REFERENCE

^ 0F Co*

v> J

NBS TECHNICAL NOTE 844

***f„u of *

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

of Small Groups of Standards

in the Presence of Drift
NATIONAL BUREAU OF STANDARDS

Rational Bureau of Standards NOV 8 1974 M&53 The National Bureau of Standards ' was established by an act of Congress March 3, 1901. ,. «*, C Q iA. 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 promote 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, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent 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 a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear

Sciences * — Applied Radiation 2 — Quantum Electronics 3 — Electromagnetics * — Time

and Frequency 3 — Laboratory Astrophysics 3 — Cryogenics 3. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized materials needed by industry, commerce, educational institutions, and Government; provides

advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard

Reference Materials and the following divisions: Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor

Radiation — Physical Chemistry. THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote

the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations leading to the development of

technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Institute consists of a Center for Building Technology and the following divisions and offices: Engineering and Product Standards — Weights and Measures — Invention and Innovation — Product Evaluation Technology — Electronic Technology — Technical Analysis — Measurement Engineering — Structures, Materials, and Life Safety' — Building Environment * — Technical Evaluation and Application * — Fire Technology. THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective

utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing equipment, techniques, and computer languages. The Institute consists of the following

divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology. THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units: Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations.

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

1 Part of the Center for Radiation Research.

• Located at Boulder, Colorado 80302.

• Part of the Center for Building Technology.
•O. y\CXJ KJSfVLQ^ &UJZM-UM. &L ^CUAj3~CUT-j3LS>

DESIGNS FOR THE CALIBRATION OF SMALL GROUPS OF STANDARDS

IN THE PRESENCE OF DRIFT

by

Joseph M. Cameron and Geraldine E. Hailes

The process of calibrating a small number of "unknown" standards

relative to one or two reference standards involved determining differences among the group of objects. Drift, due most often to temperature

effects, or a "left-right" polarity effect can bias both the values

assigned to the objects and the estimate of the effect of random errors. This note presents schedules of measurements of differences that eliminate

the bias from these sources in the assigned value and variances and at the same time gives estimates of the magnitude of these extraneous components.

The use of these designs in measurement process control is discussed and

a computer program in BASIC is presented.

Key Words: Calibration; calibration design; experiment design;

instrumental drift; measurement process; statistical analysis; trend

elimination

1. Introduction

In very few processes can the effect of time be ignored. Instability

in the object being measured, inability to maintain constant conditions

or procedures, and variations in the detector or comparator all con- tribute to changes with time. A number of approaches have been

suggested for reducing or eliminating the effects of these temporal

effects on the validity of one's measurements. One way is to make

measurements far enough apart in time (usually with some formal

randomization procedure to guarantee statistical independence of the

measurements) that the cumulative effects from the various sources appear in the random error component. At the other extreme, one can go

to great lengths to eliminate these time dependent effects by achieving

better environmental control, better instruments, better procedures,

etc. If the measurements are to be transferred, as with instrument

calibrations, then the first procedure leads to error bounds in which

the random error limits include a between-time component whereas the

latter procedure suppresses such a component. Neither of these represent the conditions of use adequately.
A compromise consists of arranging the experiment under its normal

conditions so that it is as nearly as possible free of time dependent

effects. The classic example of this is afforded by the calibration of

thermometers in a bath with a gradually rising temperature using the

schedule whose structure is as follows for a standard S, and 4 unknowns,

V V V V

S T T T T S T T T T S

If the measurements are evenly spread in time, then the average of the

bath temperature for all thermometers are the same (see [ 3 ] for a discussion of this practice) . A similar procedure has been followed in

weighing where in the substitution method one measures in scale units

A, B, B+S, A+S

to obtain the difference A-B and the deflection corresponding to the

sensitivity weight, S.

The calibration of a small number of "unknown" objects relative to one or two reference standards involves determining differences among

the group of objects. Instrumental drift, due most often to temperature

effects, or a "left-right" polarity effect can bias both the values

assigned to the objects and the estimate of the effect of random errors. This note presents schedules of measurements of differences that eliminate the bias from these sources and at the same time gives estimates

of the magnitude of these extraneous components. The use of these

designs in measurement process control is discussed and a computer

program in BASIC is presented in this report.
2. Measurement as a Process

A single isolated measurement, like a single event in history, is

difficult to interpret unless it can be regarded as a part of a continuing

process. When the measurement is looked upon as the output of a process

a production process whose output is the measured values—then one can attribute to the single measurement the properties of the process from

which it arose (for a discussion of this approach, see Eisenhart [ 2 ]).

Just as with any production process, the operating characteristics are determined by building some redundancy into the system. Redundancy is needed to assure oneself that he has indeed measured the

sought after quantity, uncontaminated by extraneous factors related to the operator, instrument, environment, or other items.

Among the characteristics of the process are these associated with

the ability to repeat a measurement both in the sho t term and in the

long term. Repetitions made within a few hours, such as with designs

having more observations than unknowns, usually exhibit less variation

than those made at long time intervals. This additional long-term

component of variance can be measured from the agreement among repeated

measurements on the same quantity. In addition to these properties

related to variability, one needs to incorporate checks on the systematic

errors which may possibly affect the process, and, if possible, measurements that provide information as to the adequacy of the assumptions in

the underlying physical model.

In calibration it is often convenient to measure a check standard

along with the calibration of one or more unknowns. One thus has a value for monitoring the process that is on an equal footing with the

unknowns. By tracking its long-run performance, one can determine not

only the presence of components of variance, but also by recording

ancillary information on environmental and other factors one can develop

information for assessing the adequacy of the assumed physical model and

for setting bounds to the effect from known sources of possible systematic

error. This "check standard" need not be the value of a single item but

may take the form of a difference between two such items or some linear

combination of several.

The effect of some sources of systematic error can be eliminated by

"balancing out" the effect by repeating the measurement of a difference,

(x - y) in the reverse order, (y - x) . Time dependent effects can be

balanced out by using the techniques of this report. For others, it is

sometimes possible to alter the conditions to levels of a factor beyond

that known to have been in effect at the time of the measurement and to use the changes produced in the output at these extremes as a bounds to

the effect of the factor.

In all cases one has to continually monitor the process output just as one does with an industrial production process if he is to have assurance

that the calibrations are correct.
3. Substitution Weighing

Consider first the simple situation of scale deflections produced on

a balance by adding weights A and B and a sensitivity weight S. One

could use either of the following sequences.

