Notes on the Fundamentals of

Measurement and Measurement

as a Production Process

Paul E. Pontius

Institute for Basic Standards

National Bureau of Standards

Wastimgton, D. C. 20234

September 1974

Final

Prepared for Optical Physics Division

. istitute for Basic Standards

IIY)

ational Bureau of Standards

Zl-^ ashington, D. C. 20234

No.7^[-5i|5 mr . 2

NBSIR 74-545

NOTES ON THE FUNDAMENTALS OF

MEASUREMENT AND MEASUREMENT

AS A PRODUCTION PROCESS

Paul E. Pontius

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

September 1974

Final

Prepared for Optical Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington, 0. C. 20234

U. S. DEPARTMENT OF COMMERCE. Frwterick 3. Oem. Set^etary

NATIONAL BUREAU OF STANDARDS, Richard W. Robarts. Oir«ctor
INDEX

1.0 Introduction

2.0 Measurement "Rules"

3.0 Conceptual Measurement Process

4.0 Variability - Two Approaches

5.0 Measurement as a Production Process

6.0 The Unit

7.0 The Practical Measurement Process

7.1 Verifying the Algorithm

7.2 Performance Parameters

7.3 Unit Error

7.4 Measurement Requirements

8.0 Summary

Page

1

2

7

10

20

30

37

37

46

49

54

60

List of Symbols

S,E. Systematic Error

Within Group Variability W

Total Process Variability

Between Time Component of

Variability

first used on page 11

" 12

" 12

" 12

tl It If

II II II
1.0 IntroducCion

Ac a very early age, most humans are aware of concepts

such as size, distance, quantity, force, time, hot and

cold. From the beginning, man has used these, and

similar concepts, to understand and to shape the

environment and society in which he lives. In so doing, more or less formal means of quantizing these

concepts have evolved. The resulting procedures,

called measurements, are now an accepted part of every

day life. Because measurements are only made to

support the accomplishment of a variety of tasks, the

interest of the individual is more often in the task

itself rather than in the measurement detail. It is

only necessary that the measurement procedure which one

uses, whatever they might be, will produce adequate

results for the task at hand. As long as the criteria

of the individual are satisfied, the procedural detail

is of little consequence. The situation changes,

however, as soon as the task is beyond the ability of a single individual to perform.

For the complex task, many value judgments are made by

many different people, frequently at various stages of

completion as well as on the completed work. Elements

from many sources are assembled to make the whole. For

functional reasons as well as an aid to communication,

it is essential to establish acceptable limits for

measurement error for all of the measurements necessary

to accomplish the desired end. For each contemplated

measurement, in addition to acceptable error limits, one must also assess the consequences of failure to

meet the requirements as well as the benefits, if any,

which might be obtained with additional measurement

effort. This evaluation is not always easy to do.

Often the dividing line between success and failure is

not well defined. One may not be aware of failure

until long after the completion of the task. In the

end, the best measurement process is that which

produces adequate measurement data with a minimum

expenditure of measurement effort. As a consequence,

the formulation of a measurement process starts with

the establishment of realistic estimates of acceptable

limits of error. Measurement process analysis provides

realistic estimates of the process variability, which

in turn provide the assurance that the results are adequate for the task at hand.

- 1 -
2.0 Measurement "Rules"

Philosophers of science consider two types of

measurables called extensive magnitude quantities and

intensive magnitude quantities [1, 2].* Extensive

magnitude quantities are those for which there is a realizable addition operation, as is the case with mass

and length. Addition permits replicas and subdivisions

of the unit to be combined to construct any desired

magnitude of the particular property. Three "rules"

are sufficient for the measurement of such quantities.

Intensive magnitude quantities are those which do not

have a realizable addition operation. For these

quantities, one must subdivide an interval between

defined fixed states at, or near, the maximum and

minimum of the range of interest. The measurement of

intensive magnitude quantities is based on five

"rules." In reality, however, it is not always clear

which set of "rules" is applicable to a given

measurement. This is particularly true in the case of

extensive magnitude quantities which are normally

discussed in context with the "three rule" scheme, but,

without exception, are measured in accordance with the

"five rule" scheme associated with intensive magnitude

quantities

.

The three "rules" for extensive magnitude quantities

are:

1. The unit rule.

2. The additive rule.

3. The equality rule.

Measurement attempts to establish a one-to-one

correspondence between a set of numbers and the

magnitude of a particular property. In order to do

this, some magnitude, however arbitrary, must be

defined to correspond to a particular number. In most

cases, the number is the "unit" corresponding to the

numeral "1", however, it can be any convenient number.

When many people are concerned with the same quantity,

communications are considerably simplified if there is a uniform acceptance of a definition of the unit. For

consistency of measurement, however, it is only

necessary that there be defined relations between the

various units in common use.

* The numbers in brackets refer to similarly numbered

references at the end of this paper.

- 2 -
The addition rule, in essence, states the manner in

which the multiples or subdivisions of the unit are constructed, as appropriate to a particular

measurement. For example, accepting the length of a line interval as a length unit, with a simple set of

dividers, one can step off intervals equal to the unit

along a straight line, or subdivide the unit as appropriate. A line interval of any desired length

relative to the unit can be constructed by making the

"starting" termination of each added increment coincide

with the "ending" termination of the preceding

increment

.

The equality rule states the circumstances under which

one announces the magnitude of a property embodied in an object as being the same as the magnitude of the

property embodied in the unit, or some multiple and

subdivisions in summation. The equality in magnitude

is expressed by assigning the same number to the

magnitude of the unknown as has been assigned to the

appropriate accessible embodiment, or extrapolation, of

the unit.

The "rules" for intensive magnitude quantities are:

1. Rule for ordering, i.e., is A greater than

or less than B?

2. The "zero" rule.

3. The "unit" rule.

4. Rule for subdividing the interval between

"zero" and "unit."

5. The equality rule.

Temperature is an example of an intensive magnitude

quantity. For a large temperature difference, our senses can easily tell us that A is hotter than B,

therefore, if temperature is related to hotness, it is

logical to assume that the temperature of A is higher

than the temperature of B. To establish a temperature

interval, one must define a "zero" state and a "unit"

state, as for example, the triple point of water and

the steam point ^. Having defined a reasonably useful

^ The triple point of water, a state where ice, liquid,

and water vapor exist in equilibrium is approximately

0°C. The steam point, the maximum temperature where

water and water vapor exist in equilibrium, is

approximately 100°C.

- 3 -
temperature interval, one needs a temperature

proportional transducer which has an output suitable

for the construction of a number scale, i.e. the change

in height of a mercury column due to the differential

expansion of mercury and glass in a normal

thermometer^. With the instrument scale interval

terminators marked for both the "zero" state and the

"unit" state, one must then define the manner in which

the instrument interval is to be subdivided^. Finally,

for the definition of equality, the temperature of the

environment surrounding the thermometer is the

instrument scale reading as defined by the height of

the mercury column.

One normally learns about the measurement of quantities

such as mass within the context of the simple three

rule scheme of measurement. Everyone is familiar with

the accepted mass standards such as the kilogram and

the pound. The addition "rule" is merely stacking the

required number of weights on the balance pan. The

major area of difficulty is associated with the

equality rule. For example, one classical definition

of equality of mass is: "Two masses are equal if they

can be interchanged on the pans of a "perfect" balance

without disturbing equilibrium." This statement has no meaning for the situation in which the two objects

being compared, A and B, are such that the mass of A is

not equal to the mass of B.

There are at least two possible courses of action. One

can obtain the prescribed equality condition by adding

small auxiliary weights as appropriate, or by altering

the mass of one of the objects. For either course of

action, at best, one cap state only that:

|a - b| < c

where e accounts for the minimum size weight which can

be manipulated, or the minimum amount of material which

can be added or removed from one of the objects, and

for the operator value judgment relative to having not

disturbed "the equilibrium." The value judgment, in

turn, depends upon the skill and perhaps political

motivation of the operator, as well as the sensitivity

of the instrument and a host of other factors. For an

Of all the human senses, vision is the most

sensitive. With adequate reference points, the eye can easily detect changes on the order of a few thousandths

of an inch.

o A common example of different ways of doing this is

the Fahrenheit and the Celsius temperature scales.

- 4 -
e which is small relative to the manner in which the

object is to be used, the numerical value, of the mass

of one object may be assigned to be the same as the

number value assigned to be the magnitude of mass

embodied in the other. The disturbing result is that

identical numbers are assigned to objects which are not

equal in mass. The whole operation can be a value

judgment where one takes refuge in terms such as "exact", "accurate", "right on....", and the like.

With the addition of a small weight, C, with a known

mass relative to the "standard", A, one can quickly

determine

(A + C) > B > A

If, with (A+C) on the balance pan, the instrument

produces an indication 0^ , and with A on the balance

pan, an indication of 0^ , with the "unknown" B on the

balance pan, one will obtain an indication 0^ such that

Q^>Q^>0 . By subdividing the instrument indication

interval (0^-02), one can relate the indication 0

directly to the mass of B. Further, for any object

with mass in the interval of (A+C) and A, one can obtain directly from the instrument indication a verifiable estimate of the mass of that object. With

this procedure, while the process variability remains,

there are no value judgments other than that associated

with reading the instrument scale. These procedures

clearly follow the intensive magnitude quantity rules,

as is always the case where one relies on the

instrument to subdivide some increment of the

accessible unit. In mass measurement, one can hardly

see an object with mass of one microgram, let alone the

task of manipulating such an object (about the size of

the smallest visible dust particle) . In length

measurement, a microinch is about one-twentieth of the

wavelength of the light source from a helium-neon

stabilized laser. With the amplification of the

measurement instrxjment, such small increments are clearly discernible on the instrument reading scale.

