviations will be different. An example of this usage is

Design E.l given in Tables 9 and 10. See [4] for details of the analysis.

13
EXAMPLE OF CALIBRATION DESIGN IN MASS CALIBRATION

As an example of the calibration process employed in the mass

laboratory at NBS , consider six weights (50, 30, 20, 10, 10 and 10 grams)

which are to be calibrated using Design C.2. The complete analysis for

this design is given in Table 5 in Appendix B.

The sum of the 50, 30 and 20 gram weights, estimated from a previous

series, forms the restraint, and one of the 10 gram weights is an NBS

standard.

If the calibration is done as a double substitution weighing on a

one pan balance, the given observations after converting the balance

readings to milligrams are:

Y1 = .370

Y

2 = -.499

y

3 = -.074

y

4 = -.079

y

5 = .395

y

6 = .395

y

? = -.454

Y

8 = .405

Y

g = .495

y

10

= .095

yn

= .490

The restraint given is m = 0.862

Denote the value of the correction to nominal size of the weights

50, 30, 20, 10, 10 and 10 grams by 6,, . . . 3 & respectively. Using

14
the multipliers given under "Parameter Values,"

the least squares estimates of the 3.'s as functions of the observations

are: K

= 920 <100(yi4Vy3*y4

) + 60y

5 - 20(y6+y 7+y8+

y

9 +y10+yll ) + 460m} = * 395

/\ i 6 = {-68(v,+y +y +y,) - 4v + 124(v +y +y ) - M

2 920 yl

y

2 y3 y 4

; y

5

y 6y

7 y 8

60(y

9+Y10+yil ) + 276m} =

- 254

*3

= 920 { - 32 (Wy

3 +y4

) - 56y

5 " 104 (Vy7+y8

) +

80(Y9 +Yl0+y

il ) + 184m} =

* 213

^4 = 920 ^ 119y

1+4y2- 111y3+4y

4 - 108y5+128y6

- 102(y

7 +y8

) -

125(y

g+y10

) - 10y + 92m} = .069

^5

= 920

^-111y1

+119y2 + 4(y

3 +V - 108y5 " 102y6 + 128Y 7 "

102y

8

- 125y

g - 10y

1() - 125yu

+ 92m} = -.357

^6 = 920 <4y1 -llly

2 +119y

3 +4y

4

-108y

5 -102(y

6+y

7

) + 128y

8

-

10y

g

- 125(y -+y- ^J + 92m} = .070

A A ^ Note that $ , 3_ and $ necessarily sum to m.

The deviations or differences between the observed and calculated

values can be computed using the multipliers given under "Deviations" in

Table 5. For example, the deviation corresponding to y. is given by

dSV

l = 184 {98yl " 17(Y 2 +Y 3

) " 40y

4

" 24y

5 " 38y6 +

54y

7 + 8(y

8+y

g

) + 31y

1£) - 15y11

) = .016

15
= {^ Z dev2

}

15 = .013

Similarly, &ev7 = ~*001

dev = -.003

dev„ = -.007

4

dev = -.005

dev,. = -.002

6

dev = .001

dev = .007

o dev = -.006

dev = .021

dev = -.010

The standard deviation for this calibration is

11

6 i=1 l

and would be used as an estimate of the process standard deviation if

that were not already established.

From the "Inverse" of normal equations we can compute the variance

for each estimate or linear combination in terms of the process

standard deviation Q. For example,

/\ on V < 62' =^°

64 9 v(6

3

) =4ja*

v(B

i

;b 2» = 555 {5 ° + 82 " 34 " 34}°

2 = ^°2

The usual correction for buoyancy has not been applied to these data

for the purpose of this example since all the weights have the same

density with the exception of the check standard.

16
EXAMPLE OF DIRECT READING DESIGN

In order to eliminate time dependency in a mass series, a schedule

was developed as in Table 10 in Appendix B which balances out the drift

effect. In this example, the corrections to four kilogram weights are

to- be computed using n = 8 observations. Since n is even, the drift

can be represented by -7A, -5A, -3A, -A, A, 3A, 5A, 7A. In addition to

four parameter values, |3, , 3?» 3o and B-, for the weights and A for

drift, a value for the tare weight T is computed. T always appears

negatively in the design because a direct reading is actually the difference between the unknown weight and the tare weight in the balance.

The correction to the third weight is taken as known, namely m = 11.906,

and assuming for the purpose of this example, that any effects due to

temperature or air density are negligible, the observations are:

y

1 = 39.112

y = 44.697

y

3 - 38.655

y. = 44.150

4

y = 44.150

y

6 = 38.685

y

? - 44.778

y - 39.207

o Using the multipliers given under "Parameter Values" in

Table 10, the least squares estimates of the parameters are given by:

17
K

= 2" (Y 1 " y 3 " Y

6 + Y8

} + m = 12 - 3955

^2 = 2" (y 2 " Y 3 " Y6 + Y

7

} + m = 17 -9735

2 = 11.9060

3

3 = m

3 4 = |(-y3 + Y4 + V

5 - y

6

) + m = 17.3860

1

T = 2"(-y

3 - Y

6

) + m = -26.7640

*

= l?8 ( - 7y

i " 5y

2 " 3y3 " y 4 + y 5 + 3y

6 + 5y7 + 7y8> = -°°69

Associated variances in terms of the process standard deviation O are

as follows:

v(6

1

) = a

2

v($

2

) = a

2

v(B

3

) = o

v(3

4

) = a

2

1 2 V(T) = J

°

V(A

* lo > =ik°

The deviations or differences between the observed and calculated

values can be computed using the multipliers under "Deviations" in

Table 10. For example,

dSV

l = li8 {35(y

l " y2 + y7 " V + 21(y

6

" Y 3

) + ?(y

5 " y4 )} = - 001

Similarly, dev_ = -.006

dev

3 = .006

dev

4 = .007

18
dev

5 = -.007

devr = -.006

6

dev_ = .006

dev

8 = -.001

The standard deviation s for this experiment is

i 8 1/2

s = {= Z dev. } = .009

3 i=l x

and would be used in an F-test to verify that this run had a standard

deviation consistent with the process standard deviation O.

19
APPENDIX A

LIST OF DESIGNS

A. Designs for Nominally Equal Groups

A.l All Distinct Intercomparisons

A. 1.1. k = 3 n = 3

*A.1.2. k = 4 n = 6

A. 1.3. k = 4 n = 8

(Cameron-Hailes)

Trend Elimination

A. 1.4. k = 5 n = 10

A. 1.5. k = 5 n = 10 (Cameron-Hailes)

Trend Elimination

A. 1.6. k = 6 n = 15

A. 2 Subsets of All Distinct Intercomparisons

A. 2.1 k = 6 n = 8

A. 2.

2

k = 6 n = 12 Trend Elimination

A. 2.

3

k = 7 n = 10

A. 2.4 k = 7 n = 14 Trend Elimination

A. 2.

5

k = 8 n = 8

A. 2.

6

k = 8 n = 12

A. 2.

7

k = 8 n = 15

A. 2.

8

k = 8 n = 16 Trend Elimination

A. 2.9 k = 9 n = 12

A. 2. 10 k = 10 n = 10

A. 2. 11 k = 10 n = 12

A. 2. 12 k = 10 n = 15

Complete analysis for 1,1,1,1 design given in Tables 1 and 2

in Appendix B.

20
A. 3 Designs Involving Grouping of Weights. v = no. of weights

in each group.

A .3.1 k = 6 n = 10 V = 3 (Bose-Cameron)

A .3.2 k = 6 n = 15 V = 2 (Bose-Cameron)

A .3.3 k = 7 n = 7 V = 3 (Bose-Cameron)

A .3.4 k = 8 n = 7 V = 4 (Bose-Cameron)

A .3.5 k = 8 n = 14 V = 2 (Bose-Cameron)

A .3.6 k = 9 n = 9 V = 4 (Bose-Cameron)

A .3.7 k = 9 n = 12 V = 3 (Bose-Cameron)

B . Z , Z f . . , 1,1, . • . Designs

B.l 2,1,1,1 k = 4 n = 6

B.2 2,1,1,1,1 k = 5 n = 13

*B.3 2,2,1,1 k = 4 n = 6

B.4 2,2,1,1,1 k = 5 n = 10

B.5 2,2,1,1,1,1 k = 6 n = 15

B.6 2,2,1,1,1,1,1 k = 7 n = 10

B.7 2,2,1,1,1,1,1,1 k = 8 n = 16

B.8 2,2,2,1,1,1 k = 6 n = 12

B.9 2,2,2,1,1,1 k = 6 n = 15

B.10 2,2,2,1,1,1,1 k = 7 n = 12

B.ll £tfi£f£-f4~f-l-fj- k = 6 n = 11

*Complete analysis for 2,2,1,1 design is given in Tables 3 and 4

in Appendix B.