Sequence 1 Sequence 2

A A

A+S B

B B+S

B+S A+S

In high precision woi c one invariably finds a change in balance response

with time so that the value for the difference (A-B) will obviously be

contaminated by whatever time effects exist for Sequence 1. If

Sequence 2 is used, it may be represented as follows.

Quantity Eff'eel: of Drift Sea]Le Divisions

A -3A X

l B - A X

2 B+S A X

3 A+S 3A X

4 and the quantity

i< X

l

" X

2 " X3 + V

can be seen to give an unbiased value for (A-B) because the drift

effect (a 2A change in scale reading between each observation) cancels

out. The least squares values for A-B, S, and A in scale divisions

are

(a2b) = I

(X

1 " X2 - X3 + X

4

)

S = ^j. " 3x

2 + 3x

3 - x

4

)

A = iC-x, + x

2 - x3 + x4

>
4. Thermometry

At NBS, the calibration of liquid in glass thermometers is usually

carried out in a controlled bath which is continually heated so as to give a slight temperature increase with time. The temperature of the

bath is measured by resistance thermometry at the start, middle, and

end of the run with the test thermometers being run once each in the

first interval and once again in reverse order in the second. The time

sequence for the resistance measurements R]_, R2

, R3 and the two series

of test thermometer values denoted by T^ T2

• • • Tk are as follows:

R

l T

l T

2

' * ' \

R

2 Tk

' ' T

2 Ti R3

If equal time intervals are maintained between readings of the test

thermometers, then one would expect an increase, AT in temperature with

each interval except perhaps the middle one in which the resistance

thermometer reading, R2

, is made. The analysis of this form of data is

given in Appendix A.

5. Polarimeter Data

In determining the optical rotation of a quartz control plate used

as reference standards in polarimeters, one measures the voltage response

of a synchronous detector as the angle is varied. However, the response,

y, of the system has a nearly linear drift with the angle so that one can represent this drift effect relative to the centroid of the data as being

either • • • -3A, -2A, -A, 0, A, 2A, 3A, • • with A being the increment

to the response added in each time interval. [For even n it is convenient

to use • • • -3A, -A, A, 3A • • • or 2A increment per time interval.]

In the polarimeter experiment the response is a linear function of

angle so that the observation becomes

y. =a+Bx. + (i- ——) A + random error

1 1 2

where the x^ are evenly spaced deviations from the nominal angle, e.g., x = 0", 10", 20", 30", • • • If the usual estimate of a and 3 are to remain unbiased and unchanged in precision, then one must have

- , . n+1

Ex. (1 —) =

1 2 so that the estimates are orthogonal to the drift in the detector. The

following orderings have this property:
n = 4 n = 5 Measurement Quantity to Setting for Quantity to Setting for

Number be Measured

a + 26

Polarimeter

20"

be Measured

a + B

Polarimeter

1 10"

2 a 0" a + 46 40"

3 a + 36 30" a + 23 20"

4 a + 6 10" a 0"

5 — a + 36 30"
6. Calibration Designs

The term calibration design has been applied [ 1 ] to experiments

where only differences between nominally equal objects or groups of

objects can be measured. Perhaps the simplest such experiment consists

in measuring the differences between the two objects of the n(n-l)/2

distinct pairing that can be formed from n objects. Ordinarily the

order in which these measurements are made is of no consequence. However, when the response of the comparator is time dependent, attention

to the order is important if one wishes to minimize the effect of these

changes. When this effect can be adequately represented by a linear

drift, it is possible to balance out the effect by proper ordering of

the observations. As with the polarimeter data, this drift can be

represented by the series • • • -3A, -2A, -A, 0, A, 2A, 3A, • • • if

n(n-l)/2 is odd or by • • • -5A, -3A, -A, A, 3A, 5A. . . if n(n-l)/2

is even.

For n = 4, n(n-l)/2 = 6, and it turns out that it is not possible

to balance out the drift effect with 6 measurements. However, with 8 measurements the balance can be achieved by the following order,

denoting the four objects by A, B, C, D.

Observation Observation is a Measurement of

A B C D A

Y

l Y

2 Y

3 Y

4 Y5

y

6

y7

y

8 The notation used here, the plus and minus signs, indicate the items_

entering into the difference measurement. Thus, y is a measurement of

the difference (D-A)

.

To see how the drift effect is balanced out, consider item C which

occurs in the third, fourth, seventh, and eighth observations. In the

third and eighth observations the item occurs positively and the cor- responding drift effects are -3A and 7A respectively. For the fourth

and seventh observations, item C occurs negatively while the corresponding

+ - -7

- + -5

+ - -3

+ - -1

+ - 1

- + 3

+

- 5

_ + 7
drift effects are -A and 5A. The overall effect can be represented by

the sum of cross products of the columns for C and A, namely

[1] (-3A) + [-1] (-A) + [-1] (5A) + [1] (7A) =

using square brackets to denote the coefficient attached to the direction

of the difference and parenthesis for the drift effect. For A, one has

[1] (-7A) + [-1] (-5A) + [-1] (3A) + [1] (5A) =

In general, if the cross products sum out to zero, then the drift

effect is said to be completely "balanced out" or "orthogonal" to the

items being measured.

7. Restraints

In calibration designs only differences between items are measured

so that unless one or more of them are standards for which values are

known, one cannot assign values for the remaining "unknown" items.

Algebraically, one has a system of equations that is not of full rank

and needs the value for one item or the sum of several items as the

restraint to lead to a unique solution.

In the design of Section 6, for example, if one has a single

standard and three unknowns, the standard can be assigned to any one

of the letters. (The same would be true of three standards and one

unknown.) If there are two standards and two unknowns, the choice of

which pair of letters to assign for the standards is important in

terms of minimizing the uncertainty in the unknown.

It turns out that the pairing of A with D or of B with C is slightly

less efficient (see Appendix B) than the other pairings A with B or C

with D. This results from the fact that the observation on the differences (A-D) and (B-C) are repeated and it is usually better (to

achieve smaller variance for the test items) to measure differences

between standards and unknowns than between pairs of standards.
8. Use of Calibration Design in Gage Block Calibration

The calibration design of Section 6 is used in gage block calibration

at the National Bureau of Standards and the analysis and interpretation

of the design for this application is representative of the principles

involves in the use of the design in other applications.

At NBS two master sets of gage blocks are maintained for transferring length calibration to users gage blocks, these are designated

A and B and their sum is designated by K. These are combined with two

sets of unknowns, designated C and D. The difference (A-B) is used as the check standard.