All direct reading instruments, regardless of the

quantity being measured, are based on the intensive

magnitude quantity rules. For the most precise

measurements, the instrument may subdivide some small

increment, as is the case with most precise mass measurement instruments. In using the substitution

weighing procedure, one compares the object with an appropriate standard and a "sensitivity" weight.

- 5 -
relying on the instrument to subdivide the mass of the

sensitivity weight. For practical measurements, the

instrument is usually designed to subdivide the maximum

capacity of the instrument. The instrument also may

effectively assume the role of the "unit", or standard.

In some cases, the embodiment of the unit is actually

changed, as is the case for most multiple lever scales

where a fixed weight and a variable lever arm replaces

many summations of "units" of known value. While the

use of intensive, quantity magnitude "rules" provide the

means for practical measurement, the problems

associated with providing assurance in the adequacy of

the end result become complex.

- 6 -
3 . 0 The Conceptual Measurement Process

In discussing a measurement process, one must keep con- stantly in mind the dual nature of human activity.

With our hands, we manipulate objects, operate

equipment and the like, but with our minds we manipulate conceptions of those objects, or actions.

We observe what happens when we take certain actions

and, as long as the results confirm what we think

should happen, we are satisfied with our conceptions.

When the results do not agree, we must either change

what we do or change our mental conceptions.

In a conceptual measurement process, repeated

measurements of the same thing, or the difference

between two things, agree exactly or within predictable

limits. This concept applies to processes which are

used to order, or sort, similar things relative to a specified magnitude of a particular property as well as

to processes which assign numbers to represent the

magnitude of the embodied property. The prediction

limits relate to the details of the process. In order

to achieve the result, the conceptual process utilizes:

a model concept associated with the object or property

to be measured; an algorithm concept which includes all

of the instrumentation, manipulative procedures,

computations, and the like, necessary to make the

measurement; and a unit concept which relates to the

way in which the unit is introduced into a particular

measurement^. For most measurement processes, the

results are to be passed on to other persons. In some

cases, the measured object and the assigned value

become the accessible unit for another measurement

process. In other cases, results from measurements on selected items, or material samples, determine the

disposition of large numbers of similar items, or quantities of material. In all cases, the area of

doubt, or uncertainty associated with the individual

measurement result is the basis for a judgment

concerning the adequacy of the measurement effort.

The area of doubt, or uncertainty, associated with the

result reflects the disparity between the various

concepts and the performance of the measurement process

in the real world. Realistic uncertainties are based

^ The unit-algorithm-model concept suggested by Volodarski,

Rozenberg, and Rubichev [20] is, in essence, a regrouping

of the elements of a measurement method and process discussed by Eisenhart in reference [3], and more extensively

in reference [4 ]

.

- 7 -
on the variability of the results from repeated

measurements. Two sources of such variability are model ambiguities and algorithm errors. Model

ambiguity, where the conceptual model differs from the

actual object, will cause "perfect" measurements to

disagree because the object is not behaving like the

conceptual model. In contrast, algorithm errors cause measurements on "perfect" objects to disagree because

the algorithm is not proper. Model ambiguity and

algorithm error can be reduced to insignificance by

either of two methods: one can refine the conceptual

model and adjust the algorithm accordingly, or the

object can be refined to fit the existing conceptual

model.

To illustrate, a conceptual model of mass might be

simply the property of an object. The algorithm could

be simply the act of achieving an "exact balance" on a suitable instrument, the "exact balance" condition

implying equality of mass. Repeated measurements of

the difference between two similar objects, i.e. mass increment required for "exact balance", would produce a sequence of numbers with a characteristic variability.

If one of the objects is changed, for example, a stainless steel kilogram is compared with an aluminum

kilogram, the variability of the collection of repeated

measurements increases drastically relative to the

comparison of two stainless steel kilograms. This

behavior is a clear indication that either the model

concept, or the algorithm, is not correct.

There are two courses of action. One can restrict the

range of density of the material to be compared, thus

"outlawing" aluminuml With such action, the

variability of the collection of repeated measurements

would always be well behaved. In this case, the object

has been refined to fit the conceptual model. However,

in order to obtain meaningful mass measurements over a variety of materials, one must change the concept of

the model to either the property of a "point" or the

property of a body in the vacuum of space. In either

case, for each object, the displacement volume, and

perhaps temperature and coefficient of volumetric

expansion must be known. The algorithm must be

modified to account for the buoyant force of the

environment in which the measurement is made. With

these actions, the variability of repeated differences

between stainless steel and aluminum will also return

to normal.

- 8 -
For all measurable quantities, there must be some defined unit. This unit, however arbitrary, is

accepted as having zero error by definition. The unit,

in some form or other, must be introduced into each

measurement. With few exceptions, the accessible unit

is the output of some previous measurement process, and

thereby defined by a model-algorithm combination. If

alternative unit model-algorithm combinations do not

produce compatible results, in addition to accepting

the defined unit, all must also accept a unit modelalgorithm combination^. The unit error, that is, the

disagreement between the unit as realized and as expressed by the assigned number, is frequently beyond

local control. In processes where the "unknown" is

compared with an artifact reference standard, the unit

error is the uncertainty from the measurement process

which was used to establish the number assigned to the

standard. In the case of direct-reading measurements,

the unit error may include additional components

relating to the instrument design.

If for example, mass measurements based on the con- servation of momentum, such as used in atomic physics,

would produce results which were not consistent with

the results from traditional mass measurement

processes, one would have to accept one method or the

other to define the measurement system. As an alternative, one might define the domains within which

each of the two methods are to be used.

- 9 -
4.0 Variability - Two Approaches [5]

As a point of departure, all measurement processes are,

directly or indirectly, comparative operations. Even

the most simple concept of such a measurement contains

certain implicit assumptions:

(a) a constancy in the basis for the ordering

or comparing; and

(b) a stability in the equipment, procedures,

operator and the like which are used to

make the easurement

;

(c) a stability in the object, effect or property being observed.

Quantitative ordering implies an invariant basis for

the ordering, thus a long term constancy in a standard

unit and a stability in the realization of a standard

unit, is necessary. In a similar manner, the property

to be measured must also be stable. If a measurement

process detects a difference between two things, it is

expected that repeated measures of that difference

should agree reasonably well. In the absence of severe external influence, one does not expect things to

change rapidly.

There is a difference between stability and constancy

in context with the above. Repeated measurements over

time can exhibit a random like variability about a constant value, or about a time dependent value. In

either case, if the results are not erratic- (with no unexpected changes), the process is considered to be

stable. The objects being compared may have constant

values, or may be changing at a uniform rate, or may be

changing at different rates. For continuity, time

dependent terms must be included in quantitative

descriptors for both objects being compared. Stable

changes with time can be extrapolated in the same manner that one "extrapolates" a constant value over

time. The extrapolations can be verified whenever

desired by making additional measurements. Constancy,

then, merely means that the coefficients of time

dependent terma are essentially zero. This is not to

say that features such as constancy are not desirable

for certain usage, but only that such features are not

necessary restrictions on the ability to make good and

useful measurements.

- 10 -
Two quantitative descriptors are used to describe the

process variability, and ultimately, to establish the

bounds for the limit of error. A given measurement

process is continually affected by perturbations from a variety of sources. The random like variability of the

collection of repeated measurements is a result of

these perturbations. One descriptor, designated random

ewovt includes effects from both cyclic perturbations

such as might be associated with the environment and

variability associated with operating procedures. The

random variability, expressed as a standard deviation,

will for a process in control (see [3]) imply a low

probability that the range of variability in the

collection will exceed certain bounds. The second descriptor, designated systematia error, S.E., includes

the use of constants which are in error as well as discrepancies from certain operational techniques. The

S.E., expressed as a single number, is an estimate of

the offset of the measurement result from some defined

process average. These two descriptors, called the

process performance parameters, are factors in

assessing the worth of a result relative to a particular requirement.

The random error estimate reflects the effects of

cyclic perturbations which are constantly changing

whether the process is being used or not. These

effects can be grouped into two categories; short term

effects which vary through one or more cycles in the

course of a single measurement or measurements made

over a short time interval, and long term effects in

which the period of the effect is at least as long as the time required for a given sequence of measurements.

A second category of short term effects are those which

are instantaneous, or step-like, in nature. In many

cases, "shocks" on the instrument, or variations in

manner in which various objects are introduced to the

instrument, cause changes in the instriiment

configuration which affect the instrument indication.

The effects appear as minute, and sometimes not so minute, instrument reading scale shifts^. For example,

the manner in which a large weight is placed on a

In this case one is not concerned with transient

changes as for example when a meter needle "jumps" and

immediately returns to the original reading. The

operator can be instructed to ignore such changes.

Neither is one concerned with slow "drifts" which occur

in the course of a single measurement. In most cases,

these can be accounted for in the algorithm.

- 11 -
platform scale may effectively shift the scale several

readable units. Off center loading on either a balance

or a large weighing instrument may cause a change in

lever ratio which has the same effect on the reading

scale. The variability from these sources may be

random in both magnitude and direction.