21
C . 5,3,2 ,1 and 5,2,2,1 Series

C.l 5,3,2,1,1 k = 5 n = 8 (Hayford)

*C2 5,3,2,1,1,1 k = 6 n = 11

C.3 J|J;Z^ZfX;-L k = 6 n = 11

C.4 5,3,2,2,1,1 k = 6 n = 12

C.5 *D / ~J p -J) f £ i jl f

A- k = 6 n = 12

C.6 5,5,3,2,1,1,1 k = 7 n = 15

C.7 5,2,1,1,1 k = 5 n = 7 (Hayford)

C.8 5,2,2,1,1 k = 5 n = 7 (Hayford)

C.9 5,2,2,1,1 k = 5 n = 8 (Benoit)

**C10 5,2,2,1,1,1 k = 6 n — 8

D. Binary and Miscellaneous Series

k = 5 n = 6

k = 5 n = 10

k = 5 n = 11 (Benoit)

k = 6 n = 10

k = 5 n = 9 (Benoit)

k = 6 n = 9

k = 7 n = 14

*Complete analysis for 5,3,2,1,1,1 design in Tables 5 and 6 in Appendix B,

**Complete analysis for 5,2,2,1,1,1 design in Tables 7 and 8 in Appendix B.

D.l 4,2,2,1,1

D.2 4,2,2,1,1

D.3 4,2,2,1,1

D.4 4,2,2,1,1,1

D.5 4,4,2,1,1

D.6 4,4,2,2,1,1

D.7 4,4,2,2,1,1,1

22
D.8 4,3,2,1,1 k = 5 n = 8 (Hayford)

D.9 4,3,2,1,1 k = 5 n = 9 (Benoit)

D.10 10,4,3,2,1,1 k = 6 n = 10 (Hayford)

D.ll 10,5,2,1,1,1 k = 6 n = 8 (Hayford)

D.12 10,5,2,2,1,1 k = 6 n = 8 (Hayford)

D.13 10,5,3,2,1,1 k = 6 n = 10 (Hayford)

D.14 5,4,3,2,1 k = 5 n = 7 (Hayford)

D.15 10,5,4,3,2,1 k = 6 n = 10 (Hayford)

D.16 6,5,4,3,2,1 k = 6 n = 9 (Hayford)

D.17 10,6,5,4,3,2,1 k = 7 n = 12 (Hayford)

Direct Readings

* E.l 4 nominally equal weights Trend elimination

E.2 2,2,1,1 Trend elimination

Complete analysis for direct reading design 1,1,1,1 is given in Tables 9

and 10 in Appendix B.

23
USES OF THE DESIGNS

Description of Designs

Each design lists k, the number of weights; n, the number of

measurements; and d.f., the degrees of freedom associated with the

standard deviation.

The identification or nominal size of each weight 3. is given next

to the heading "Observations." Y(l), Y(2), . . . denotes the measurements

where + indicates the weight is present positively, and - indicates the

weight is present negatively.

For example, Design A. 1.1 involves three equal weights. If used as

a starting design for a 1 kg to 1 mg set, the three weights would be

kilograms, and the first observation Y(l) would be the difference between

the first and second kilograms.

Two restraints are listed although others are possible. Usually

"Restraint A" is appropriate for descending series involving, for

example, 5+3+2 which would be calibrated as a "Ten" weight in the

higher series involving 50, 30, 20, 10. For ascending series, "Restraint

B" is usually a single unit weight.

In the case of three equal weights, Restraint A takes the sum of

two kilograms as known; Restraint B takes the single third weight as the

reference standard.

"Factors for Computing Standard Deviations" give the multipliers needed

to calculate the standard deviations of linear combinations of the weights.

"Wt" identifies the total weight combination or load L. where

k

L. = £ £ . 3 . , £ . = or 1

i i=1 i i' i

24
The next two columns under Restraints A and E list the factors D where

D. is calculated so that

1

2 2 Variance (L.) = D. Q

l l

The remaining columns, under the nominal sizes of the weights, show the

actual weights involved in the combinations, i.e. a + indicates I. = 1

and a blank indicates £. =0.

Selection of the Appropriate Design

The selection of the appropriate design is usually dictated by the

set of weights to be calibrated and the reference standards one has

available. There is occasionally some flexibility as, for example, if

one were working up from 1 kg to 100 kg from a single reference kilogram.

If either the 1,1,1,1,2 Series (Design B.2) or the 1,1,1,2,2 Series

(Design B.4) were usable one would make the decision on the basis of the

standard deviation of Z5, the 5 kg summation. In design B.2 the factor

for computing the standard deviation is 1.67 33, whereas in B.4 it is

2.1909. [Even if one compares the two standard deviations on an equal

number of observation basis (n = 13 vs. n = 10) one would have 1.6733 vs,

1.9215 ( = 2.1909 /10/13) ] . The advantage clearly is with design B.2

because in the next series, 15, 5, 5, 5, 10, the uncertainty associated

with the starting standard £5 is smaller.

When the standard deviation for the weight combinations of interest

(computed using the standard deviation associated with the measurement

process) exceeds the tolerance given by the requirements, then one has

to consider the feasibility of repeating a design to reduce the standard

J_ _1_ deviation factors by rr (or y— if n repetitions are made)

.

25
DESIGN A. 1.1

CESEfiVAT IONS

V<1> Y<2>

Y(3» RESTRAINT A RESTRAINT B

THREE EQUAL WEIGHTS

FACTORS FOR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .7071 .0000

1 .4082 .8165

1 .40e2 .8165

2 .0000 1.4142

3 .7071 1.4142

OESI GN A . 1 .

3

OESERVAT IONS

Y<1 ) Y(2)

Y(3)

Y(4>

Y(5)

Y(6)

Y(7>

Y(6) RESTRAINT A RESTRAINT B

FOUR EQUAL WEIGHTS TREND ELIMINATION*

(CAMERON-HAILES

)

1

N= 8

• F .= 5

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A .5204

.5204

.3227 .3227

.0000 .5204 .8165

B .0000

.6455

.5774

.6455

1.0 40 8 1.5275

1.5275

*See page 13.

jM\\%%%%%%%%%X%%%X%%%%%%%%%%%%%%%^%%«%%Wa%%%%%%%%%%%%%«^^

DESIGN A. 1.2

OESERVAT IONS

Y( 1) Y<2>

Y<3>

Y(4)

V(5)

Y(6) RESTRAINT A RESTRAINT B

FOUR EQUAL WEIGHTS

FACTORS FCR COMPUTING ST OEVS

T RESTRAINTS

A B

1 .6124 . 0000

1 .6124 .7071

1 .3536 .7071

1 .3536 .7071

2 .0000 1 .2247

3 .6124 1.7321

4 1 .0000 1.7321

DESIGN A. 1.4

OBSERVATIONS

Y( 1) Y(2)

V<3)

Y(4)

Y<5)

Y(6) Y<7>

Yte)

Y<9)

Y( 10) RESTRAINT A RESTRAINT B

FIVE EQUAL WEIGHTS K= 5 N=l D.F.= 6

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .5477 .0000

1 .5477 .6325

1 .5477 .6325

1 .3162 .6325

1 .3162 .6325

2 .0000 1.0954

3 .5477 1 .5492

4 .8944 2. 0000

5 1.2247 2. 0000
DES IGN A .1 .5

CESERVATICNS

Y( 1) Y<2)

V(3)

r ( 4) Y(S)

Y<6> Y<7>

Y18>

Y(9)

Y( 10) RESTRAINT A RESTRAINT B

FIVE EQUAL WEIGHTS TRENO ELIMINATION *

( CAMERON-HAILES)

K= S N=10

.F.= 6

1 1 1

FACTORS FOR COMPUTING ST DEVS

WT RESTRAI NTS

A B .5477 .0000

• S477 .6325

.547"? .6325

.3162 .6325

.3162 .6325

2 .0000 1.0954

3 .5477 1 .5492

A .8944 2. 0000

5 1 .2247 2.0000

See page 13. !