/\ /\ /\ /\ If we denote the values determined for ABC and D by A B C D in

accordance with the statisticians' practice of distinguishing the value

from the experiment from the sought-after or long-run value, we may then

write

£ " l4 (5y

l " 2y2 " y 3 - 2y4

- 3y

5

- 2y6 + 3y

7 + 2y8> + I

S = 34 ( " 5y

i + 2y2 + y3 + 2y4 + 3y5 + 2y6

" 3y

7

- 2V + I

A-B = -^-(lOy-L - 4y

2

- 2y

3 - 4y4

- 6y

5 - 4Y(_ + 6y? + 4yQ

)

C = y~Yl * 2y

2 + 5y

3 - 6y

4 - y5 + 2y

6 - 7y? + 6y

8

) + §

D = ^-(y

1 + 6y

2 - 5y

3 - 2y4

- 7y

5 + 6y

6

- yy + 2y

8

) + |

where A + B necessarily sum to K.

These values have the following standard deviations in terms of the

long run precision as represented by the process standard deviation 0.

s.d. (A) = s.d. (B) = a / 4s"

s.d. (A-B) = O / —

s.d. (C) = s.d. (6) = a /i|

One also obtains values A for A where

1 = lk ( - 7y

i - 5y2 " 3y3 " Y4 + y5 + 3y6 + 5y7 + W

s.d. (A) =a/ I

i

8

-
Because this is an overdetermined system (more observations than

unknowns) the deviation between observed and computed value is, in

general, different from zero and reflects the random errors of measurement. These deviations, d^ d2 • • dg are as follows:

d

l =

168 (49y

l " 7y

2 _ 7y3 + 2lY4 + 49Y5 + 49y6

" ?Y7 + 21y

8

)

d

2 = li8 ( ' 7yl + 8?y2 + 13y3 " 5y4 + 33y5 ' 41y6 + 53y7 + 35y8

)

d

3 = lie ( " 7yi + 13y2 + 89Y3 + 25y4

" 39y

5 + 3?y6 + 5?y7 " 7y

8

)

d

4 = 1?8 (21y

i

- 5y

2 + 25y3 + llly4

- 2?y

5 + 3y6

" 23y

7 + 63y8

)

d

5 = 1?8 (49yi + 33y2

" 39y

3 - 27y4 + 97y5 + 25y6 + 9y7 + 21y8

)

d

6 = lis (49yi " 41y2 + 3?y

3 + 3y

4 + 25y5 + 103y6 + 13y

7 " 21y

8

)

d

7 = ik ( " 7y

i + 53y2 + 5?y3 " 23y

4 + 9y5 + 13y

6 + ?3y

7

"

?V

d

8 = ife (21y

i + 35y

2 - ?y3 + 63y4 + 21y5 ~ 21y6

" ?y

7 + 63V

These deviations provide the information needed to obtain a value,

s, which is the experiment's value for the process standard deviation,

a.

, Z (dev) 2

. . s = v degrees of freedom = 4

The number of degrees of freedom results from taking the number of

observations less the number of unknowns then adding one (for the

restraint) to give 8-5+1=4.

10
9. Example

Routine calibration of gage blocks is carried out with two NBS

master blocks (designated S. and S..) and two test blocks (designated

X and Y) . The blocks are placed close together on a metal platen for

a sufficiently long time to insure temperature equilibrium. A mechanical

intercomparator is used to determine the difference between the blocks

by first determining a reading for the block indicated by "+" then

following with the block indicated by "-". The difference between

these two readings is the observation, y (all values are in micro-inch)

.

For a set of 0.101 in. blocks, the following data was obtained.

DATA FROM NBS CALIBRATION OF FOUR 0.101 INCH GAGE BLOCKS

i Schedule

+ -00

Difference

Measured

S.-S..

First

Reading

52.0

Second

Reading

52.5

Difference

y(i) Deviation

1 -0.5 0.029

2 - + Y-S. 45.2 52.1 -6.9 -0.046

3 + - X-Y 50.0 45.1 4.9 0.113

4 + - S. .-X 53.1 50.0 3.1 0.571

5 + - S. .-Y 52.3 45.2 7.1 -0.238

6 - + Y-S. 45.1 52.0 -6.9 -0.079

7 + - S.-X 52.0 50.1 1.9 -0.154

8 - + S. .-X 50.1 52.3 -2.2 0.304

S.+S.. = 6.4 used as restraint

S.-S.. = -0.133 used as check standard

a = . 32 accepted standard deviation

The values for the blocks and the drift effect, A, are

S. =^-[5(-0.5) - 2(-6.9) - (4.9) - 2(3.1) - 3(7.1) - 2(-6.9) + 3(1.9) +

2(-2.2)] + i£#-

= J7(-6-0) + 3.2 = 2.9500

S..= —[-5(-0.5) + 2(-6.9) + (4.9) + 2(3.1) + 3(7.1) + 2(-6.9) - 3(1.9)

2(-2.2)] +

{6 - 4)

2

= -^-(^.O) + 3.2 = 3.4500

11
S.-S. . = -0.5

X = ^-[-(-0.5) + 2(-6.9) + 5(4.9) - 6(3.1) - (7.1) + 2(-6.9) - 7(1.9) +

6C-2.2)] + iSji2-

= tj(-54.8) + 3.2 = 0.9167

Y = ^-[(-0.5) + 6(-6.9) - 5(4.9) - 2(3.1) - 7(7.1) + 6(-6.9) - (1.9) +

2(-2.2)] + %i2.

= ^-(-170.0) + 3.2 = -3.8833

A = ~-[-7(-0.5) - 5(-6.9) - 3(-4.9) - 1(3.1) + (7.1) + 3(-6.9) + 5(1.9) +

.Lbo 7(-2.27)

= 7Tf(.7) = 0.0042

168

The accepted standard deviation for the process is 0.32 y-in so that

one can compare the observed standard deviation, s,

s = / ZdevV4 = / J 5208 „ „,., = 0.361

4

to the accepted value by computing

_ ,s 2 _ 0.1302 _

F

~

{ F ~ 0.1024 " 1 ' 21

Had the ratio (s/O) exceeded 3.32 (the critical value for the 1% probability level of the F distribution) , then the measurements would be

regarded as being "out of control" and would be repeated. The other

check on process performance is provided by the check standard for

which the difference between (S.-S..) and its accepted value should be

less than 3 times the standard deviation of (sr-S..). See Section 10

for a discussion of this test.

The drift term, A, has a standard deviation of a//l68 or 0.025. The

statistical significance of A can be judged by forming the ratio A

CT//168

If this ratio exceeds 3, then A would be regarded as significant.

However, because the design has eliminated the effect of drift on the

12
values of the blocks, one would not be concerned about a "significant"

A unless it was greatly in excess of previously encountered values.