In terras of measurement process performance, the

within-gr^up variability expressed as a standard

deviation , ay, reflects the combined short term

effects both cyclic and step. In many cases,

represents an optimum process performance. The withingroup variability of the measurement process is the

most familiar process parameter as it is easily

demonstrated in a repeated sequence of measurements of

the same thing in a very short time interval.

Practically all important measurements are repeated

several times. The magnitude of the within-group

variability is generally established by the degree to

which certain types of perturbations are controlled and

by factors such as the operator skills, quality of the

instrument, and attention to detail procedure. In most

. cases one cannot identify sources of perturbations

which contribute to within-group variability. Process

improvement in terms of reducing Oy is obtained perhaps

more frequently by trial and error than by design. The

adequacy of a given process relative to a particular

requirement is often judged on the basis of the withingroup variability. Such a judgment, however, may be

erroneous

.

The total variability is the variability of a long

sequence of data which reflects the effects of all

possible perturbations. Repeating a given measurement

over a time interval sufficiently long to reflect the

influence of all possible perturbations establishes a total process standard deviation, a^, which reflects

both the short term and the long term random

variability^

.

^ Standard deviation is used only as an index of variability,

with no restriction on the distribution intended.

2 The total prcaess variability ^ o^, can be thought of as the sum

of the variabilities of all of the perturbations that affect the

process, that is, o^^=aj^+02^+ (^n^ • class of

perturbation with variabilities which are those with

very short periods and with nearly equal amplitudes, it may not

be possible to identify the individual perturbations. The

variability from these perturbations combine to form a threshold

variability Other perturbations, with variabilities to

0^, may be identifiable if the magnitudes are sufficiently large.

These effects combine to form a between tine component of

variability an. The total variability is then Oj^^o^^+c ^.^

.

- 1 ? -
With a sufficiently long sequence of data, one should

be able to identify the sources of the largest

perturbation through supplemental measurements and

correlation studies. Having identified the source of

the largest perturbation, the magnitude of its effect

on the measurement can be minimized, with a consequent

reduction in the magnitude of a^. Frequently one is

tempted to idealize the process in order to reduce the

total variability, that is, to establish a carefully

controlled environment and use only selected artifacts.

Such actions are self-defeating in terms of

understanding the measurement process. A more

appropriate action, provided one has sufficient

motivation and resources, is to modify the process to

account for the variability associated with all the

perturbations that can be identified. Since a large

perturbation will "offset" the single value from the

process average, these effects, if uncorrected, are frequently called Systematic Errors.^

There are several different classes of Systematic

Errors. Perhaps the most familiar class of S.E. is

associated with instrument reading scale offset. Such

S.E.'s are not present in comparative measurements

provided that the instrument indication can be related

to the measurement unit, and provided that the instrument response is reasonably linear over the range of

difference which must be measured. A second class of

S.E.'s is associated with supplemental data such as barometric pressure, temperature and relative humidity

measurements which are in turn combined to determine

air density, index of refraction and the like. Each of

the supplemental measurements is, in essence, a separate distinct measurement process with both random

variability and systematic effects. The random

variability of the supplemental measurements is, of

course, reflected in the total process variability.

The S.E.'s associated with supplemental data must be

carefully considered.

One action, which is rarely practical, would be to

"randomize" the S.E. by using different instruments,

operators, environmental or other factors., in which

case the variation from these sources becomes part of

^ If the large perturbation is truly cyclic, corrective

action will frequently reduce the magnitude of with

only minor change in the process average.

- 13 -
the random error. A more practical procedure is to

evaluate the S.E. associated with an instrument (or

other factor) by direct experiment. When the change in

response, such as for example that introduced by a temperature error of 0.1 degree, is a small fraction of

the standard deviation of the process, a rather large

number of measurements is required to establish the

effect with a reasonable degree of assurance. Bearing

in mind that an average of n measurements has a standard deviation of iZ/rT times that of the original

measurements, in order to determine an effect of size

one standard deviation with an uncertainty (3 standard

deviations) of half of its size one would need about 36

measurements. (If one relaxes the uncertainty requirement for the average to a value equal to the standard

deviation of the process, then 9 measurements would be

required .

)

With evidence that the individual supplementary

measurements are satisfactory, the next concern is the

manner in which the supplementary data are combined and

used to adjust the observed data. For example, having

adjusted the data for thermal expansion, one would not

expect a collection of values over time to correlate

with the temperature measurements for each individual

value in the collection. A collection of values from

repeated measurements should be tested for dependence

on each of the supplementary measurements, and their

various combinations, as appropriate. If dependence is

indicated, either the supplementary measurement is not

being made at the appropriate location, or the manner

in which the supplementary measurements are combined

does not describe the effect that is actually

occurring. Corrective action is necessary. No dependence does not necessarily indicate that there are no S.E.'s present, but only that for the supplementary

measurements which have been made, the magnitude of the

combined S.E.'s is not large relative to the total

standard deviation of the process.

^ For example, in the simple case of measuring the

width of a piece of paper with a rule, a practice which

permits setting the "0" of the rule to one edge of the

paper will introduce the bias of the operator in

setting "0" as well as the bias associated with the

location of the printed scale on the rule. By placing

the rule at random on the paper, both terminators of

the interval are estimated, thus eliminating both

sources of bias.

- 14 -
There may be long term systematic error effects from

sources not associated with the current supplemental

measurements. It is relatively easy to demonstrate the

presence or absence of such effects, but it may be

difficult to reduce their magnitudes. If one has

available a collection of values over a long time span, one can compare the standard deviation as computed for

small numbers of sequential values over short time

spans with the standard deviation of the total

collection.^ While reasonable agreement is expected,

frequently such is not the case. If the magnitude of

the effect is sufficiently large, the collection of

values may indicate grouping, with the group means appearing as random variability about the process

average. If the distribution of the collection of

values appears to be bi-modal, one should look for a large long term cyclic effect. Until the source of

such variability is identified, and appropriate action

taken to modify the process, the total standard

deviation must be used as the descriptor of the random

variability of the process.

The reason for making measurements is to assign numbers

representing the properties of interest in such a way

that the numbers will be useful to others. The reason

for characterizing the measurement process is to assign

meaningful error bounds, or uncertainties, to the

numbers representing the properties. The magnitude of

the uncertainty is established by the error bounds of

the local measurement process and the error of the

accessible unit. In most mass and length measurements,

access to the unit is through an artifact which has

been assigned a length, or mass, value by another

measurement process. In the case of mass, for example,

the international prototype kilogram is defined to have

zero unit error. With a process operating in a state

The use of comparison designs, such as described in

reference [6], facilitates this type of analysis. The

within group variability, o^, is computed for the

prescribed sequence of measurements. Each measurement

sequence includes in effect a "check standard" which is

measured over and over again with similar measurement.

The total standard deviation is computed for the

collection of values for the "check standard." The

inequality o^>KOy is taken as evidence of the existence

of a long term systematic effect, perhaps as yet

unidentified. The term K in this relation accounts for

the fact that the "reported" value of the "check

standard" from the observations required by the design

is not the result of a "single measurement" but, in

effect, is the weighted average of "n" measurements in

the design sequence, while is the standard deviation

of a "single measurement."

- 15 -
of control, that is, with no known systematic effects

unaccounted for, and with the international prototype

kilogram to introduce the unit, the uncertainty is only

a function of the process standard deviation, either

or a^.^

The reported value may be the result of a single

measurement, or the average of n independent

measurements. Both results are only estimates of an expected long term process average, and, as a consequence, the "reported" result is always offset from

the true process average by some amount. This offset,

as determined by the process standard deviation, can be

either plus or minus. If an object, as measured above,

and its assigned value are used to provide an accessible unit to another process, this offset is a systematic error associated with the unit. That is,

the results from the following process, which uses the

object as the accessible unit, may be biased by the

offset or systematic error of the first process. The

uncertainty of the result from the second process must

include this systematic error associated with the unit

in combination with the random variability of the

second process. Whenever a fixed value is assigned to

the accessible unit, a S.E. component of magnitude

equal to the uncertainty of the assigned value is

introduced. For all well characterized measurement

processes operating in a state of control, the S.E.

associated with the accessible unit should be the only

S.E. component in the uncertainty for the result, all

other identifiable S.E.'s having been accounted for in

the process.

A measurement process is said to be operating in a state of control when: (1) sequences of independent

measurements support a single valued limiting mean; (2)

the collection of values is free from obvious trends or

grouping; and (3) each new measurement verifies the

validity of prediction limits based on historical

It is important to note that only the assigned value

is assumed exact. Model ambiguities associated with

the accessible realization of unit are reflected in Or^. In the case of the International Prototype Kilogram,

variability associated with stability in mass is not

well known. Equipment wich sufficient precision to

evaluate this stability has only recently been

developed [7]. For length, the stability of the Iodine

stabilized laser is on the order of 1 part in 10^^,

thus in all practical measurements, components of

variability from this source are essentially zero [8].