DESIGN A .2.

1

observations

y(i ) Y<2I

Y(3)

Y(4)

Y(5)

Y(6)

Y(7) Y(8) RESTRAINT A RESTRAINT B

SIX EQUAL WEIGHTS K= 6 N= 8

D .F .= 3 111111

FACTORS FOP COMPUTING ST DEVS

T RESTRAINTS

A B .7071 .6 236

.7071 .6236

.""07 1 .6236

.7071 .6236

.3536 .4249

.3536 .4249

2 .0000 .4714

3 .7071 .707!

4 1 .0000 . 74 = 4

5 1.2247 • 6236

6 1 .4 142 . 0000

'V'^%.'^'%.'^^^%.'^^%.'^%.'^^«-'^'V'^'«.'%.'%.'%^^

DESIGN A. 1.6

OeSEPVATICNS

Y<1)

Y«2)

Y<3)

Y(4) YJ5)

Y(61

Y(7)

Y( e> Y(9)

Y( 10) YUl ) Y< 12 ) Yt 13) V(14l

Y( 15) RESTRAINT A RESTRAINT B

SIX EQUAL WEIGHTS K= 6 DESIGN A .2.2

N=l 5 F. = l 4 OESERVATIONS

Y( 1 ) Y(2>

Y(3>

Y(4 > Y(5>

Y(6)

Y( 7) Y<8 ) Y( 9) Y( 10)

Y( 1 1 ) Yl 12 )

FACTORS FCR COMPUTING ST DEVS

WT RESTRAI NTS

A B .5000 .0000 .5000 .5774

.5000 .5774

.5000 .5774

.2887 .5774

.2887 .5774

2 .0000 1 .0000

a .5000 1 .4142

4 .8165 1 .8257

c 1 .1180 2 • 2361

6 1.4142 2 .2361

SIX EQUAL WEIGHTS TREND ELIMINATION

K= 6 N=l 2

• F.= 7

RESTRAINT A RESTRAINT 3

FACTORS FCR COMPUTING ST

WT RESTRAINTS

A B 1 .5401

. S401 .5401

.5401 .3536

.3536

2 . 000C

3 .5401

4 .8165

5 1.1365

6 1.4142

*See page 13.

. 00C0

.7 07 i .645 5 .6455

.6455

.6455

1. 0801 1.5546

2. 0000 2.5495

2.5495
DESIGN A. 2.

OBSERVATIONS

»{ 1 ) V( 2> Y<3>

Y{4)

Y<5)

Y(6)

Y(7)

V(8) Y<9)

Yl 10) RESTRAINT A RESTRAINT B

SEVEN EQUAL WEIGHTS

1

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B .7071 .6389

.7071 .6389

.7071 .6389

.7071 .6389

.7071 .6389

.3 162 .3886

.3162 . 3886

2 .0000 .4518

*a .7071 .7284

4 1.0000 .8207

5 1.2247 .7954

6 1 .4142 .6389

7 1.581 1 . 0000

"IN=10 f D.F.= 4

:|i i i J

+ *

: I

+ + + a

DES IGN A .2.5

CESERVAT ICNS

Y( 1) Y(2)

Y(3)

Y(4)

Y(5)

Y(6) Y<7>

Y( 8) RESTRAINT A RESTRAINT B

EIGHT EQUAL WEIGHTS K= 8 N= 8 ,F.= 1

FACTORS FOR COMPUTING ST DEVS

T RESTRAINTS

A B .8660 8101

.8660 8101 .8660 8101 .8660 8101 .7906 8101 .7906 8101 .7906 8101

.7906 .8101

2 .7071 7906

a .7906 .9520

4 . OCCO 7 07 1 5 .8660 .9520

e 1.2247 1 ,0607

7 1 .3229 .8101

8 1 .4142 ,0000

NMiwniWMNiwvNMiwvMNMMnNMnNNNnNnMMniVNnnMnftNnnnMnNnaiMNnniViKvsHKwvwKwwwwwwwwwwwwwwwKww^

OESIGN A. 2.

OESERVAT ICNS Y<1)

Y( 2) Y(3)

Y(4)

Y(5)

Y(6>

Y(7)

Y(8)

Y(9)

Y( 10)

Y( 1 1) Y( 12 ) Y( 13)

Y< 14) RESTRAINT A RESTRAINT B

SEVEN EQUAL WEIGHTS

TREND ELIMINATION *

FACTORS FCR COMPUTING ST DEVS

T RESTRAINTS

A B .5694 .4447

.6157 .4447

.6504 .4447

.6157 .4447

. 5694 .4447

.3231 .4447

.3231 .4447

2 .0000 .6112

3 .5694 .7110

4 .9945 .7110

5 1 .4507 .6112

6 1 .8337 .4447

7 2. 139 4 .0000

•See page 13.

N=14 p D.F.= 8 4

1 \1 1 J :l+ j

+ + + j

DESIGN A. 2.

OBSERVAT ICNS

V( 1) Y(2)

Y13)

Y<4)

Y(5>

Y(6)

Y(7)

Y(8)

Y(9)

Y<10)

Y( 1 1 ) Y( 12) RESTRAINT A RESTRAINT B

EIGHT EQUAL WEIGHTS K= 8 N=12

.F.= 5

FACTORS FCR COMPUTING ST OEVS

T RESTRAINTS

A B .7071 .6495

.7071 .6495

.7071 .6495

.7071 .6495

.7071 .6495

.7071 .6495

.2887 .3608

• 2ee7 .3608

2 .0000 . 4330

•a .7071 .7395

4 1 .0000 . 866

5 1.2247 .8927

e 1.4142 .8292

7 1 .581

.6495

8 1.7321 .0000

28
HI I- 3 <

Z

-J >- < z

D -

J

HI W

h Q 1 z O UJ

1 + +1

3i/)£Dooooooooc?imr'-!T )o*o

o — <\i "i < m o u (/> IL UJ O O <f 3'OO'C*0OO<->C0m^^- ir^NiMov^Dnioi? ^o » <u c\J o

t\j n ^ uj \o i> cd

+ + + + + + + +

+ + + + +

l/)tDl/)li)l/)lfilflO, a, 0'N*ir)'0«lf)0

z ir)intf")ir>inrororo<t-*yo>0v0ino

c\j n ^ i/> -o n co > * > > > > > >>>>>>>

^q'-*-*^-^^ — -* o <* w o s o

i>r-. IJsr*-N'0iO'O'Oo>Np*omc7»

i/)ini/)i;)ifi(*)P)r^r')ou)CDOHW

i\i p) c «) >u MD

29
o o

I?

a k

s z o «

u <

IT a iDDooeototDcocoaJcDictooowioot/jcocotoo OOOOOOOOOO-^^CJil^On^^oO

G1 (?> Q> J»0*C, 0"'(7*0*^0*OONOOo\^o

U* a» y (^ (J>

<r-*<t tf ^ * •$ * o r- <*cofM—« ^ <j -* ^ * ^ 44ocuuinu<H

<DC0C0^CDCDItDO(J>Off)|/')lfi

i/inr)P)P)i')(n(,i(roi (?coNCOwaiirio

k aia»C7v o^oi <?, oooor-rrim«NO>o

z <o»o\o ,0'0'Oifiifiin'Oif) |BOu, ooiOo

U V) <t <t <t <t *

U. W < U» 10 U! If) U) U> NNNNt-N-*^ tf

•*« ^*0^<jNh- ,U U * A| W l») cr n

o n o c\j •* u> r

IM PI » tf) *U (*- DO U*

30
o in >o

n IT ii x z .

i +

+ + + + +

+ + + + +

+ + + + + + + + + +

H OUJCOaJIOCDtDlDCOlDOO^^N^OtDo

OT </>

*t»n)\o»W'-«^-«^^io ,iJ -> o >o ir> o r- o

Nf-r-r~N(\jc\j<\j(\if\iooc\jor-cucoorff I/)* MI) pio

17 Ii* ii * z •

u.o -

+ + + + +

I z

o «

u <

OOOOoO-^-tf 444^0)4OCVOO oiAiomiftiniAiOiototi^nNiADin^ino **4**4hPir)noo^n*Ha)*o

~ — — (- o ina

co ct> o

-« \0 o (\i n

-^ D in »o r^

<\jr»)^uivf^cu(?io

31
DESIGN A. 3.1

OBSERVATIONS

Y< 1 > Y<2) Y<3>

Y<4)

Y(5>

Y<6)

Y<7)

Y(8)

Y(9)

Y(lO) RESTRAINT A RESTRAINT B

SIX EQUAL WEIGHTS

( BOSE-CAMERON) 11111

FACTCRS FCR COMPUTING ST DEVS

WT RESTRAI NTS

A B .3536 . 0000 .3526 .4082

.3536 .4082 .3536 .4082

.2041 .4082

.2041 .4082

£ .OOCG .7071

3 .3536 1.0000

4 .5774 1 .291

B .79C6 1. 581 1 6 1 .0000 1 .581 I

DESIGN A. 3.