The deviations are computed as shown in Section 8, for example, for

the deviation corresponding to yg is given by

(dev) = T^-[21(-0.5) + 35(-6.9) - 7(4.9) + 63(3.1) + 21(7.1) - 21(-6.9)

8 J.DO

7(1.9) + 63 (-2.2)]

= ik[51 - 1] = - 304

13
10. Process Control

As mentioned in Section 2, continued monitoring of the measurement

process is required to assure that predictions based on the accepted

values for process parameters are still valid. For gage block calibration, the process is monitored for precision by comparison of the

observed standard deviation to the accepted value, aw , by means of the

F-test. In the case of the design of Section 6, the square of the

ratio of the two standard deviations is compared to the critical value,

F(4, °°, a), which is the a probability point of the F distribution for

degrees of freedom 4 and °°. [For calibrations at NBS, a is chosen as

.01 to give F(4, °°, .01) = 3.32].

The check for systematic error is given by comparison of the

observed value of the difference, S. - S.., between the two standards

with its accepted value. The uncertainty of this difference is given

by aT =/(5/12)aw

+ Cfg where aw

is the "within run" standard deviation

and aBis the component of variance arising from variations from run-torun. The value of (J T

is obtained directly from the sequence of values

of S. - S.. arising in regular calibrations. The check standard test

is therefore,

| observed (S. - S..) - accepted (S. - S..)[

< 3

i.e., t is compared to the critical value 3.0 which would correspond to

the .003 probability level for the normal distribution.

If both the "precision" (F-test) and "accuracy" (t-test) criteria

are satisfied, the process is regarded as being "in control" and values

for the unknowns, X and Y, and their associated uncertainties are regarded as valid. Failure on either criterion is an "out-of-control"

signal and the measurements are repeated.

When the between run component, ag, is present, the standard

deviation associated with the values for the unknowns are given by

a(S.) = a(S..)

a(x) = a(Y)

= Jl^i + a

2 = /a2 - i^a2 48°w + °B "

CTT 48^w

= 4i°w + °b = M - igl

The value for the drift serves as an indicator of possible trouble if

it changes markedly from its usual range of values. However, because

any linear drift is balanced out, a change in the value does not of

itself vitiate the results.

14
If the uncertainty attached to the restraint value is not negligible,

this will lead to a possible systematic error in all measurements based on

this restraint. Therefore, as a bound to this error one should, for the

design of section 6, add to the uncertainty from random error an allowance

of one-half the uncertainty in the sum (S. + S..). This is shown in the

computer example.

11. Computer Program

Appendix C lists a computer program in BASIC for carrying out the

calculation for the gage block example. The program can be used with

any design provided one has the arrays of coefficients for the determination of the values of the unknowns and the deviations corresponding

to the two arrays given in Section 8 for the gage block example.

The program calls for input of:

a) Administrative data—designation of blocks, operator, date,

etc.

b) Process parameters—standard deviations, value for check

standard, etc.

c) Comparator readings

The computer programs provide in the output:

a) Deviations, s.d.

b) Values for unknowns, drift, and associated undertainties.

c) Statistical tests as to whether process can be regarded as "in control": on standard deviation and on value of check

standard

15
12. Other Designs for Elimination of Drift With Order of Measurement

The number of observations over which a linear drift could be

expected to be valid varies with the type of measurement, but experience

indicates that it is unusual if it is as large as 20. For all distinct

pairings of n items n(n-l)/2 exceeds 20 for n > 7. The table below gives

designs for n = 5, 6, 7 which are balanced for linear drift.

n = 5 n == 6 n = 7 10 Observations 18 Observations 21 Observations

+ - 1-2 + - 1-2 + - 1-2

+ - 2-3 + - 2-3 + - 2-3

+ - 3-4 + - 3-4 + - 3-4 + - 4-5 + - 4-6 + - 4-5

- + 5-1 - + 6-5 + - 5-6

- + 5-1 + - 6-7

- + 4-1 - + 7-1

+ - 2-4 + - 3-6

0-0 + 5-2 - + 4-1 + - 2-4

+ - 3-5 + - 2-5 + - 3-5

+ - 1-3 - + 6-3 + - 4-6

+ - 1-4 + - 5-7

- + 5-2 +

+

6-1

7-2

- + 5-4 + - 1-3

- + 6-3

- + 4-2 - + 7-3

- + 3-1 - + 6-2

+ - 2-5 - + 5-1

+ 1-6

+

+

+

+ 4-7

3-6

2-5

1-4

An alternate form of displaying the design is shown for n = 5 and is used

for the other designs.

16
Designs that involve a subset of all possible pairings are given

below:

n = 6 n = 7 n = 8 n = 9 12 Observ. 14 Observ. 16 Observ. 18 Observ.

1-2 1-2 1-2 1-2

5-1 2-3 2-3 2-3

2-3 3-4 3-4 6-5

4-6 4-5 4-5 3-1

3-4 5-6 5-6 5-4

6-5 6-7 6-7 8-9

7-1 7-8 4-7

2-4 8-1 9-6

4-5 3-1 7-8

6-2 5-3 4-1

3-1 7-5 7-4 7-1

1-6 2-7 2-7 4-6

5-3 4-2 5-2 9-7

6-4 8-5 1-4

1-6 3-8 3-9

6-3 5-8

1-6 6-3

2-5

8-2

17
REFERENCES

1. Bose, R. C. & Cameron, J. M. , The Bridge Tournament Problem and

Calibration Designs for Comparing Pairs of Objects, NBS J. of Res,

69B (1965) , 323-332.

2. Eisenhart, C, Realistic Evaluation of the Precision and Accuracy

of Instrument Calibration Systems, NBS J. of Res. 67C (1963)

,

161-187.

3. Swindells, J. F., Calibration of Liquid-in-Glass Thermometers,

NBS Monograph 90, GPO, 1965.

4. Zelen, M. , Linear Estimation and Related Topics, Chapter 17 of

Survey of Numerical Analysis edited by J. Todd, McGraw Hill,

New York City (1962), 558-584.

18
APPENDIX A

Thermometer Calibration

Liquid in glass thermometers are calibrated at NBS in a controlled

bath in which the temperature is increasing in a nearly linear fashion

with time. The temperature of the bath is measured by platinum

resistance thermometry at the beginning, middle, and end of a run with

the test thermometers being read once in the first interval and again

in reverse order in the second interval. The time sequence for the

resistance measurements, R^, R2, R3 and the two series of thermometer

values denoted by T| and T^ are as follows:

R

l T

i T2 K Ri Ti, k 2 k T2 Tx R3

with uniform time intervals between the thermometer readings. Figure

shows a schematic of the situation with the increment to the bath

temperature being A for each time period except for the middle reading

involving resistance thermometry where a step of a in temperature is

assumed.