- 16 -
performance data. Most processes can be made to

operate in a state of control. A process is said to be

well characterized when the process performance is

independent of the objects which have to be measured

and the environments in which the measurements must be

made. While this is a desirable goal, particularly

with respect to understanding the process, it is not

always achievable. To obtain such performance, the

unit-model-algorithm combination must be refined and

generalized so as to be applicable to each set of

conditions. Sources of S.E. which can easily be

identified can be taken into account. However, the

condition that the within-group variability, c^^, be

identical to the total variability, c^, for all

conditions of measurement is seldom achievable. As a practical matter, for the well characterized process

one must be satisfied with realistic estimates of Oj.

Refinements to reduce the magnitude of are both

costly and time consuming.

Fortunately, most measurement processes for a given

property are similar so that the characterization and

documentation of a typical process over the range of

objects and environments in which the measurements are usually made substantially shorten the time required

for characterizing other processes. As a practical

matter, few can afford the time and effort to identify

all perturbations related to the between-group

variability of S.E. components as previously discussed.

For each practical measurement it is only necessary

that the uncertainty of the result be adequate for its

intended usage. In the end, the uncertainty associated

with a sequence of operations defined to be a

^ The restrictions for control listed here apply to a variety of measurement situations where a "repetition"

is a repeated measurement after a relatively long time

interval. These measurement situations usually involve

the properties (physical, optical, electrical, etc.) of

objects or systems. A more general condition for being

"in control" is that the process behaves as the output

of a probabilistic model. Situations involving cor- related measurements may be regarded as "in control" if

the output is predictable in the sense that it can be

considered as producing random variables from the

assumed mathematical model. In the latter case, the

measurements are usually concerned with characterizing

a time dependent phenomenon such as the output of an oscillator. A detailed discussion of such a situation

is given in reference [22].

- 17 -
measurement is determined in part by the larger of Oy

and Orj, and by the S.E. components associated with the

unit. The uncertainty statement must also Include the

S.E.'s which are not accounted for in the measurement

process for reasons of convenience.^

For some measurements, the difficulties associated with

making the measurement, the time Involved, and the

cost, make It Impossible to consider sequences of

repeated measurements under varying conditions. In

these cases one usually relies on an "error budget" to

establish an estimate of the uncertainty of the

reported results. The typical error budget assumes the

algorithm-model concept of the measurement to be exact.

A term by tern analysis establishes the effect of the

variability of each term with respect to the announced

result. For each term, a listing is made of all known

sources of error, with an estimate (usually based on theoretical analysis and "engineering judgment") of the

magnitude of the expected variability. The estimates

are usually combined in some appropriate manner to

obtain the error bounds for the final result. While

the error budget analysis is helpful in many kinds of

measurement, it is not unusual to find measurements of

the same thing which disagree in excess of the expected

limits of error based on extensive error budget

analyses. In such cases, the disagreement is strong

evidence that the algorithm-model concept used in one

or both of the measurements does not reflect the real

life situation.

The choice between the two approaches to measurement

variability depends upon the detail of the algorithm.

In both cases, one is trying to establish a realistic

error limit for the process result. The reliance on an error budget does not negate the need for experimental

verification of the appropriateness of the error limit.

The instrument Included in the algorithm is, in

essence, an amplifier, with input signal being

proportional to the property of interest, and including

"noise" from perturbations which affect the process

^ In many cases acceptable limits relative to a particular usage are large with respect to measurement

process capabilities. In the interest of conserving

measurement effort, detailed corrections for S.E.'s are frequently ignored. When such is the case, the effect

of the ignored S.E.'s must be Included in the

uncertainty statement.

- 18 -
performance. As the "gain" is increased in order to

detect smaller and smaller changes in the property of

interest, the "noise" is also amplified. For the most

precise processes, the gain is adjusted so that the

noise is clearly observable. For these processes,

sequences of repeated measurements produce sequences of

non-identical numbers which reflect the process variability. In the case of many practical measurement

processes, the gain is adjusted so that noise is not

observable. For these processes, sequences of repeated

measurements always produce the same number, and, as a consequence, the error limit must be established by some other means. In most cases, one can purposely

introduce changes of known magnitude into the

measurement process to verify the minimum incremental

change which the process can detect.

- 19 -
. 0 Measurement as a Production Process

The concept of a measurement system requires that

values assigned to represent certain characteristics of

objects be reasonably unique and repeatable over time

and changes in location. It is expectd that sequences

of measurements of the same thing made at various times

and at different locations show evidence of convergence

to the same limiting mean. Uncertainty statements are,

in essence, predictors of the degree to which such

agreement can be attained.^ Failure to agree within

uncertainty limits is an indication that the two

processes are fundamentally different, or that the

uncertainty statement does not adequately describe the error bounds. For a practical measurement, the

measurement algorithm, or the mathematical model of the

measurement process, cannot possibly reflect all of the

sources of variability. The instrument or comparator

cannot differentiate between a real change and all of

the perturbations which change the indication in the same manner as a change in the object. None the less,

it is important to know the bounds for the variability

which occurs in the course of making measurements.

Redundancy, either by repeated measurements or incorporated in a particular measurement process,

provides a means for assessing this variability.

In order to illustrate the nature of a measurement

process, consider first the collection of simulated

^ The word "closure is used in the following sense:

One does not expect the results for the same measurement by each of several processes to be

identical. On the other hand, if the property being

measured is stable, one would expect that collections

of measurements by each process would support the same limiting mean and, for each process, one would expect

the uncertainty limits centered on the result to

encompass the limiting mean. While one may not know

the limiting mean value, under these conditions the

results from several processes can be compared by

centering the appropriate uncertainty limits on the

respective results. If the areas so defined close, or "overlap", the results are considered to be in

agreement within the capabilities of the processes

involved

.

- 20 -
measurement in figure 1. The data shown reflects the

effects of variability from both cyclic and step

sources over time. For the 300 measurements shown, the

data has the appearance of coming from a reasonably

well behaved measurement process. The measurements

tend to cluster around the central line—the process

average or limiting mean. Confidence that the process

has a single limiting mean is strengthened as the

length of the record is increased. With process

performance as shown, a predictive statement concerning

the next, but as yet un-made measurement, can be

considered

.

70

60

• • •

• •

•

• •

•

•

• • •

» • • • 50

UJ 40

• V *— —

-1<

> • • • • • • • • 30

• • • * • •

20

• • • «

•

10 1 100 200 300

RUN NUMBER

FIGURE 1

It seems clear that the predictive statement cannot be

exact but will have to allow for the scatter of the

results. The goal is a statement with respect to a new

measurement, a measurement that is independent of all

those that have gone before. Such a statement should

For the purpose of illustration, the data shown is

from a simulated measurement process. The "process"

includes a simulation of both cyclic and step variability. Four sinusoidal functions, with amplitudes al,

a2, a3 and a4, and with periods differing by a factor

of 10, simulate the cyclic variability. A random

choice between +b, 0 and -b simulates the step

variability. The value shown is the sum of four cyclic

functions, sampled at random times, and the step

function, with appropriate "scale shift" adjustment.

For the "improved" process, as shown in figure 3, all

the amplitudes of the cyclic terms and the step

function are equal. For the "unimproved" process,

shown in figure 1, the amplitude of one of the cyclic

functions has been doubled.

- 21 -
be based on a collection of independent determinations,

each one similar in character to the new observation,

that is to say, so that each observation can be

considered as random drawings from the same probability

distribution. These conditions will be satisfied if

the collection of points is independent, that is, free

of patterns, trends and so forth; and provided it is

from a sufficiently broad set of environmental and

operating conditions to allow all the random effects to

which the process is subject to have a chance to exert

their influence on the variability.

From a study of a sequence of such independent measurements, control chart techniques can be used to set up

limits within which the next value should lie. For an extremely long sequence, limits can be marked off on either side of the mean so that some suitable fraction,

say 99 percent, of the observations are within the

interval delined. The probability is also 99 percent

that the limiting mean will be within the interval

established by centering these same limits on any

observation chosen at random. This will be true of the

next observation as well, provided it is an independent

measurement from the same process. The probability

statement attaches to the sequence of such statements.

For each individual new observation, the statement is

either true or false but in the long run 99 percent of

such statements will be true.

Assuming that the limits are based on large numbers of

observations, for a process operating in a state of

control, very nearly the intended percentage of all

such limit bands, centered on the observed values,

would in' fact overlap the mean. This will not be true

for points in the area outside of the control limits.

This is expected in only 1 percent of the cases. More

frequent occurrence is a clear indication of either

loss of control or that the limits were not properly

set

.

If, over the sequence of the 300 measurements shown,

the variability of the collection reflects the maximum

excursions of each parameter, the s.d. of the

collection is the total s.d. of the process, a^,

computed in the usual manner:

- 22 -
°t'\ (n-l)

where - individual result

y - V

n = no. of y's in the collection

The 3orj. limits shown in figure 1 appear to be adequate

bounds for the "process" variability.^ One would

surely expect the next "single" measurement result to

be within these prescribed limits. Further, if the

next measurement was defined to be the average of n independent "single" measurements from this process,

one would expect this average to_ agree with the average

of the collection within (3o^//n). Without knowledge

of independent parameters which are known to be

proportional to the magnitude of each source of

variability, there is no way to further analyse this

data, and the random component of the uncertainty of

the result, however defined, would be a function of a.^,. If this performance of the process, according to this

particular algorithm, is adequate for the intended use.

^ Unpublished material furnished by Eisenhart,

indicates that 3 s.d. limits are appropriate for .99

probability for all distributions. For 02^5.6 the

probability is slightly less than 0.99; for 62<5.6, a bit more than 0.99. For distributions with 62<3, the 3

a.d. limits are somewhat pessimistic, however, the

simplicity of using 3 s.d. limits- far outweighs the

complexity of determining the appropriate distribution.