2

OBSERVATIONS

Y( 1) Y<2)

Y(3>

Y(4)

Y<5)

Y(6)

Y<7)

Y<8)

Y(9)

Y<10)

Y{11) YX121

Y( 13) Y<14>

V( 15) RESTRAINT A RESTRAINT B

SIX EQUAL WEIGHTS

( BOSE-CAMERON

)

1 1 1

FACTORS FOR COMPUTING ST DEVS

T RESTRAI NTS

A B .3536 .0000 .3536 .4082

.3536 .4082

.3536 .4082

.2041 .4082

.2041 .4082

2 .0000 .7071

-a .3536 1. 0000

4 .5774 1.29 10

E .7906 1. 581 1 6 1.0000 1.581 1

K= 6 N=l J D.F.= 5 5

I

\ "1

K= 6 4 N= 1 5 '

| D.F.=10 J

i

DESIGN A. 3.

3

CBSERVAT IONS

Y( 1 ) Y<2» Y<3)

Y(4)

Y(5)

Y(6)

Y(7) RESTRAINT A RESTRAINT B

SEVEN EQUAL WEIGHTS

< BOSE-CAMERON) 11111

K= 7 N= 7 D.F.= 1

1

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B 1 1 .4629 .4629 .4629

.4629

.4629 .2673

.2673

.0000 .4629

.75 59 1.0351 1.3093

1 .5811

. 0000 .5345

.5345

. 5345

.5345

. 5345

.5345 .9258 1.3093

1.6903

2. 0702 2.4*95

2.4495

DESIGN A. 3.

4

CBSERVATIONS

Y< 1) Y<2> Y(3)

Y(4)

Y<5)

Y(6)

Y(7) RESTRAINT A RESTRAINT B

EIGHT EQUAL WEIGHTS

( EOSE-CAMERON

)

11111

K= 8 N= 7 D.F.=

FACTCRS FCR COMPUTING ST DEVS

T RESTRAINTS

A B .4330 »OOO0 .4330 .5000

.4330 .5000 .4330 .5000 .4330 .5000

.4330 .5000 .2500 .5000

.2500 . 5000

2 .0000 .8660

3 • 433C 1.2247

4 .7071 1.5811

£ .9682 1.9365

e 1.2247 2. 2913

7 1 .4790 2.6458

8 1.7321 2.6458

32
DESIGN A .3.

5

EIGHT EQUAL WEIGHTS K= 8 N=l 4

( eOSE-CAMERON) D.F.= 7 06SERVAT IONS

Y( 1 t Y(2>

Y(3)

Y(4>

Y(5)

Y(6)

Y<7)

y( e) Y<9)

Y( 10) Yd 1 ) Y( 12)

Y( 13 ) Y( 14)

1 1 1 1 1

RESTRAINT A RESTRAINT B

FACTORS FOR COMPUTING ST DEVS

WT RESTRAI NTS

A B 1 1 .4330 .0000 .4330 .5000 .4330 .5000

.4330 .5000

.4330 .5000

.4330 .5000

.2500 .5000 +

. 2500 .5000 +

2 .OOCC . 866 + + 3 .4330 1.2247 + +

4 .7071 1. 581 1 + +

c .9682 1.9365 + +

e 1.2247 2.2913 + +

7 1 .4790 2.645e + +

E 1.7321 2.6456 + +

33
- II I I + + I

II II II J- II * z •

Q - I I + + I

I +1 + + +

^ m - i + + + |)| + 3 < O U

HI I ~ + + II || ,,

w in z o - m _, + z w

i +i +i + +

+ + + +

!!>»>•*>•>>>>>>-

+ + + + + + +

+ + + + + + + + +

O tf * <? •*

o -

u <

a

^^^^^vO-SOinO^CUcO

h-f'-NNNf-iU)CFC\J>oc>c\Jc\J ^^^^^CO-«^CO-«^cOCO

-* ** -h c\j (\j cm c\j tMWOJ(\J(MCUWNI^Ot\Jrv(3>l^stoa3 <Qjmajtoai(K(Dii)mo'Ooivifl^f)o

**tf-tf^-3-*r\lc\JO-* ,O(7>-«r'|v0C0

a ^ -

ll II ii v z

u.

+ I + I + I + I

I I + I I + +

+ + I I I + I +

+ + I I I

t/> -. I I I + + + I

UJ z * o

a. j li - | + + + i + i i < z o u

UJ I - + I I + I + + I UJ

in in z o

"- m -i+ i + ii+ + +

»• + r + + i i i in id o h h

I- o I"- N i Z O •» -*

++++++

+ + + + + + +

+ + + + + +

+ + + + + + + +

-!.-.„_ _, o»oiho»u;hj Ni^r-rv i^^u)c>iM"0^iMr\j ^*-*^<tco^<tco^^coco

h- niN m m

> > * > > > > * > 2

iUJa

<\!ojc\i{\joj(\jc\jr-i>-oc\jh.mi^^oco cocococuiucocoioujocuvDiy^r^i'jo ooooooororooo0-«tnc7roN

im r*) ^ m ^) r- co (j»

34
CESI6N B.l 2,1.1.1 I

K = 4 f DESIGN B.3

(HAYFORD)

CESEFVATICNS 2 111 4 OESERVATIONS

Y<1> + "

- v <»> V<2) 5 V(2)

. . . _ . a — — r* v f i *

2,2,1.1 K= 4 N= 6 j> N= 6 F.= 3 D.F.= 3

Y<3) g Y(4)

Y(S)

Y(3)

Y<4)

* V(5)

A B

1 .4243 .0000

1 .4243 .7071

1 .4243 .7071

£ .3464 1.0000

3 .4690 1.5811

1 .4243 2.1213

£ .0000 2.1213

A B

1 .4564 .0000

1 .4564 .5774

2 .2673 .9512

2 .2673 .9512

3 • 52e9 1 .3801

4 .0000 1 .8257

s .4564 2 .2361

6 .7071 2 . 2361

Yi6i " *<e> +

s RESTRAINT A + + + + / RESTRAINT A + +

RESTRAINT B + t RESTRAINT B +

\ FACTORS FOR COMPUTING ST DEVS 4 FACTORS FCR COMPUTING ST DEVS

*T RESTRAINTS WT RESTRAINTS

2 111 S

*

+

* * * If. +

[

_. ! ...... N=13 4 N=10 D.F.= 9 O.F.= 6

s CeSERVATICNS 2 1111 J OESERVATIONS 2 2 111 YI1I + VIII + - - + V<2> + - +

J Y(2) + - + Y<3) + - - + - Y<3> + - + Y<4> * - ~ - + % y(4) + Y<5> + 4 Y(S) + - - Y(6) + - Y<6) + Y<7> + 5 r <7) Y(8) 4 y ( e> Y(9) + Y<9 ) Y(lO) + " Y<10>