0)u

+J

(0u

<DIID

T T

i

(k-1) intervals

~ *s— *\

n,

R, T ' T

1 1 2 T ' R T

k 2 k

Time

Figure Thermometer reading at fixed time intervals in a bath with

linear drift.

19
The average (T| + T^)/2 will be the indication of the i-th thermometer

at the temperature implied by (Rj_ + R2 + R3_)/3. The differences, d^ =

Ti

~ Ti wiH be a measure of a + 2 (k - i) A so that the observational

equations may be written

E(d) = E

T

2

- T

1 k-1 k-1

Tk~ T

k

a + 2(k - 1)A

a + 2(k - 2)A

a + 2A

a

2(k - 1:

1 2(k 2)

1 2

1

" N. \ * \ a = X a

A A

>• ' *» A

where X stands for the indicated matrix, and where E( ) stands for the

"expected value of," i.e., the limiting value if the effects of random

error were eliminated.

The least squares estimates of a and A are given by the solution to

the normal equations

(X'X)

A

= X'd Ed

2ld (k-i)

where the inverse of the matrix of normal equations is

(X'X)

-1

k k(k-l)

k(k-l) 2k (k-1) (2k-l)/3

k(k -1)

2 (k-1) (2k-l) -3 (k-1)

-3 (k-1) 3

2

' The estimates of a, A and , the variance of the observations are given

by

a =

A =

k(k+l)

3 k(k2 -l)

[3lid - (k+l)Zd]

[ (k+l)Zd - 2£id]

a

2 = -^- [Ed

2 - aid - 2AE(k-i)d] = -^- I (dev)

2 k-2 k-2

where dev. = d. - a - 2 (k-i) A.

1 1 The standard deviation of the value for the test thermometer is

a//2 and for a and A,

20
s.d. (a) = a/2(2k-l)/k(k+l)

s.d. (A) = a/3/k(k 2 -l)

Control on the measurement process is maintained by two forms of

redundancy—one to check on the process average and the other to check

on process variability. The first of these is provided by incorporating

an NBS standard thermometer among the k thermometers and requiring that

its value be within random error of its accepted value. The variability

check is given by comparing G with the long run value established for

the process. When these conditions are satisfied, then one can regard

the process as being in a state of control.

A typical set of data for this type of calibration is given in the

following table. For simplicity the resistance measurements have been

suppressed and the temperature reported directly.

Calibration of Thermometers

Data From NBS Calibration of 22 August 1972

Provided by J. Wise, NBS, Thermometry Section

Thermometer

Reference (PRT)

T' (check standard)

Reference (PRT) T.

Observation

39.9378

39.983

39.913

39.966

39.840

39.9422

39.842

39.969

39.917

39.991

Averages

T 39.9870

T 39.9150

T 39.9675

T. 39.8410

4 PRT 39.9434

PRT - OBS = Correction at 40°

-0.0436

0.0284

-0.0241

0.1024*

*accepted value is 0.1000

Reference PRT 39.9501

d = Predicted

i

-0.008

a

1

A

6 Ed = -0.077

d dev.

1 -0.0071 -0.0009

2 -0.004 1 4 Eid = -0.033 -0.0052 0.0012

3 -0.003 1 2 2/k(k+l) = 1/10 -0.0033 0.0003

4 -0.002 1 3/k(k2 -l) = 1/20 -0.0014 -0.0006

Edev2 = 0.00000270

21
a =

Y5" [3(-0.033) - 5(-0.017)] = -0.00140

A = — [5(-0.017) - 2(-0.033)] = -0.00095

a = /Edev2 /2 = /0. 00000135 = 0.00115

~ / 7 s.d. (a) = ov— s.d. (average T) = a/v2

s.d. (A) = a/i22
APPENDIX B

Least Squares Analysis of Calibration Designs

In this appendix the least squares analysis is presented in matrix

form for those wishing to prepare a general analysis. Each formal

statement will be illustrated by its application to the calibration

design of Section 6.

It is assumed that the expected value of the observations represented

in vector form as y' = (yi Y2 • • • Yn^ nave expected values E(y) = X$

where $ is the vector of parameters and X is the design matrix. It is

also assumed that the errors of measurement are uncorrelated and have

equal variance, i.e., V(y) = 0^1.

For the design of section 5,

f 1-10

-10

1 1-1

10

-10

10-1

0-11

The matrix of normal equations is given by (X'X)p = X'y which for

calibration designs is not of full rank.

\

-7

1 -5

1 -3

-1

1 1

1 3

5

7 )

/ N A

B

C

D

A

x'xB = N4 -1 -1 -2

-1 4-2-1

-1 -2 4-1

-2 -1-1 4

168

,- X'y = 0-1 10

-10 11

1-10

1-1 0-1

7 -5 -3 -1 1 \

-1

-1 1

1

3 5 7/ In order to solve this system, a restraint in the form h'B = K is

imposed leading to the augmented equations (see Zelen [ 4 ] ),

23
\ (^ s X'X h

h' X

X'y

K

IS /

For the design as used in the calibration of gage blocks the restraint

is that A + B = K, giving h

1 = (110 0) and the augmented equations

4 -1 -1 -2 1

1 4 -2 -1 1

1 -2 4 -1

2 -1 -1 4

168

1 1

The solution for the parameter values 3 are

<. > \ X'X h

h'

X'y c

X'y

K

V < \

g X'y

K s /\ h' K

*v S \ J \ /

where C is the indicated KxK matrix arising in the inversion process n. For the example

\

l336 35 -35 - 7 7 168 x'y

-35 35 7 - 7 168 K

- 7 7 91 21 168

7 - 7 21 91 168

2

168 168 168 168

1 168 35 -14 - 7 -14 -21 -14

35 14 7 14 21 14

7 14 35 -42 - 7 14

7 42 -35 -14 -49 42

7 - 5 - 3 - 1 1 3

'