For the simulation shown in figure 1, 62=2.58, and for

figure 3, 62=^2.36. With sufficient measurement effort,

it may be possible to achieve an algorithm for a particular measurement process which will produce a collection of values with distribution approaching

62=^3. For a collection of differences between two

nichrome kilograms in excess of 300 and over

approximately a 10-year period, 62«3.02. In this case,

considerable effort was made to assure that the

algorithm accounted for the effects of all known

sources of variability. (32 is the usual kurtosis

parameter, having value 3.0 for a normal distribution.)

- 23 -
In this process simulation, there are identifiable

parameters which are proportional to the effects o^

sources contributing to the process variability.

Recording the parameter values along with each

measurement result permits the use of correlation

studies to further evaluate the process. In figure 2,

the parameter for each source of variability is plotted

PARAMETER A

PARAMETER B

30

UJ

-J 40 <

>

JO

PARAMETER C

70

50 .

40

MIJO -

10"- PARAMETER 0

In the simulated process, the parameters are the

individual values of the sinusoidal functions which, in

summation , define in part the resulting "measurement".

- 24 -
against the appropriate measurement result. For parameters B, C and D, there is little evidence of

correlation. While the variability of these parameters

contributes to the process variability, one cannot

differentiate between their respective contributions.

Clearly, there is a correlation with parameter A. This

correlation indicates that a "between time" variability

associated with parameter A is influencing the measurement results. The effect is systematic, that is, the

result is high when the parameter value is high, and

vice versa. It should be noted, however, that in spite

of the existence of the systematic effect, the initial

Sorj, limit is still an appropriate bound for the process

variability. On the other hand, having identified the

source of variability, "corrective" action can be taken

to reduce the magnitude of this particular systematic

effect, with a resulting decrease in the "process"

total s.d., as shown in figure 3. The results from the

new algorithm are now free from identifiable sources of

systematic variability.^

70r60

50

UJ

_j 40

<

>

30

20

10 100 200 300

RUN NUMBER

FIGURE 3

The random component of the uncertainty statement, 2a,^,

relates to the limiting mean of the process. It is not

only a quantitative statement with respect to a

^ The corrective action was to set all coefficients of

the cyclic terms to the same value. (This action is,

in essence, changing the algorithm to correct for* the

identified source of variability.) Under this

condition, the term by term correlation studies did not

show strong evidence of correlations. The plots were

similar to those from parameters B, C and D in figure

2.

- 25 -
"aingle" measurement and the process limiting means,

but also with respect to the expected agreement between

any two "single" measurements. If the process limiting

mean is within 3o^ of any "single" measurement, then

for any two measurements, the 3o^ limits, centered on the individual values, should overlap most of the time.

If a quantitative uncertainty statement is intended,

one must accept the process limiting mean as the "best

estimate" of the process output.^ It is seldom

practical, however, to establish a process limiting

mean tor each measurement which has to be made. One

must look for other techniques if one intends to make

an operationally verifiable uncertainty statement. Two

such techniques involve redundancy and a "check

standard" concept. When used in combination, these

techniques will provide the desired data.

The use of redundancy in measurement is not new. Most

important measurements are repeated several times, the

announced value being the average of the results

obtained. Some assessment of the process performance

is made by computing the standard deviation of the

collection of these repeated values. This standard

deviation, however, reflects the process performance

over a relatively short time, and as such is not

necessarily a valid estimate of a^. By defining the

announced value to be the average or n measurements,

for a process in a reasonable state of control, the

standard deviation of the n single measurements should

also be well behaved. These standard deviations for

each sequence of similar measurements can be combined

to obtain a long term accepted within process standard

deviation, a^. First estimates of may be based on only a few measurement sequences, however, in time the

collection will provide a stable value for a , which is • w

a characteristic of the particular process, or algorithm. For each sequence of measurements, the

computed within s.d. can be compared with the accepted

to verify that the process is performing as expected

.

The "check standard" concept provides a means to

accumulate, usually at little cost in terms of

A measurement effort, a collection of data which will

^ If the magnitude of the property being measured is

temporally stable, the estimate of the limiting mean is

the average of available collections of values. A

somewhat different approach is used when the magnitude

of the property is changing with time. This approach

will be discussed later.

- 26 -
establish the accepted total standard deviation, a^.

One form of a "check, standard" is the inclusion of an extra object in each sequence of measurements used to

establish the announced value. This object is chosen

to be similar in all respects to other objects which

must be measured. With a particular "check standard"

in each similar measurement, the collection of values

obtained will, in time, be similar to the collection

shown in figures 1 or 3. For a long sequence of

values, the standard deviation of the collection is a measure of Oj. The total s.d., o^, determined in this

manner is appropriate for use in expressing the random

variability of the process. One could, for example,

interchange the "check standard" with one of the

"unknowns" and, after the same number of repeated

measurements on the "unknown", the characteristics of

the collection of values obtained would be the same as those which were from the original collection of values

for the "check standard^."

In some cases, the collection of repeated measurements

will indicate that the magnitude of the quantity of

interest is not constant, but slowly increasing or decreasing. In this situation, the limiting mean or average value of the collection is not the best

estimate of current, or future, values. One must

predict an appropriate value for a particular time,

together with the uncertainty of the predicted value.

The prediction must be valid over some reasonable time

increment in order to be useful. This must be done

with care. A typical error in judgment is declaring

the magnitude of the property to be changing without

knowledge of either a or a^. Usually significant

rates of change are readily apparent in the collection

1 It should be noted that the Initial assignment to the

"check standard" is of little importance. The value

assignment which is made in accordance with the

procedures of the algorithm, can be easily changed,^ one way or another, provided that the "check standard" is

stable. The variability of the collection of values,

however, is of major importance. The long term

variability of the result is a characteristic of the

particular process. Confidence in the total s.d., Oj,

increases as the collection of values for the "check

standard" increases. The accepted value for Oq> can be

used to assess the process performance for each

sequence of measurements relative to the value obtained

for the "check standard" in that measurement.

- 27 -
of historical data As the data base increases,

confidence in the ability to predict both the value and

the rate of change will increase.

To illustrate, in figure 4(a), a prediction line as a function of time has been fitted to three typical

measured values. The predicted value is that expected

one time increment beyond the existing data base. With

a reasonable estimate of the process standard

deviation, the uncertainty of the predicted value can

be computed from the formula [23]:

t " time/date of the prediction

t average time/date (location of the

centroid of time span covered by the

measurement, that is t - Ht^/n

t^« time/date associated with each of the

n values

a " process standard deviation (s.d. about

the fitted line)

0 =» s.d of predicted value at time t.

The uncertainty of the predicted value is large because

the extrapolation interval is large relative to the

time span of the data base.

With three additional data points, as shown in figure

4(b), a new prediction line is established. The

uncertainty of the new predicted value is somewhat

smaller since the extrapolation interval is a small

fraction of the new data base. The uncertainty of the

previous predicted value overlaps the. prediction line

as expected. Again, with three more data points, a new

In addition to the obvious evidence which may be apparent in a control chart, there are several other ways

to verify the existence of time dependent changes. If

the s.d. about a "fitted" line is clearly smaller than

the s.d. about the average value, there is a time

dependent change occurring. Normally, in the process

of fitting a set of data to a linear fimction of time,

X, by the equation y - ax + b, the s.d. of the rate of

change coefficient a, can be determined along with the

value of a. The significance of the rate of change, a, can be determined relative to the s.d. of a.

where n the number of points in the collection
prediction line is shown in figure 4(c). Again, for

both of the previous predicted values, the uncertainty

overlaps the new prediction line. As the historical

data base increased, the uncertainty of the prediction

over a relatively small time interval approaches the

"uncertainty of the mean", 3o-(1/k^). At this point,

the rate of change should be well known.
6.0 The Unit

With the process parameters, and o^, known, to

complete the measurement, that is, to assign a number

which relates a property of an "unlcnown" to some defined "unit", a representation of the "unit" must be

included in the measurement process^. The process

compares the "unknown" with this accessible "unit".

Error bounds for agreement, or "closure", within a system of similar measurements must account for the

"unit" error. Difference measurements are determined

by the response of the process to the real magnitudes

of the properties involved in the "unit" and the

"unknown" while number assignment "assumes" the value

assigned to the "unit" Is correct.

Unit errors, unlike process performance parameters,

cannot be operationally identified with a single

measurement process, or with several measurement

processes which use the same accessible "unit" and the same "equality rule", or measurement algorithm.

Accessible units are a part of all practical

measurement processes, few of which have access to the same realization of the unit. For a process operating

in a state of control, the uncertainty is the sum of

the limits for random error and the unit error, the

unit error being a systematic error, or bias, beyond

the control of the local process^.

For each measurement process, the value assigned to the

accessible unit is the product of one or more prior

measurements which, in a geneological sense, extend

back to some accepted definition of one or more units.

While the accepted units are usually the units of the

SI system or units which have exact relations to the SI

^ A general interpretation of "unit" includes artifact

replicas or other realizations of the units of the SI

system, replicas of production objects, standard

reference materials, and the like [9, 10].