Y(ll) + + -_ J V(12) + - + - RESTRAINT A V<13» + - - + 4 RESTRAINT A RESTRAINT B * !f

FACTORS FCR COMPUTING ST DEVS 4

I

RESTRAINT B

FACTORS FOR COMPUTING ST DEVS

T RESTRAINTS

A B

1 .3347 • osoo

1 .2683 .4472

1 .2683 .4472

1 .2683 .4472

2 .2191 .6325

3 .2966 1.0000

4 • 26e3 1.3416

5 .0000 1.6733

e .3347 1.6733

T RESTRAINTS

A B

1 .4382 .ooco

1 .4382 .6325

1 .3347 .6325

2 .2710 .8634

2 .2710 .8634

3 • 271C 1.3 94 8 4 .3347 1.6733

£ .0000 2. 1909

6 .4382 2. 6833

7 .6066 2.6833

35
DESIGN B.5

OBSERVATIONS

Y<1 ) V<2> Y(3)

V<4)

Y(SJ

Y<6)

Y(7)

Y(8)

Y<9)

Y( 10 > Y{ 11 ) Y( 12 ) Y{ 13)

Y< 14)

Y( IS) RESTRAINT A RESTRAINT B

2.2.1.1.1.1 K= 6 N=l 5

.F .=1

FACTCRS FCR COMPUTING ST DEVS

T RESTRAI NTS

A B

1 .2614 .0000

1 .2614 .3536

1 .2614 .3536

1 .2082 .3536

2 .2051 .5496

2 .2051 .5496

2 .2051 .8229

4 .2082 1. 0408

5 .0000 1.3070

6 . 2614 1 .5679

7 .3851 1.8257

8 .4899 1 .8257

DESIGN B.6

OBSERVATIONS

Y< 1 ) Y<2)

Y(3>

Y(4 ) Y(S)

Y<6)

YI 7) Y(8)

Y(9)

Y< 10) RESTRAINT A RESTRAINT B

2.2.1.1,1.1,1 K= 7 N=l

.F .= 4

FACTCRS FCR COMPUTING ST DEVS

WT RESTRAINTS

A .4000 . 0000

.4000 .5774

.4000 .5774 .4000 .5774

.3055 .5774

2 .2708 .7958

2 • 27C8 . 7958

3 .2708 1 . 2780

4 .3055 1 .5275

5 .0000 2 .0000

6 .4000 2 .4495

7 . £538 2 . 8868

e .6623 2 .3166

9 .7483 3 .3166

36

DESIGN P.

7

OBSERVATIONS

Y( 1 ) Y12)

Y(3)

Y{4) Y<5>

Y(6)

Y(7>

Y(8)

Y(9)

Y( 10)

Y( 1 I ) Y( 12) Y113)

Y( 14)

Y ( 15)

Y( 16) RESTRAINT A RESTRAINT 8

K= 8 N=l 6 D .F .= 9

FACTOR S FCR

REST

A .2550

.2550 .2550

.2550

.2550

.-20C0 .1871

. 1871 .1871

. ?000

. occc

.2550

.3674 .4563 .5385

.61 24

CO

RAJ MPUTI NTS

. COCO

.3536

. 3536

.3536

.3536 .3536

. 5244

. 5244

.8062

.0000

.2748

.541 1

. S028

.0616

.3184

,31 84

+ +

+ +

NG ST DEVS

2 2 11
DESIGN B.8

OBSERVATIONS

Y<1)

¥(2)

Y(3)

Y<4)

V(5)

Y<6)

V{7)

Y(8)

VC91

Y( 10 ) Y( 11 > Y< 12) RESTRAINT A RESTRAINT B

2.2.2.1.1.1 K= 6 N=l 2 D.F.= 7

FACTORS FCR COMPUTING ST OEVS

T RESTRAINTS

A B

1 .5069 . 0000

1 .5069 .8165

1 .5069 .8165

2 .5C00 1.0541

2 .2887 1.0S41

2 . 2887 1.0541

3 .7682 1.7638

4 .5774 2.0276

5 • 82C7 2.7285

6 .5000 2.0000

7 .7683 3.6966

e .8819 4. 3589

9 .9014 4.3589

DESIGN B.10

OeSERVAT ICNS

Y( 1) Y(2)

Y(3)

Y<4)

Y<5> Y(6>

Y<7>

Y(8)

Y(9)

Y(10)

Y( 11 ) Y( 12) RESTRAINT A RESTRAINT B

2.2 .2 . 1 .1 .1 .1 K= 7 N=12

,F.= 6

FACTORS FCR COMPUTING ST DEVS

T RESTRAI NTS

A B

1 .4183 .0000

1 .4183 . 5774

1 .4183 .5774

1 .4183 .5774

2 .5477 . 8944

2 .3162 .8944

2 • 3162 .8944

2 .7583 1.34 16

4 .6325 1.6733

5 .8216 2.1 134

6 .5477 2.4495

7 .7583 2.8868

e .9309 3.3166

9 1 .0840 3.7417

10 1 .2247 3.7417

^v^^^'^%."»."^"^'^^'^^'^^'^'^'^^-^'^'^»^'»^^^^'^^'%.-^'^^

DESIGN B .9

CESERVATICNS

Y( I ) YI2)

Y(3)

Y(4)

Y(5)

Y(6)

Y<7)

Y<8)

Y<9)

Y( 10)

Y( 1 1 ) Y( 12)

Y( 13 ) Y( 14) Y(15) RESTRAINT A RESTRAINT B

2.2.2.1.1.1 K= 6 DESIGN E.ll N=15 D.F.=10

V OBSERVATIONS

Y(l)

Y(2> YC3)

Y<4> Y(5I

Y(6)

V(7)

Y<8) Y<9>

Y{ 10)

Y( I 1 )

FACTORS FCR COMPUTING ST DEVS

T RESTRAINTS

A B

1 .3819 . cooo

1 .3819 .5774

1 .3819 .5774

c • EOOC • eies

£ .2887 • ei6S

£ .2687 .8165

3 .6922 1.2910

A .5774 1.5275

5 .7500 2.0000

£ .5000 2.2261

7 .6922 2.7080

a • eies 2. 1622

9 .9014 3.1623

2.2.2.2.1.1 K= 6 N=l 1 • F.= 6

RESTRAINT A RESTRAINT B

FACTORS FOR COMPUTING ST OEVS

til RESTRAINTS

A B 2 2 .0000

1. 0000

1. 1832

1 .1832 1.1832 +

1. 1832 + 2.0976

2.2804

2.1937 3.3764 + 4.2895

4.4721 + + £.3852 + + 5.3852 + +

1 .5701

1 .5701

2 .5477

2 .5477

2 .3162

2 .3162

2 .8515

4 .8944

5 1.151

6 .9487

7 1.1927

6 . 8944

9 1.1511

10 1 .2247
( HAYFORD) OESERVAT IONS 5 "a 2 V( 11 + - - V(2) - - Y<3> + - - V(4) + - YI5) + - Y<6) + - YC71 + v( «) RESTRAINT A + + +

N= 8 .F.= 4

6

\ OESIGN C.l 5.3.2,1,1 K= 5 f DESIGN C.3 5.3.2,2.1,1 K= 6

9. N = l 1

f CESERVATIONS

Y<1 ) 5 Y(2)

\ Y<3> Y<4>

Y(5)

Y<6)

Y<7>

J Y<9>

5 Y(10)

4 Y(ll>

\ RESTRAINT A RESTRAINT B +

i FACTORS FOR COMPUTING ST DEVS 4 RESTRAINT B T RESTRAINTS

A B

1 .4175 .0000

1 .4175 .5345

2 .3546 .9258

3 .3338 1.309 3 4 .5237 1 .6903

e .26 73 2.0702

6 .5237 2.4495

7 .3338 2.9277

e .3546 3.3381

5 .5632 3.7033

.0000 4.1748

1 .3592 .0000

1 .3592 .4714

2 .4259 .7553

£ .3625 .7553

3 .3835 1.2536

3 .6116 1. 1 123

4 .6130 1 .4254

£ .2266 1.7412

6 .4790 2. 0893

7 .3835 2.4199

8 .3625 2.9493

9 .4711 3.2708

.0000 3.5922

5 WT RESTRAINTS

i

\ CESIGN C.2 5.3.2.1.1.1 K= 6 2 OESIGN C.4 5,3.2.2.1,1 K= 6 N=ll ![ N=12 D.F.= 6 f. D.F.= 7

8 OBSERVATIONS 5 3 2 111 j| CBSERVATICNS 5 3 2 2 1 1 Y<1> + - - + v<1) + Y«2> + + y <2 ) + - Y(31 + + 5 Y(3 ) + - Y<4) + 4 Y<4) + - Y<5> + - { Y(5) + " " V(6I + - + Y(6) + - Y<7) + - - +