21 14 84

21 -14 84

49 42 84

7 14 84

5 7

for which C

1 336

S

35-35-7 7

-35 35 7-70

-7 7 91 21 7-7 21 91

\

2

24
2 The variances of the parameters are given by Cj^a and of linear

functions, £'$, the variance is £'c£a 2 . For the example

V(A) = V(B) = CO2 = CO2 = 350

2/336 = 5Cf 2/48

V(C) = V(D) = c^^a2 = c a

2 = 9la

2/336 = 130

2 /48

33 44

V(A-B) = (C + C - 2C

12

)a

2 = 140a2/336 = 50

2/12

V(A+B) =

V(C+D) = (C,_ + C A

. + 2C, Ja2 = 2240 2/336 = 2a

2/3 33 44 34

V(C-D) = (C„ + C AA - 2C J(J

2 = 140O2/336 = 5a

2/24 33 44 34

v(A) = c rr a

2 = 2a

2/336 = a2/i68

55

25
FOOTNOTE 1

If one had assigned the two standards to positions B and C instead

of to A and B as was done, then one would be repeating the measurement

of the difference (B-C) , and of the difference (A-D) . These differences

are internal to the pair of standards over the pair of unknowns and

one might suspect that they add little to the transfer from standard

to test item. This is confirmed by examination of the inverse of the

matrix of normal equations, s /

4 -1 -1 -2 -1 1 168

56 28 84

-1 4 -2 -1 1 14 -14 84

-1 -2 4 -1 1 -14 14 84

-2 -1 -1 -14 28 56 84

168 1

1 1 /

84

S

84 84 84

The variances for the standards are

V(B) = V(C) = 14CT/168 = a /12

V(B-C) = 56a2/168 = a

2 /3

which is smaller than for the restraint A + B = K.

However, for the test items the variances are

V(A) = V(6) = 56a

2 /168 = a2/3

which is larger than that with the restraint A + B = K for which the

corresponding variance is 130^/48.

The estimate for the test item, A, is

A = i [8Yl - 4y

2 - 4y

3 - 4y

5 - 4y6 + 8y?

] + §

which does not involve y4 and yg which are measurements of the difference

between the two standards, i.e., of B-C. Thus, there is a gain in efficiency in the calibration of the test block by using positions A and B for the standards, the efficiency factor being (a

2 /3)/(13a2 /48) = 16/13.

26
FOOTNOTE 2

If there were but a single standard, A, the inverse of the matrix

of normal equations would be

/

-1 _ _1_

168

4 -1 -1 -2 1

-1 4 -2 -1

-1 -2 4 -1

-2 -1 -1 4

168

The variances of the test items are

168

70 42 28 168

42 70 28 168

28 28 56 168

10

\

168 168 168 168

/

V(B) = V(C) = 7oa

2/i68 = 5a2/i2

V(D) = 56a2/l68 = a

2/3

FOOTNOTE 3

If the sum of all four were taken as the restraint, the inverse of

the matrix of normal equations would be

\

4 -1 -1 -2 1

1 4 -2 -1 1

1 -2 4 -1 1

2 -1 -1 4 1

168

-1 1 336

49 -21 -21 - 7 84

-21 49-7 -21 84

-21 - 7 49 -21 84

- 7 -21 -21 49 84

2

/

84 84 84 84

\ /

The variances of all four test items are the same

V(A) = V(B) = V(C) = V(D) = 49<J 2/336 = 7<J 2/48

27
APPENDIX C

Computer Program for Analysis of Gage Block Data

5 *****»*« **»&G AGE BLOCK CALIBRATIO N**»*»*#**»

10 e THIS PROGRAM COMPUTES THE VALUES OF THE UNKNOWN GAGE BLOCKS,

15 * AND PERFORMS TWO STATISTICAL TESTS, THE F-TEST AND THE T-TEST,

20 * T6 DETERMINE IF THE PROCESS IS IN CONTROL.

2 5 ****««•»«• *«*»*«***** B-»**-fr**e**»o*****fl- »»»***»# ***»*» *«*«•» ««»»«#«*«»#«

30 * INPUT

35 * 40 * THIS PROGRAM CALLS FOR A USER CREATED DATA FILE WHICH CONSISTS

45 * OF THE FOLLOWING:

50 * (1) (C$( I ), I »1 ,3 ) - DATE, OBSERVES, INSTRUMENT

55 «• (2) K - VALUE OF RESTRAINT! M ICR6INCHES ) 60 * (3) NO - NOMINAL SIZE OF TEST BLOC KS( INCHES

65 * ( 4 ) S5 - ACCEPTED VALUE OF CHECK STANDARD! M ICROI NCHE S ) 70 * ( 5 ) B5 - ACCEPTED S.D. OF THE INSTRUMENT! M IC ROI NCHES ) 72 * ( 6 ) B6 - ACCEPTED TOTAL S.D,

75 * ( 7 ) M2 - UNCFRTAINTY IN THE RESTRAINT

80 # (8) X(I),Y(I) - OBSERVED READINGS (EIGHT PAIRS)

85 6»ttnmtKnnttHf*5tttittt ******* »*«***»**«**«* *»*»****«*» «»«««»««»»«<-

90 * 95 * 100 *»»«»»*»»****»*»»**»**»«•*»»»***»*»»»****» »** **»*»* «*»e- »****«»

105 #

110 * DATA VALUES WHICH ARE DETERMINED BY THE CALIBRATION DESIGN,

115 * AND WHICH ARE STOKED WITHIN THIS PROGRAM IN DATA STATEMENTS,

120 * ARE AS FOLLOWS:

125 # (1) B(I,J) - LEAST SOUARFS COEFF TO COMPUTE THE UNKNeWNS

130 * (2) M( I , J ) - LEAST SQUARES COEFF TO COMPUTE THE DEVIATION

135 * ( 3 ) E( I ) - VARIANCE FACTOR

138 * (4) F( I > - DRIFT VECTOR

140 * (5) CI - MATRIX DIVISOR

142 * (6) GS( I ) - BLOCK DESIGNATION

145 * 150 » OTHER VARIABLES:

155 * (1) N - NO. OF BLOCKS IN THE CALIBRATION (N - 4)

160 * (2) Gl - NO. OF OBSERVATIONS ( Gl - 8)

165 «• (3) F4 - F RATIO (CRITICAL VALUE FOR P --.01); ( F4 « 3,32t

170 * (4) A4 - NO. OF DEGREES OF FREEDOM ( A4 » 4)

1 75 *****•»*«*»*«»»»* *»»****» »««»««»»««*»»»«*»«*** »*»*« »»**** *«**«***«**

180 * 185 * 190 ******** «»*»*»««*»»»»**»«»>«*«»»»«*»»»»*>*«»«**»»«>» «»*«•«*«»»*