^ The condition for operating in a state of control

were discussed earlier. A process is said to be

operating in a state of control when, for each defined

sequence of measurements, the computed s.d. is in

agreement with the long term accepted c^, and the value

of the "check standard" obtained is in agreement with

the long term accepted value within the limits

established by a .

1 - 30 -
units, it must be emphasized that an SI unit definition

is only the specification of the phenomenon, or artifact, which represents the unit [11]. There is no attempt to specify the manner in which the units are to

be made accessible to the many measurement processes

which must use them. The phenomena have been selected

on the basis of temporal stability, and are subject to

change as appropriate and agreed upon by the General

Conference of Weights and Measures.

With the exception of mass, the only remaining artifact

standard, the recommended realizations of the units are quite complex, and, once realized, are not in a form

easily adaptable to practical measurement. For

simplicity, mass will be used to illustrate unit error,

but it must be remembered that with other units,

several basically different processes, or algorithms,

must be studied in the same manner. In the case of

length, for example, one must go from the defining

vacuum wavelength to the wavelength of the available

light source in the environment in which the measurement is to be made, to the length of the objects such

as gage blocks, to the length of scales such as meter

bars, to dynamic "fringe counting" interferometers, and

then to a wide variety of practical measuring equipment

including both instruments and reference shapes.

All mass measurement algorithms, from those used with

the defining unit to the most crude measurements, are based on the same principle, differing only in the

degree of refinement. This is both convenient and

practical; convenient because of the inherent

simplicity of the process, and practical because of the

availability of a wide variety of weighing equipment.

The algorithm is based on relating the gravitational

force acting on the "unknown" to the gravitational

force acting on some representation of a suitable unit.

Algorithms based on other principles are possible,

however, suitable equipment has not been developed

sufficiently to compete with the accepted algorithm^

.

Starting with a comparative measurement process, the

realization of a particular algorithm-model concept,

and the unit as embodied in the defining artifact, unit

error can be illustrated by the manner in which values

are assigned to replicas of the defining artifact.

Since the result is defined by the particular

algorithm, a considerable degree of confidence is

gained when measurement of the same thing by two

basically different algorithms are in agreement. Such

a case in length measurements will be discussed later.

- 31 -
Assuming that the replicas are similar in all respects

to the artifact, and that the mass differences between

the replicas and the artifact are within the "on scale"

capacity of the instrument for a measurement process

operating in a state of control, the results, y^^, of a large number, n, of difference measurements between the

artifact and any one replica might be distributed as shown in figure 5(a).^ Studies of the collection, y^,

such as correlation studies mentioned before, may

result in algorithm improvements so that for the next

series of measurements, the results, y^ , might be

distributed as shown by (b) . Now, being satisfied with

the process performance, one can define the announced

result to be the average of n "single" measurements.

The distribution of the results, defined in this manner

might be as shown by (c). Note, however, that the

measurement effort to achieve this distribution

requires m x n "single" measurements. In all cases, if

m is sufficiently large, the limiting mean will not be

changed^

.

The limiting mean, or "0" in figure 5, compares to some number relating to the difference in mass between

the defining artifact and the replica. For a process

operating in a state of control, the "best estimate" of

the limiting mean is an average of the available

measurement results:

n

- 111.

The uncertainty of thd limiting mean is:

Unc(y) « 3a^(l//n)

1 Normal distributions are shown here for convenience.

The only requirement on the distribution is that it be

reasonably symmetric.

^ For a given set of conditions, the limiting mean

reflects the "average" of all of the cyclic

perturbation. Gross changes from "average" for

perturbation of major influence on the process, if

uncorrected in the algorithm, will obviously shift the

limiting mean.

- 32 -
The expected shift in the computed average by virtue of

adding one more measured value to the collection is on

the order of ±3o^/(n+l).

-« -5 -4

Distribution curves for three distributions,

a, b and c, with the same mean ^nd with

distributions a=2.5 for a, <t =2 . 5/ Vl2= • 72 for

b, and <y=.72/Vl2-.21 for c.

FKiURE 5 The value reassigned to certain defining artifacts,

such as the international prototype kilogram, is

"exact" by definition. For replicas of such artifacts,

the assigned value is an estimate of some process

limiting mean and, as such, can never be "exact." The

announced value for tlie "unknown" replica is the sum of

two numbers, the number assigned to the defining

artifact and the number assigned to the measured

difference. One could use the result from a "single"

measurement by the crude process, as shown by

distribution (a), or from the improved process, as shown by (b) , or an average of n "single" measurements,

as shown by (c) . Most of the time, each value, from

whatever process, will not lie farther away from the

limiting mean than 3o^, or 3o^//n", as appropriate where

Crr. is the total s.d. of a single measurement. Whether

the value used is too big or too small cannot be

determined, but, given sufficient time, it can be

demonstrated that the limits are appropriate. A

collection of replicas, "calibrated" in the above

manner, together v^ith the announced values and

uncertainties serves to extend the unit to other,

facilities

.

Figure 5 can also be used to illustrate the results of

measurements of the difference between a particular

- 33 -
"calibrated" artifact and unknown by three different

processes. Because the process responds to the actual

magnitude of the properties embodied in the artifact

and the unknown, the results from all three processes

should tend to group about the same limiting mean. For

all three processes, however, the number assignment for

the unknown will be in error because the accessible

unit, or the number assignment to the artifact, is not

quite correct. The results from all these processes

would be biased, or offset, in one direction or the

other, by an amount which would rarely exceed the

uncertainty of the value assigned to the artifact, the

"unit error." The importance of the magnitude of the

"unit error" depends upon one's ability to detect that

it exists.

A measurement system consists of many different

measurement processes, each with some suitable access

to a unit. Some of these processes have

characteristics like distribution (c), others have

characteristics like distributions (a) and (b) . The

measurements from well characterized processes,

regardless of the magnitude of the total s.d., would be

considered in agreement if the results of measurements

on a given object tend to support a single limiting

mean. That is, the uncertainty limits associated with

each reported value should tend to overlap a common

limiting mean. If the local "unit errors" are large

and not accounted for in the uncertainty of the stated

result, for processes with the characteristic of

distribution c, this condition would almost never

occur. Each stated result would be biased by the

magnitude of the local "unit error." The results from

some facilities would always be high, and from others,

always low with respect to the average of all of the

results from similar facilities.

For realistic prediction limits for the agreement

within the system, the local "unit error", that is the

uncertainty of the value assigned to the "calibrated"

replica, must be considered as a systematic error and

added to the estimate of the local random error, 3a.j..

For processes with characteristics similar to

distribution a, the local "unit error" is still a bias

which should be included in the uncertainty statement.

In this case, however, if the magnitude of the local

"unit error" is small relative to the random

variability of the process, few could afford to make

the number of measurements necessary to verify its

existence.

- 34 -
The accessible unit can take many forms. It may be the sum of the values assigned to two objects, or the

average of the values assigned to two objects, or the

average value for a group of objects. Such an embodiment of the unit permits monitoring the relative

stability of the items which in summation establish the

unit. For most measurements, the accessible unit is a part of some instrument or measuring device, which in

turn is some convenient realization of a unitalgorithm-model concept. Because of the simplicity of

the manipulative procedures, and the ability to read,

encode, or print out the desired quantity directly from

the instrument, the importance of the unit-algorithmmodel concept is frequently overlooked. It must be

emphasized that the results from any process, be it a comparative process or a direct reading process, are a realization of particular unit-model-algorithm

concepts. The usefulness of the result depends upon

how these concepts relate to particular requirements.

To illustrate, consider the substitution balance and

the large multiple lever scale, both of which can be

used in a comparative mode, but are normally used as direct reading instruments. With the substitution

balance, the direct reading measurement is a "comparison" between the object in question and the

"calibrated" built-in weights of the instrument which

are manipulated by the operator. For the multiple

lever scale, the direct reading measurement is a "comparison" between the object in question and the

position of a poise along a graduated beam, the mass of

the poise and the lever ratio of the instrviment being

in essence a multiple of the mass unit. In the first

case, the instrument assumes the role of the "unit" over the incremental difference between the built-in

weights. In the second case, the instrument assumes

the role of the "unit" over the capacity of the

instriiment. In both cases the task of establishing the

"unit error" is a substantial one.

The instrument manufacturer normally assumes the

responsibility for assuring that the initial instrument

"unit error" is within some specified limit. Usually

this can be accomplished by adjusting the instrument

indications relative to the values of "known" weights

made from materials which are reasonably similar to the

materials of the built-in weights or the poise and

beam. The interpretation of the instrument indication

with respect to the object at hand, however, is the

responsibility of the user. Failure to account for all

of the factors in the algorithm may result in

unexplainable discrepancies.

- 35 -
The unit error concept is applicable to all measurements which must be Judged in accordance with

established bounds for the limit of error. For direct

reading instruments, it is prudent to have one or more

"known" reference standards over the range of the

instrument and in the neighborhood of the expected

measurement result. In most cases, it is within the

ability of the user to construct such standards, and in some cases, such action is necessary. For a wide

variety of common measurements, suitable reference

standards are available in the fonn of well

characterized objects, instruments, and materials in

which a considerable amount of effort has been soent to

formulate the appropriate algorithms, and to reduce the

magnitude of the unit error. The use of these items,

in accordance with the appropriate algorithm and with

measurement processes operating in a state of control,

result in both an economic saving of measurement effort

and an increase in the consistency of results. In many

cases, the time and effort spent in trying to resolve

inconsistent results is far greater than the time

involved in the actual measurements

.