W

4 Y(7) + Y<8> + - - - + 4 »|8) + - - YJ9I + J Y<9) + V<11» + Y<11> RESTRAINT A + -» +

4 RESTRAINT B + 4

Y( 12) RESTRAINT A

FACTORS FOR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .3551 .0000

1 .3551 .5000

1 .3551 .5000

2 .2638 .7802

3 .2985 1.0 8 85

4 .4778 1.4781

5 .2331 1.7846

6 .4299 2. 1644

7 .2985 2.5216

8 .2638 2.8284

9 .4616 3.2016

10 .0000 3.5509

4 RESTRAINT B

4 FACTORS FCR COMPUTING ST DEVS

4 WT RESTRAINTS

3127 1.0541 2848 1.4530

! i A B

ji 1 .4796 .0000

,' 1 .4796 .8165

2 .4055 1.0541

i

2 f

~ 4 3 .6173 1.7638

ij 4 .4372 2.0276

' 5 .2887 2.3805

6 .588 8 3.1091

[ 7 .S60 8 3.3 83

! 8 .3127 3.8006

I, 9 .6119 4.5216

10 .0000 4.7958

38
DESIGN C.5 5.5.3.2.1,1

observat ions y<i> Y<2>

YJ3>

r<4)

Y<5) V(6)

Y<7)

Y(8)

YI9)

Y( 10) Y<1 I ) Y( 12) RESTRAINT A RESTRAINT B

FACTORS FOR COMPUTING ST DEVS

K= 6 N=12

,F.= 7

WT RESTRAINTS

A B

1 .3536 .0000

1 .3536 .5345

2 .33 84 .7319

-a .3281 1.1802

4 .4236 1. 5924

5 .4366 1.7743

S .3010 1.7743

6 .4790 2.2051

7 .3281 2.4213

e .3364 2.9041

9 .4473 3.3174

1C .0000 3. 5363

DESIGN C.6

OBSERVAT IONS

Y( 1 ) Y<2)

Y(3)

Y<4)

Y<S)

Y(6)

Y(7) Y«8 ) Y(9)

Y( 10)

Y( 1 1 > Y( 12)

Y( 13)

Yl 14 ) Y( 15) RESTRAINT A RESTRAINT B

5>5.3.2.1 .1.1 K= 7 N=15

-.= 9

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .4967 . 0000

1 .4967 .8165

1 .4967 .8165

2 .3464 1. 0465

"a .3251 1.5584

4 .5640 2.2573

£ .2673 2.4689

B .4629 2.4689

6 .7198 3. 1773

7 .5529 3.4572

8 .5904 3.9940

9 .7862 4. 6853

10 .3000 4.9666

39
CESIGN C .7 5,2,1.1.1 K= 5 1£ DESIGN C.9 5.2.2.1.1

N= 7

( HAYFORO) OESERVAT IONS 5 2 1 V( 1 ) + - - Y(2> + - Y(3> + - V(4> + Y(S) + Y(£) + Y(7) RESTRA INT A + RESTRAINT B• F.= 3 ( BENOIT

)

CESERVAT ICNS 5 2 2 Y( 1) + -

Y( 2) + - Y(3) + — Y(4) + — Y(5) + - Y(b) + Y(7) + Y(8) RESTRAINT A

FACTORS FCR COMPUTING ST DEVS

T RESTRAINTS

A B

1 .4690 . 0000

1 .4690 .7071

1 .4690 .7071

2 -5292 1.0000

5 .3000 2. 3452

3 .7616 1.581

1

4 .9055 2.1213

e .4690 2.9155

7 .5292 3.2404

8 .7616 3.8079

9 .9055 4.3589

1 .0000 4. 3589

f RESTRAINT B

A FACTORS FOR COMPUTING ST DEVS

4 WT RESTRAINTS

\A 1 .3780 .5345

1 2 .2361 .9258

A 4 .4140 1.7728

4 £ .37e0 2.0702

A e .3381 3.2950

4 9 .3760 3.7796

!

1 .4140 .0000

3381 .9256 4140 1.3093

»]lO 2. 4495

»140 2.9277

10 .0000 4.1 40"

%%%%%%%%%%«,%W%%%%\%%»%%%%%%%1»%%^%^^^^«^^«MM%\%%^^^

i DESIGN C .8 5.2.2.1.1 K= 5

ta DESIGN CIO

N= 7 A

• F.= 3 ( HAYFORD) \ OESERVAT IONS 5 2 2 V( 1 ) + - - Y(2) + - Y(3) + - Y(4) + - Y(b) + Y<6) + Y<7) RESTRAINT A + + + RESTRAINT B

OBSERVATIONS

Y( I ) Y(2)

Y( 3) t

Y ''») Y( 5) Y(£)

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .4583 . 0000

1 .3786 .5774

2 .3805 .9512

2 .3805 • 9E12

3 .4413 1.3801

4 .5416 1.8257

c .5000 2.4495

6 .5416 2.8868

7 .4413 3. 3022

e .3805 3.7289

s .3786 4.1633

1C .0000 4.5826

A Y( 7) 4 Y(8)

\

4 RESTRAINT B

\4 WT RESTRAINTS

4 1 .4326 .5345

| 2 .3854 1. 1339

I 2 .3854 1.1339

I 3 .37cl 1.4639

4 .5555 2.2039

5 .3273 2.1712

6 .5555 2.5355

RESTRAINT A

FACTORS FCR COMPUTING ST CLVS

A P,

1 .4645 . 0000

1 .4645 .5345

\4 7 .3761 3.2514

: 8 .3654 3.5857

>'! 9 .4326 4.3260

A 10 .0000 4.6445

I

40
DESIGN D.l

OBSERVATIONS

Y<1>

Y(2>

V(3)

Y<4»

V<5)

Y<6> RESTRAINT A RESTRAINT B

4, 2.2, 1 ,

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

I .4677 .0000

1 • 4677 .5000

2 .4743 1. 1619

2 .4743 1.1619

a .5646 1.4491

4 .0000 1.8708

c .4677 2. 1794

6 .4743 2.9749

7 .5646 3.2558

a .7071 4.0620

9 .6847 4.3301

10 .7906 4.3301

DESIGN D.2

OBSERVATIONS

Ytl)

YI21 Y(3)

VC4)

YCS)

V<6) YJ7)

Y<8>

YC9) VC10I RESTRAINT A RESTRAINT B

4.2.2.1 .1

FACTORS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .3536 .0000

1 .3536 .5000

2 .3536 .7906

2 .3536 .7906

3 • 50C0 1.1726

4 .0000 1.4142

c .3536 1.6028

6 .3536 2. 1506

7 .5000 2.5249

8 .5000 2.8723

9 .6124 3.2404

10 .7071 3.2404

-.1 N= 6 j D.F.= 2

I

\ K= 5 \ N=10 f D.F.= 6

DESIGN D.3 4.2,2,1.1

( BENOIT ) C6SERVATI0NS 4 2 2 Y( 1) + - - Y<2> + - - Y(3) + - - Y(4) + - Y<S) + - Y<6> + - YI7) + - Y(8) + - Y(9) + Y< 10) + Y(ll >

K= 5 N=l 1

• F.= 7

RESTRAINT A RESTRAINT B

FACTCRS FOR COMPUTING ST DEVS RESTRAINTS

7

8

10

A .3354 .3354

.3536 .3536

.4873

.0000

.3354 .4873

.5000 .6021

.7071

B .0000

.4472 .7583

.7583

1. 0840

1 .3416 1.6733

2.361

2.7295

3. 0414

3. 0414

DESIGN D .4

CBSEPVAT IONS Y(l)

Y<2>

YI3)

V(4I

YIS) Y<6>

lf(7) Y«8)

Y<9)