195 * OUTPUT

200 «• 205 * (1) DATE, OBSERVER, INSTRUMENT

210 * (2) COMPARATOR READINGS

2 15 * (3) OBSERVED DIFFERENCES

2 20 * (4) DEVIATIONS

225 # (5) VALUES OF THE UNKNOWNS

230 * (6) OBSERVED STANDARD DEVIATION

235 * 17) STATISTICAL TESTS

240 * (8) UNCERTAINTY STATEMENT

245 **»» **»» &»*********»*»****»*********»******»*«*«******»* «»»««««**««

28
250 DIM (:$( 3 ),B( 4,9),M< 8,8), A( 9 ) , X( 8 >,Yt 8 >.G$(4 ) 255 DATA 35, -14,-7,-14,-21 ,-14,21 ,14,84

260 DATA -35, 14, 7, 14, 21, 14, -2 1,-14, 84

265 DATA -7, 14,35,-42,-7, 14,-49,42,84

270 DATA 7,4 2,-35,-14,-49,42,-7,14,84

275 DATA 49, -7, -7, 21, 49, 49, -7, 21

280 DATA -7,87,13,-5,33,-41, 53,35

285 DATA -7, 13,89,25,-39,37,57,-7

290 DATA 21, -5,25,1 1 1,-27,3, -23,63

295 DATA 49,33,-39,-27,9 7,25,9,21

300 DATA 49, -41 ,37,3,25,103, 13,-21

3 05 DATA -7, 53,57,-2 3,9, 13,73,-7

310 DATA 21 ,35, -7,6 3,21 ,-21,-7,63

312 DATA -7,-5,-3,-1,1,3,5,7

315 DATA ,31 250, .312 50,, 1458 33, ,145833

32 DATA 168

322 DATA o • « 3« • t X 9 I 330 N-4

335 Gl -8

340 F4-3.32

345 A4-4

350 Rl -Gl l

355 * READ COEFFICIENTS USED TO COMPUTE VALUES 6F THE BLOCKS

360 FOR I - 1,N

365 FOR Jl - 1,R1

370 READ B( I , J 1 ) 3 75 NEXT Jl

380 NEXT I 3 85 * READ COEFFICIENTS USED TO COMPUTE THE DEVIATIONS

390 FOR ][ - 1,G1

395 FOR Jl- 1,G1

400 READ M( I ,J1 ) 405 NEXT Jl

410 NEXT I 411 * READ DRIFT VECTOR

412 FOR [ - 1.G1

413 READ F( I ) 414 NEXT I 415 * READ VARIANCE VECTOR

420 FOR I - 1,N

425 READ E( I ) 430 NEXT I 435 * READ MATRIX DIVISOR

440 READ CI 441 * READ BLOCK DESIGNATIONS

442 FOR ]C - l,N

443 READ G$( I ) 444 NEXT I

29
445 » DEFINE USER DATA FILE - DATA1

45 FILES DATA1

455 * READ ADMINISTRATIVE DATA AND PROCESS PARAMETERS

460 READ#1 ,CS( 1 ),C$( 2 ),C$( 3 ) 465 READ*1 ,K,N0, S5, B5.B6.M2

470 » READ COMPARATOR READINGS AND COMPUTE THEIR DIFFERENCES

471 * ALSO, COMPUTE DRIFT -Dl, AND S.D.( DRIFT) » SI

473 Dl -0

475 FOR I - 1.G1

480 READ#1,X( I ),Y( I ) 4 85 A( I )-X< I )-Y( I ) 487 Dl -Dl *A( I)*F( I ) 490 NEXT I 492 D1-D1/C1

493 SI -B5»( 1 ,/Cl )t.5

495 » SET A(9)- RESTRAINT

500 A(9)-K

505 * COMPUTE VALUES - V( I ), S. D. - Z(I), AND UNCERTAINTY - C( I ) 510 FOR I - 1,N

515 Yl -0

520 FOR J1-1.R1

525 Yl -Yl »B< I, Jl >*A< Jl ) 530 NEXT Jl

535 V( I )-Yl/Cl

540 Z( I )-( B6T2-B5T2*E{ I ) )t.5

545 C( I )-3»Z( I )».5*M2

550 NEXT I 555 » COMPUTE CHECK STANDARD

560 C5-V( 1 )-V( 2 ) 565 * COMPUTE THE DEVIATIONS AND THE OBSERVED S.D.

570 SO'-O 575 FOR I - 1,G1

580 DO-0

585 FOR J1-1.G1

590 D0-D0*M( I,J1 )*A{ Jl ) 595 NEXT Jl

600 D( I )-D0/Cl

605 S0-SO*D( I >*2

610 NEXT I 615 S-{ S0/A4 )t.5

620 » PERFORM STATISTICAL TESTS

625 F-( S/B5 )f2

630 T-( C5-S5 )/B6

30
635********* ***************0 U T P U T*»************»******* *** 640 PRINT, 906

645 PRINT DATE , C$< 1 ) 650 PRINT OHS. , C$( 2 ) 655 PRINT INSTR.,C$(3)

660 PRINT, 905

665 PRINT OBSERVATIONS

670 FAR I - 1,G1

675 PRINT X( I ),Y( I ) 680 NEXT I 685 *PRINT OBSERVED DIFFERENCES AND DEVIATIONS

690 PRINT, 695

695 FMT //,X18, A( I ),X8, DEV( I ) 700 FOR I - 1,C?1

705 PRINT, 71 0, A( T ), D( I ) 710 FMT X14,F9 ,3,X4,F9.3

715 NEXT I 720 * PRINT VALUES OF THE BLOCKS

722 PRINT, 723

723 FMT //,X58 .UNCERTAINTY

725 PRINT, 730

730 FMT X19, NOM. ,X8, CORR, ,X1 0,S. D, , X4, 3 ( S. D , ) . 5( S . E . ) 735 FOR I -1 ,N

740 PRINT, 745, G$( I ),N0,V( I ),Z( I ),C( I ) 745 FMT F12.6.F1 1 ,2,X7, F8.5,F15,5

750 NEXT I 755 * PRINT STATISTICAL INFORMATION

760 PRINT, 765

765 PRINT OBS. S. D, ACC. S. D. , F TEST, F RATIO, D.F.

770 PRINT, 775, S,B5,F,F4,A4

775 FMT F7.4,X5,F9.5,X3,F8.3,F12.2, 110

780 IF F>F4 THEN 790

"*85 G0 TO 800

790 PRINT, 795

795 PRINT ************S. D. IS NOT IN CONTROL**********************

800 PRINT, 805

805 PRINT OBS, CHECK, ACC, CHECK, T TEST

810 PRINT, 81 5,C5,S5,T

815 FMT F10,5,F11.5,F12.5

820 IF ABS( T > > 3 THEN 830

825 GO TO 840

830 PRINT, 835

835 PRINT *****»******CHECK STANDARD IS NOT IN CONTROL****** ********

840 PRINT, 905

845* PRINT DRIFT AND S, D, (DRIFT)