- 36 -
7 . 0 The Practical Measurement Process

7. 1 Verifying the Algorithm

The importance of the algorithm-model concept and the

realization cannot be over emphasized. Model concepts

and algorithms must be adjusted until the realization

will provide consistent results which are adequate for

the intended usage, for all materials which must be

measured and for all environments in which the

measurements must be made. This is not to say that all

measurements should be made in accordance with the most

complex algorithms, but rather that all of the factors

in the algorithms have to be considered relative to

each set of measurement requirements. When this is

done, simplified instrumentation and procedures, if

adequate, can be used with full confidence. When

inappropriate algorithms are used, one is plagued with

both real and imagined new sources of variability.

It is not enough to demonstrate consistency within one

facility. The most severe test on the measurement

system is to maintain consistency between facilities

and environmental changes. For such tests on the

system to have real meaning, each participating

facility must first establish measurement processes

which operate in a state of control. For example,

figure 6 shows sequences of values for each of three

"check standards" used in the NBS process for assigning

21 —

20=^

lOlNCH STEEL

28i

27

26

2S

10 INCH CERVIT

0 —

.150 INCH CHROME CARBIDE Eorly Interferomefric Process Confrol Block Volues

(Y, IN MICROINCMES PLOTTED IN SEQUENCE)

FIOURC 6

- 37 -
values to long gage blocks. These particular "check

standards", or "control blocks", were chosen so that

the difference between the .15 block and the 10 in

blocks would emphasize length dependent variability,

and the difference between the two 10 in blocks would

emphasize temperature dependent variability. Clearly,

in the beginning the variability of the results for the

10 in blocks was larger than the variability of the

results for the .15 block. This suggested problems

with the part of the algorithm which converts the

vacuum wavelength to the effective wavelength.

A study of the values for the 10 in cervit block, using

the techniques mentioned earlier, shows a correlation,

figure 7, between the values and the vapor pressure

in the environment at the time of the measurement. The

corrective action, which resulted in a significant

decrease in the variability of the values for both

blocks, consisted of relocating the sensor to a position in closure proximity to the measuring position

of the blocks. Further Improvements have resulted in

achieving approximately the same variability for each

of the "control blocks."

28

27

26

29

24

5 6 7 VAPOR PRESSURE

FIGURE 7

- 38 -
Figure 8 illustrates the results of measurements on

the difference in length between two 16 in gage blocks.

The initial value for the difference was established at

NBS over the time interval Sept. 9 to Dec. 14, 1970.

In a cooperating laboratory, the differences in March

1971 were considerably offset from the initial value.

This offset was verified by repeating the measurements

aC the NBS in April. This discrepancy was not

expected, and after considerable thought, it was attributed to the difference in the finish of the nongaging surfaces of the block and the illumination level

in the two facilities. The lab at NBS is almost dark,

and the participating lab is a well lighted general

purpose facility. The action which appeared to correct

the problem consisted of wrapping the blocks in several

layers of gold-coated mylar film, so as to achieve more

nearly uniform thermal characteristics. While the

measurements of 1972 did not necessarily confirm the

action, at least the offset was no longer apparent.

CONDITION

I H I IL I H

20

18

•

16

6 _ o

o o o o

* •

•

•

• o 14

12

o — o

<? .

O o —

• o

0

•

o

10.

0

0 8

O Co

6

SEPT

19/0

NOV. OEC.

1971 APRIL AUGUST

1972 SEPT. (5-18) SEPT.

(20-22)

16 fn. BLOCKS, (2902-HI55), PROCESS R

CONDITION I

- LOW LIGHT LEVEL

CONDITION n- HIGH LIGHT LEVEL

o UNLIKE FINISH

• "LIKE" FINISH (WRAPPED)

FIGURE 8

- 39 -
Youden suggests a technique for verifying the existence

of systematic errors between facilities which are interested in treasuring a particular quantity [12] . Two

objects, or samples, similar in nature, are sent in

turn to each of the participants. Independent

measurements are made on each of the two samples, and

the two results from each facility are plotted against

each other. To quote, "Two median lines divide the

graph paper into four quadrants. In the ideal situation where only random errors of precision operate, the

points are expected to be equally niunerous in all

quadrants. This follows because plus and minus error

should be equally likely In any existing test

procedure that has come to my attention the points tend

to concentrate in the upper right and lower left

quadrants. This means that laboratories tend to get

high results on both materials, or low results on both

materials. Here is evidence of individual laboratory

biases."

Such a test, called TAPE 1, was made among the

participants in the Mass Measurement Assurance Program

[13].^ Two pairs of stainless steel kilograms,

designated XI, X2 and Yl, Y2, were circulated in such a way that a time interval from three to six months

occurred between pairs for each facility. The results,

including the results from NBS measurements with

reference to the "defining artifact" kilograms, Nl and

N2, are shown in figure 9. For each facility, the same measurement algorithm was used. The "unit error" of

the local unit was on the order of 0.1 mg, and the

process performance parameters of all facilities,

and Orj,, were reasonably well known. The results from

the first series of measurements were used, except

repeats were required if an "out of control" situation

existed. All of the results were in agreement on the

basis of overlapping uncertainty limits.

Time And Place Evaluation

- 40 -
YOUDEN PLOT OF TAPE-I RESULTS

9 —.1 mg-—

.1 mg —i o PACKAGE X

y 9 PACKAGE Y • NBS

0

X2,Y2

FIG. 9

For each measurement process, it would be expected that

a sequence of such paired values would fall within a circle, the diameter of which would be a function of

the process total standard deviation. For the NBS

values, with the exception of one outlier, this appears

to be the case. For other processes of comparable

precision, one would expect the values to group about a line of 45° slope in a manner commensurate with the

"unit error" of the local unit. Again, this is the

case. The conclusion is that, in addition to an overall system agreement better than a part in 10^, the

measurement algorithm and its realization is reasonably

correct for the items which were measured.

- 41 -
The more precise facilities participated in an extension of this, called TAPE 2, which involved

tantalum, aluminum and stainless steel kilograms [14].

The results were as shown in figure 10. The ellipse

shown is the "expected agreement" based on the previous

test. There were difficulties from the beginning in

achieving "in control" process performance. While the

results for the stainless steel kilogram were as expected, there is clearly a correlation between the

results for the tantalum and the aluminum kilograms in

which the first order dissimilarity is in displacement

volume, and also, perhaps, surface characteristics.

This suggests both local process problems in measuring

the parameters necessary to compute the air density and

algorithm problems associated with the actual

computation of the air density, and the accounting for

surface effects, if need be. These studies are continuing, primarily, in order to construct an appropriate algorithm for such a situation. Until this

is done, there is no real assurance in the ability to

assign consistent mass values to objects made from

materials other than that rrormally used to make

weights.

YOUDEN PLOT OF TAPE 2 RESULTS

.6 > 1 1 1 1 II T 1 T -I 1 1 I

.4 9

.2 mg

.2 f TA

9 - - > 09' <b O

0

-.2 —

J.2mg |-

-.4 1 1 ! 1

' ' '

' ! 1 ! *! 1 1 < -.5 -.4 -.2 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 AL

® AT LASS • VALUES AT N8S

FICUR£ 10

One form of an accessible length unit is the gage block

and its assigned length. Two different processes are involved in determining the announced value. One

process, the interf erometric process, determines the

lengths of selected blocks, the algorithm being based

- 42 -
on a model which "declares" the length of the block to

be the separation between a defined point on the top

gaging surface of the block and the surface of a platen

"wrung" to the bottom surface of the block, the platen

being made from the same material as the block. The

procedures are tedious and time consioming. The total

s.d. of the process is relatively large. Few

facilities have the capabilities for such measurements,

since many measurements are required to reduce the

"unit error" imposed on the process in which the

"calibrated" blocks will be used.

The second process, the mechanical comparison process,

can only determine the difference between the reference

blocks and the "unknown" blocks. The algorithm is

based on essentially the same model, except the block

is not "wrung" to the reference plane. The procedures

for this process are simple and comparisons can be done

rapidly. The total s.d. is somewhat less than that of

the interferometric process. There are many such

processes in every day use. The announced value can be

computed from the value assigned to the reference block

and the "unknown." The uncertainty of the announced

value,' aj: NBS, is the sum of the "unit error", or the

uncertainty from the interferometric process, and the

3aJ limits for the transfer process. This announced

uncertainty is the "unit error" for the process in

which the calibrated blocks will be used.

Confidence in the two-process system is considerably

enhanced by demonstrating that both processes are in

agreement, within their respective uncertainties. For

example, the difference in length as computed from the

values assigned to two blocks by the interferometric

process can be measured directly in the comparison

process. For similar block materials, the agreement is as shown in table 1. When different materials are

involved, both process algorithm-model concepts must be

reviewed carefully.

- 43 -
Male 1' Ocnpariacn ((.)-(..)), Proceas I and Prrweaa II, January 1974

Mcndnal Size

Process

(.)-(..)

IUNC

Process 11 {For 1- (.)-(..) Ttotal S.D. D.F.