Y( 10) RESTRAINT A RESTRAINT B

4.2,2.1.1,1 K=6

N=l D.F.= 5

FACTORS FOR COMPUTING ST DEVS

KT RESTRAINTS

A B

1 .4873 .0000

1 .4873 .6325

1 .4873 .6325

2 .5528 1.32S0

2 .5528 1 .3250

3 .5413 1.7192

4 .0000 1 .9494

£ .4873 2.4083

6 .5528 3.2180

7 .5413 3.6271

8 1.0000 4 . 4 94 4 9 .8588 4.8785

10 . 7416 5.2726

41
OESIGN D.5 4,4,2.1.1 K= 5 8 DESIGN 0.7 4.4.2.2.1.1.1 K= 7 N = 9 8 N=14 (BENOIT ) O .F .= S |k O.F .= 8 OBSERVAT IONS 4 4 2 r( i

)

+ - + V(2) + ~ - Y(3) + - Y<4> + - YI5) + - Y(6) + - V{7) + _ Y<6) + Y(9) RESTRAINT A * + + RESTRAI NT B

FACTORS FCR COMPUTING ST DEVS

T RESTRAINTS

A B

1 .3651 .0000

1 .3651 .5774

2 .3651 .8165

3 .51 64 1 .2910

4 .28C7 1.4873

4 • 28C7 1.4873

5 • 46C6 1 .9695

6 • 28C7 2.2088

7 .4606 2.6855

6 .3651 2.9439

9 .5164 3.4157

.0000 3.6515

OESIGN D.6 4,4,2.2,1,1

OBSERVATIONS 4 4 2 2 Y< II + ~ Y(2) - + - Y(3) + - - + V(4) ~ Y(S)

Y(6) + Y<7> + " Y<8) + Y(9) + RESTRAINT A + + RESTRAINT B

FACTORS FOR COMPUTING ST OEVS

T RESTRAINTS

A B

I .3578 .2560

1 .3576 .2560

2 .3907 .6345

2 .3907 .6345

•a .521 1 .6 776

3 .521 1 .6776

4 .2195 1.0 238

4 .2195 1.0238

e .4476 1.0667

c .4476 1 .0667

6 .4058 1.5539

6 .4058 1 .5539

7 .5547 1. 5796

7 .5547 1.5796

6 .0000 2.0000

9 s

OESIGN 0.7 *

5 s OBSERVATIONS 4

Y< 1 ) + Y(2) + Y(3) + Y(4) + Y(5) + Y(6) +

1

Y<7) + YI8)

Y<9)

YI 10)

Y( 1 1 ) Y( 12)

Y< 13) 8

Y( 14) 6

s RESTRAINT A + RESTRAINT El

s FACTORS FOR COMPUT

WT RESTRAINTS

A B

1 .4830 .0000

1 .4330 .7 07 1 s 1 .4830 .7071 § 2 .4472 1. 1547

2 .3651 1 . 1547

3 .5774 1.6833

4 .3237 1 .e898

4 .3237 1.689 8

5 .6362 2.4640

€ .5521 2.9841

7 .7105 3. 5220

6 .3651 3.7417

9 .7071 4.3012

10 . 0000 4. 8305

S ""TSr^MSMWffffMKMMri T

5

s 6 § DESIGN 0.8

9

4 5

s

1

OBSERVATIONS 4 Y<1 ) + Y(2) + Y(3) + Y(4 ) + Y(5)

Y(6)

Y(7)

Y(8) RESTRAINT A + RESTRAINT B

FACTORS FOR COMPUT

WT RESTRAINTS

A B

1 .4275 .cooo

1 .4390 .6892

2 .3950 .9354

a .3532 1.2649

4 . 3146 1.6956

c .4677 2.25e3

R .4677 2. 1 154

e .4231 2. 5495

7 .4180 2.9112

e .3950 3.4496

9 .4390 3.7550

1 .0000 4.2749

1

n= a

( HAYFORO) D .F .= 4

42
I DESIGN D.9 4,3.2.1.1 K= 5 K DESIGN D.ll 10.5.2.1.1,1 K= 6 n= 9 | N= e (BENOIT) D.F.= 5 fc (HAYFORD) D .F .= 3 i OBSERVATIONS 4 3 2 11 £ OBSERVATIONS 10 5 2 1 1 1 Y(l) +-_+ Y(1) +-____ V<2> + - - y(2) + Y(3) Y(3) Y<4> + - Y(4) + " Y<5> + - - - Y(5) + - Y<6> + - _ y{6) + Y(7) + - - y(7 , + Y<8) + - - Y(8) + Y(91 + 2£ RESTRAINT A + RESTRAINT A + + + + 2

t RESTRAINT B + RESTRAINT B + 3

FACTORS FCR COMPUTING ST DEVS k5 FACTCRS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .3751 .6325

2 .3502 .9309

2 .3256 1.2649

4 .301 1 1.6733

£ .2416 2.1909

e .4131 2.5430

7 .4155 2.8983

e .3502 3.4059

9 .3751 3.7506

10 .0000 4.2505

WT RESTRAINTS

A B

1 .4472 .0000

1 .44 72 .7071

1 .4472 .7071

2 .4472 1 .0000

2 .6325 1.581

4 .7071 2. 1213

5 .7071 2.3452

t .8367 2. 9155

7 .8367 2. 2404

8 • 94e7 3.8079

9 1.0000 4. 3589

10 .0000 4.4721

10

\ DESIGN D.10 10.4,3,2,1,1 K= 6 ^ DESIGN D.12 10.5.2,2.1.1 K= 6

N=l N= 8 (HAYFORD) D.F.= 5 S (HAYFORD) D.F.= 3 OESERVATICNS 10 4 3 2 1 1 S CESERVATIONS 10 5 2 2 1 1 Y<n +____ y<1)

Y<2) ____ «" Y(3) + + Y<3> Y(4) + V(0)

V(2) +

Y(5) + ^ Y(S)

Y<6) + - - YJ6)

Y(7) + J

Y(7)

Y(8) + - Y(8J

Y(9) + Y(10) + - K RESTRAINT A

2, RESTRAINT A + RESTRAINT B RESTRAINT B +3 FACTORS FCR COMPUTING ST DEVS

K

10 4 3 2 1 1 9

+ + +

+ +

+ + + 2

+ + + + + + + 8s

i

FACTORS FCR COMPUTING ST DEVS

WT RESTRALNTS

A 8

1 .3848 . 0000

1 • 4411 .6195

2 .4025 .9045

3 .3885 1. 2060

4 .4188 1.6018

5 .6552 2.0889

e .5270 2. 0226

6 .5610 2.4225

7 .6092 2.7561

8 .7696 3.2193

9 .6895 3.5661

1C .7720 4. 01 13

WT RESTRAINTS

A B

1 • 465C . 0000

1 .3916 .5774

2 .4298 .9512

2 .4298 .9512

3 .5336 1.3801

4 .6723 1 .8257

5 .7071 2.4495

6 .8083 2.e868

7 .8275 3.3022

8 .8859 2. 7289

9 .9764 4.1633

10 .0000 4.6904
DESIGN 0.13 10.5.3.2,1.1

( HAYFORD)

K= 6 N=10

D.F.= S OeSERVAT IONS 10 V<1 ) YC2) V<3>

Y<4)

Y<5)

Y{6) Y<71

YC8)

Y<9)

Y< 10) RESTRAINT A RESTRAINT B

FACTORS FCR COMPUTING ST OFVS

WT RESTRAI NTS

A B

1 .3819 .0000

1 .3819 .534 5

2 .4055 .8545

3 .3844 1.2 599

4 .5014 1.6523

5 .4410 1.9740

e .5738 2.3688

7 .6692 2.7588

8 .6481 3.1910

9 .7159 3.5713

10 .7817 3.9551

DESIGN D.15

OBSERVATIONS

Y( 1 > Y(2)

Y(3) Y<4>

Y(5)

Y(6)

Y(7 J Y(8)

Y<9)

Y< 10) RESTRAINT A RESTRAINT B

1 0.5,4,3.2.1

« HAYFORD)

K= 6 N=10

.F.= 5

FACTCRS FCR COMPUTING ST DEVS

WT RESTRAINTS

A B

1 .4096 .0000

2 .3945 .9045

3 .4006 1.3104

4 .4259 1.8340

c .4523 2.2134

6 .5218 2.2134

7 .6038 3. 0928

e .6547 3.4772

9 .7290 4.0176

10 .0000 4. 0961

T^&**m*'**^**^***""*""""**"""**"*""&**"*"""^"***"**","*r-*''r*--r^~'-***JFjr******-*JrMMA