885 PRINT, 890, Dl , SI 890 FMT DRIFT " ,F10,4/ S.D. (DRIFT) - F10.4

895 PRINT, 900,

900 FMT ///, RESTRAINT (S.*S..) - ,F8.2

901 PRINT, 903, M2

902 PRINT, 906

903 FMT SYSTEMATIC ERROR( S. E. ) IN RESTRAINT • ,F8,2

905 FMT //

9 06 FMT //////

910 STOP

915 END

31
INPUT = USER DATA FILE

10 MAY 2 8 19 74, H6WELL, FEDERAL 1 20 6. 4, .101, -.133, .32, .49. .20

30 52.0,52.5

40 45.2,52.1

50 50.0,45.1

60 53.1 ,50.0

70 52.3,45.2

30 45.1,52.0

90 52.0,50.1

1 00 50.1 ,52.3

OUTPUT

DATE MAY 28 1974

ttBS. H6WELL

INSTV. FEDERAL 1 OBSERVATIONS

52 52.5

45.2 52.1

50 45. 1 53. 1 50

52.3 45.2

45,1 52

52 50. 1 50. 1 52.3

A( I ) -.500

-6.900

4.900

3. 100

7.100

-6.900

1 .900

-2.200

DEV( I ) .029

- .046

.113

.57 1 -.237

-.079

-.154

.304

UNCERTAINTY

N6M. C6RR. S. D. 3(S.D. )».5( S.E. ) S. . 101000 2.95 .45618 1 .46854

S.. .101000 3.45 .45613 1 .46854

X . 101000 .92 .47452 1 .52355

Y . 101000 -3.88 .47452 1 .52355

6BS. S.D ACC. S. D. F TEST F RATIO D.F.

.3607 .32000 1 .271 3.32 4

6BS. CHECK ACC. CHECK T TEST

-.50000 -.13300 " • 74898

DRIFT .0042

S.D.( DRIFT) .0247

RESTRAINT (S.*S..)

SYSTEMATIC ERR0R( S. E. ) IN RESTRAINT - 6.40

.20

32
NBS TECHNICAL PUBLICATIONS

PERIODICALS JOURNAL OF RESEARCH reports National Bureau

of Standards research and development in physics, mathematics, and chemistry. Comprehensive scientific papers give complete details of the work, including1 laboratory data, experimental procedures, and theoretical and mathematical analyses. Illustrated with photo- graphs, drawings, and charts. Includes listings of other NBS papers as issued. Published in two sections, available separately

• Physics and Chemistry (Section A)

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on

standards of physical measurement, fundamental constants, and properties of matter. Issued six times a

year. Annual subscription: Domestic, $17.00; Foreign,

$21.25. • Mathematical Sciences (Section B)

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical analysis, theoretical physics and chemistry, logical design and programming of computers and computer

systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic, $9.00; Foreign, $11.25. DIMENSIONS/NBS (formerly Technical News Bulletin) —This monthly magazine is published to inform

scientists, engineers, businessmen, industry, teachers,

students, and consumers of the latest advances in science and technology, with primary emphasis on the work at NBS. DIMENSIONS/NBS highlights and reviews such

issues as energy research, fire protection, building technology, metric conversion, pollution abatement,

health and safety, and consumer product performance.

In addition, DIMENSIONS/NBS reports the results of Bureau programs in measurement standards and tech- niques, properties of matter and materials, engineering standards and services, instrumentation, and automatic

data processing. Annual subscription: Domestic, $6.50; Foreign, $8.25.

N0NPERI00ICALS Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and

industrial practice (including safety codes) developed

in cooperation with interested industries, professional organizations, and regulatory bodies. Special Publications—Include proceedings of high-level national and international conferences sponsored by NBS, precision measurement and calibration volumes, NBS annual reports, and other special publications appropriate to this grouping such as wall charts and

bibliographies. Applied Mathematics Series—Mathematical tables, manuals, and studies of special interest to physicists, engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides

quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a world-wide program coordinated by NBS. Program under authority

of National Standard Data Act (Public Law 90-396). See also Section 1.2.3.

Building Science Series—Disseminates technical infor- mation developed at the Bureau on building materials, components, systems, and whole structures. The series presents research results, test methods, and performance criteria related to the structural and environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete

in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies. Voluntary Product Standards—Developed under pro- cedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations. The purpose of the standards is to establish nationally recognized requirements for products, and to provide

all concerned interests with a basis for common under- standing of the characteristics of the products. The

National Bureau of Standards administers the Voluntary Product Standards program as a supplement to the activities of the private sector standardizing

organizations. Federal Information Processing Standards Publications (FIPS PUBS)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The purpose of the Register is to serve

as the official source of information in the Federal Gov- ernment regarding standards issued by NBS pursuant

to the Federal Property and Administrative Services Act of 1949 as amended, Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717

(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations). FIPS PUBS

will include approved Federal information processing standards information of general interest, and a complete index of relevant standards publications. Consumer Information Series—Practical information, based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable language and illustrations provide useful background

knowledge for shopping in today's technological marketplace. NBS Interagency Reports—A special series of interim

or final reports on work performed by NBS for outside sponsors (both government and non-government). In

general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Infor- mation Service (Springfield, Va. 22151) in paper copy

or microfiche form. Order NBS publications (except Bibliographic Subscription Services) from: Superintendent of Documents, Government Printing Office, Washington, D.C. 20402. BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service

(Publications and Reports of Interest in Cryogenics). A literature survey issued weekly. Annual subscription: Domestic, $20.00; foreign, $25.00. Liquefied Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00. Superconducting Devices and Materials. A literature survey issued quarterly. Annual subscription: $20.00. Send subscription orders and remittances for the preceding bibliographic services to the U.S. Department

of Commerce, National Technical Information Service, Springfield, Va. 22151. Electromagnetic Metrology Current Awareness Service (Abstracts of Selected Articles on Measurement

Techniques and Standards of Electromagnetic Quantities from D-C to Millimeter-Wave Frequencies).

Issued monthly. Annual subscription: $100.00 (Special rates for multi-subscriptions). Send subscription order and remittance to the Electromagnetic Metrol- ogy Information Center, Electromagnetics Division, National Bureau of Standards, Boulder, Colo. 80302.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington. D.C. S0234

OFFICIAL BUSINESS

Penalty for Private Use, $300

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE COM-215

*" /Si. %

7^6 -i9l6