DESIGN D. 14 5,4 3.2. 1

< HAYFORD)

OBSERVAT IONS 5 4 3 2 Yd » + - — + Y(2I + - - Y(3) + - Y<4> + - Y(S) - - Yt6> - - Y<7» + —

K= 5 N= 7 D.F.= 3

RESTRAINT A RESTRAINT B

FACTORS FCR COMPUTING ST DEVS

WT RESTRAI NTS

A B

1 .4775 .0000

2 .4775 .4663

2 .5912 .56 83

4 .5912 .6825

c .0000 .6825

6 .4775 .6825

7 .4775 .8776

8 .5912 .9292

9 .5912 1.19S2

10 .6752 1 .1952

6,5.4. 3.2, 1

< HAYFORD)

5 4 3

DESIGN D.16

OBSERVATIONS

Y< 1) Y(2)

Y(3)

Y(4)

Y(5)

YI6)

Y(7I

Y«8)

Y(9) RESTRAINT A RESTRAINT B

FACTORS FCR COMPUTING ST OEVS

WT RESTRAINTS

B 6 5

. 0000

.9181 1.4639 1.8936 2.1679 + 2.6049 + 2.6049 + 3.4310 + 4.0089

4.4657 +

K= 6 N= 9 .F.= 4

A

1 .4342

2 .4646

3 .5175

4 .4697

S .4956

e .0000

7 .4342

8 .4646

9 .5175

.4697

44
(HAYFORO)

OSSERVAT IGNS 1 6 5 V< 1) + - Y<2) + - Y{3> + - Y(4) + - Y(5> + Y(6) + _ Y{7) + _ Y<8) + Y<9> + Y(10) + V(ll) Y<12» RESTRAINT A + RESTRAINT B

!

CESIGN D.1T 10.6,5,4,3.2,1 K= 7 \ N=12 S D.F.= 6 58

3 2 1 2

: I

:

I

FACTORS FCR COMPUTING ST DEVS

5 4 3 2 1+ + 5

*

* •

|

\

DESIGN E.l FOUR EQUAL WEIGHTS * K= 5 ^ DESIGN E.2 2.2.1,1 * K= f-. N= 8 8 N = 10 Trend Elimination D.F.= 4 S Trend Elimination O .F .= 5 9 OBSERVATIONS 1111 K OBSERVATIONS 2 2 11

V<1) + - Y( 1 ) + Y(2> + - 8 Y(2) + Y(3) + - V(3) + + - Y<4) + - Y(4) + Y(5) + - Y(5) + Y(6) + - Y(6) + V(7) + - Y(71 + Y<8> + - Y(8) + + - Y<9> + RESTRAINT A + Y(10) + RESTRAINT B + RESTRAINT A + +

NT RESTRAINTS

A B

1 .4110 .0000

2 .4229 .3560

3 .4290 1.3841

4 .4203 1.8076

c .4352 2.0860

e .4889 2.4656

7 .6696 2.4656

7 • E8ie 2. e467

7 .6607 3. 1442

8 .6877 3.2365

e .6597 3.4224

9 .6819 3.797 5 9 .6090 3.8493

10 .7210 4. 2403

FACTORS FCR COMPUTING ST DEVS

^ RESTRAINT B RESTRAINTS

A B 1111 FACTORS FCR COMPUTING ST DEVS

1 .8660 .0000 + 9 WT RESTRAINTS

1 .8660 1.0000 + A B 2 2

1 .5000 1.0000 + 1 .6614 .6928

1 .5000 1.0000 + 1 .6614 .5292

2 .0000 1.7321 + + 2 .5000 .5657 + .8660 2.4495

.4142 2.4495 + + + 3 .8292 .9592

*See page 13.

1 .6614

1 .6614

2 .5000

2 .5000

3 .8292

4 .0000

£ .66 14

2 .5000 .5657

!K *See page 13

5

45

.5292

1. 0000
APPENDIX B

1

1 LIST OF TABLES

Table Weights Design

A. 1.2

Restraint

1 1,1,1,1 Sum Two of two weights

2 1,1,1,1 A. 1.2 One weight

3 2,2,1,1 B.3 Sum Four of first two weights

4 2,2,1,1 B.3 One weight

5 5,3,2,1,1,1 C.2 Sum Ten of first three weights

6 5,3,2,1,1,1 C.2 One weight

7 5,2,2,1,1,1 CIO Sum Ten of first four weights

8 5,2,2,1,1,1 CIO One weight

9 1,1,1,1 Direct reading E.l Sum Two of two weights

10 1,1,1,1 Direct reading E.l One weight

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56
REFERENCES

[1] Benoit, M. J. R. , L'Etalonnage des Series de Poids , Travaux et

Memoires du Bureau International des Poids et Mesures, No. 13,

1-47 (1907)

[2] Bose , R. C. and Cameron, J. M. , The Bridge Tournament Problem and

Calibration Designs for Comparing Pairs of Objects, National Bureau

of Standards Journal of Research 69B, 323-332 (1965)

[3] Bose, R. C. and Cameron, J. M. , Calibration Designs Based on

Solutions to the Tournament Problem, National Bureau of Standards

Journal of Research 71B, 149-160 (1967)

[4] Cameron, J. M. and Hailes, G. E., Designs for the Calibration of

Small Groups of Standards in the Presence of Drift, NBS Technical

Note 844 (August 1974)

[5] Chakravarti, I. M. and Suryanarayana, K. V., Partial Difference Sets

and Partially Balanced Weighing Designs for Calibration and Tournaments, Journal of Combinatorial Theory 13(3), 426-431 (1972)

[6] Goldman, A. J. and Zelen, M. , Weak Generalized Inverses and Minimum

Variance Linear Unbiased Estimation, NBS Journal of Research 68B,

No. 4, 151-172 (1964)

[7] Hayford, John A., On the Least Squares Adjustment of Weighings,

U.S. Coast & Geodetic Survey Report for 1892, Appendix 10, USCGS

(1893)

[8] National Bureau of Standards Circular 3, Design and Test of

Standards of Mass, 1918. This circular is out of print but

57
its contents have been incorporated in NBS Handbook 77, Precision

Measurement and Calibration, Volume III, Optics, Metrology and

Radiation, pp. 588-706. This handbook is available for reference

or circulation in most Government Depository Libraries throughout

the United States.

[9] Pontius, P. E. , Mass and Mass Values, NBS Monograph 133, 39 pages

(1974)

[10] Pontius, P. E., Measurement Philosophy of the Pilot Program for

Mass Calibration, NBS Technical Note 288, 39 pages (1966)

[11] Suryanarayana, K. V., Contributions to Partially Balanced Weighing

Designs, Ph.D. dissertation submitted to the Department of Statistics,

University of North Carolina at Chapel Hill, Institute of Statistics

Mimeo Series No. 621 (1969)

[12] Zelen, M. , Linear Estimation and Related Topics, Survey of Numerical

Analysis, Chapter 17, edited by J. Todd, McGraw Hill, New York City,

558-584 (1962)

58
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN 9522. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Designs for the Calibration of Standards of Mass

5. Publication Date June 1977

6. Performing Organization Code

7. AUTHOR(S)

J. M. Cameron, M. C. Croarkin, R. C. Raybold

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Proiect/Task/Work Unit No.

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16. ABSTRACT (A 200-word or less factual summary ol most significant information. If document includes a significant

bibliography or literature survey, mention it here.) This report presents a collection of designs for the intercomparison of sets of

weights for use in precision calibration of standards of mass. These include a number of previously unpublished designs which have an additional weight in each

set to serve as the check standard for monitoring the performance of the weighing

process. Also included are the classical designs of Benoit and Hayford. The

complete least squares analysis is presented in integer form (i.e., with a common

division) for the most widely used designs; and for the others, the standard

deviations are given for various weight combinations when used as an ascending or

as a descending series. Designs for sets of nominally equal objects, the

2 2 . . . 1 1 . . . series, the binary sequences, the 5 2 2 11 series, and

the 5 3 2 11 and some miscellaneous series are given.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by serriicolons) Design of experiments; least squares; mass calibration; statistical design;

weighing design

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