1-74) 2S.D. Mean (Delta)

5 27.95 .95 28.460 .489 90 .153 -1.29

6 32.70 1.14 33.476 .525 97 .513 - .78

7 23.53 1.37 23.562 .543 106 .546 - .03

a 43.94 1.54 44.212 .753 309 .129 - .27

10 61.95 1.97 61.373 .729 88 .231 .58

12 56.65 2,36 56.908 1.104 89 .348 - .26 16 55.64 3.26 57.282 1.072 71 .378 -1.24

20 -15.33 4.20 -16.484 1.054 94 .324 . 1.15

In the above table, the designaticn (.) ani (..) is used

to differentiate tetueen C-o specific blcxJts of the sare noninal leni^th. 'I^e Process I ((.)-(..)) has been cxnputed

fron interfercTKtric neasurenents . Ths Process II ((.)-

( . . ) ) has been measured dir-.-ctly by a rBcnanicai caiparisai

process. is tii-ie unotirtaintv of the interferotetric difference. iTie decrees of frnedan, D.F., is the number of ind^Jendent raeasuref.ients in the oollectim used to determij^

the total S.D. (Delta) illustrates the closure between the results fron the tv« different processes, (Process I (( )-

{ . . ) ) - Process II ((.)-(.. j ))

.

In the interferometric process, the "differential"

phase shift at the reflecting surfaces of the block and

the platen have been "defined" out by requiring both

the block and the platen to be of the same material.

In like manner, in the transfer process, "differential"

penetration of the probes into the surfaces is

"defined" out for like materials. Where different

materials are involved, such as quartz, cervit, and the

various carbides, the comparison process algorithm must

consider "differential" penetration. Normally one would compute the "differential" penetration

correction, to be included in the algorithm, using a form of the Hertz equations and the necessary parameters [15]. In this case, the result can be checked

with measurements using the interferometric process.

The difference between the values for cervit blocks as determined interferometrically on a quartz platen, and

as determined relative to alloy steel reference blocks

was approximately 2 microinches, an amount in excess of

- 4A -
the normal process variability . Until this

discrepancy is resolved, the uncertainty of the values

assigned by NBS to blocks from materials other than

alloy steel includes an allowance to account for the

doubt associated with differential penetration.

Fortxinately , a complete algorithm for one particular

process is usually appropriate for most similar

processes, provided that the algorithm accounts for the

range of condition, and materials, which must be made.

The same is true for processes operating in a state of

control. If it can be demonstrated that, for a particular algorithm, a particular process will operate

in a state of control over the expected environmental

changes, and with the various objects which have to be

measured, one is reasonably sure that all similar

processes which use the same algorithm can also be made

to operate in a state of control. For each process,

however, it is necessary to determine the appropriate

process performance parameters.

This is to say only that under conditions of gage

block comparison, it may not be possible to determine

the appropriate parameters for the Hertz equations. In

many cases there is no competing process which can be

used to verify the results of the algorithm

independently. In the case of the measurement of

"thread wires", the Hertz equations are used to

determine an unstressed diameter from measurement data

taken from a stressed condition. In use, the same relations are used to determine a stressed diameter

from an unstressed condition. If the stressed

condition in measurement and in use are identical, or nearly so, minor differences between the Hertz

corrections and the actual differences may be of no importance. The same is true for other corrections

which may be included in the algorithm, such as thermal

coefficients of expansion. However, if the objects are

to be used under conditions which differ significantly

from the conditions under which the values have been

assigned, it may be necessary to verify the

appropriateness of the algorithms used.

- 45 -
7.2 Performance Parameters

In the beginning, the value for and/or is

unknown, therefore the question of how much effort is

required to establish a reliable value for the process

to establish a reliable value for the process standard

deviation must be considered. One cannot normally

afford to make a sequence of repeated independent

measurements sufficiently large to determine the long

term process standard deviation. As an alternate

procedure, one relies on estimates of the standard

deviation, s, computed from collections of n repeated

measurements. When intercomparison designs are used,

estimates of are obtained from each sequence of

measurements. With single measurements, as is the

normal situation for production measurements,

initially, and at reasonable intervals, sequences of n measurements must be repeated to establish s, and

subsequently the long term process standard deviation,

O'j.. The variation in independent values s^, S2 . • . of

the estimated standard deviation is surprisingly large

if the 8^ are based on small numbers of degrees of

freedom. (The degrees of freedom are (n-1) when only

an average is involved).

Although the distribution of s is not symmetrical about

the standard deviation, sigma, for small degrees of

freedom (e.g. less than 10), the standard deviation of

the distribution of s, expressed as a percentage of

sigma, is a useful guide in determining the number of

observations required to determine a reasonable value

for s. Table 2 gives the percentage standard

deviation associated with s based on different degrees

of freedom [16].

- 46 -
Table 2

Standard Deviation of

« aa a Percentage of D«gr««« of Freedom 2

1 60.3

2 46.3

3 38.9

4 34.1

3 30.8

6 28.2

7 26.2

ao 24.5

9 23.9

10 22.1

12 20.2

15 18.1

20 15.7

25 14.0

SO 10.0

It is suggested that a sequence of 8 to 10 repeated

Independent measurements should be sufficient to

provide an initial estimate of s for the production

process. For a process operating in a state of

control, using either comparison designs, or repeated

sequences of single measurements, the estimates s^^^ can be "pooled", or combined, to determine the total

process s.d. The total process s.d. should be a reasonably stable value for cumulative degrees of

freedom in excess of 50.

The word "precision" is usually associated with the

magnitude of the total process variability. For

example, distribution a in figure 5, would be called a "more precise" process than either distribution b or d.

However, the total process s.d. for a single

measurement is a process characteristic, whereas the

variability exhibited in a collection of data and the

associated standard deviation are functions of the

process definition. For example, a collection of data

involving the same object and the same instrumentation

- 47 -
treated in several different ways is shovm in figure 11

[17]. Figure 11(a) illustrates the results from a group of measurements in which a "single" measurement"

is defined as the difference between the standard, S,

and the unknown, X. In figure 11(b) the same data has

been used but the "single measurement" has been defined

as the average of two measurements of the difference

between S and X. In figure 11(c) the result is the

value obtained from an intercomparison design.

It is immediately apparent that the variability and

thus the standard deviation associated with each

treatment is markedly different even though the total

measurement effort is the same. The uncertainty of the

value established for X relative to S relates the

correctness of the long term average of the respective

total collection of data while the precision of each

treatment refers just to the variability of the

collection of data about their central values. The

uncertainty of the average in each case is the same.

(el SINGLE SUBSTITUTION

1 -O,

I -0,

»-«'

\ Oj- 0, /

( ~ so POINTS)

* I0 -1

09 -

0' 4-. - ^ . • •

•

09 H

- 10 —

(bl 0OUBL£

SUBSTITUTION

J —»0,

I —0,

X + A —• 0, l+A—» 0,

tri (0, -Oj +0,-0j)

» - I • (0,-0,12

( ~ 30 POINTS)

(Cl DOUBLE SUBSTITUTION a

WEIGHING OCSIGN

t I, »2

+ ajd«

a* +

-09 —

0

( ~ 15 POINTS )

PToc«a( pracliton aa t function at d«ftn*4 aaaaaraMnt aachod*.

- 48 -
The above is not intended to imply that the precision

of a process is not important but rather that the

precision must be considered carefully in terms of the

process requirements. For a given algorithm, the more

precise the process, the less the measurement required

to produce satisfactory results. On the other hand, as the process precision is increased, the variability of

the data reflects more and more disturbances from

various sources. Sometimes such variability can be

handled by "rounding" but for complete explanation,

algorithm modification is necessary. For the research

process, algorithm modification is a necessity. For

the production process, accounting for variability

which is not significant with respect to the

requirements is a wasted effort.

Aside from initial estimates, the long term total

process standard deviation is the variability

associated with repeated measurements on a "check

standard." The "check standard" can be introduced into

a measurement process in a variety of ways. For a comparative measurement, the "check standard" may be

the difference between two objects used to introduce

the unit, such as a pair of mass standards or a pair of

gage blocks. It may be a selected object, similar to

other "unknowns" which is measured at frequent

intervals until such time that a reasonably stable

estimate of the total s.d. is obtained, and then,

perhaps at less frequent intervals to verify the

validity of the accepted total s.d. For a production

process, it may be a dedicated artifact, similar in

nature to the product, which is measured on a ragular

routine schedule. Properly chosen, the variability of

the collection of values is a real measure of the total

process variability. The collection of values becomes

supporting evidence for the total s.d., and, if

necessary, a measurement of the "check standard" at any

time should demonstrate that the prediction limits are valid.

7.3 Unit Error

The magnitude of the "unit error" relative to the total

s.d. is important to the proper interpretation of the

measurement result relative to the requirement. For

comparative measurements, where the unit is introduced

into the process by means of "calibrated" reference

artifacts, it may be possible to achieve a situation in

- 49 -
which the "unit error" is some small fraction of the

total s.d. In this case, the random variability

between the results from several measurement processes

can sometimes be reduced by having all processes

introduce the unit by means of the same "calibrated"

reference artifact, or perhaps a suitable well

characterized reference material.

In both of these cases, it is necessary that all of the

processes use essentially the same algorithm as was

used to establish the value assigned to the

"calibrated" artifact. Failure to do so may lead to

illusory results.

In the case of direct reading equipment, the unit is

introduced by the instrument, and as a consequence, the

determination of the magnitude of the unit error is a sizeable task. One class of instrument, such as a substitution balance, relies on a linear relationship

to subdivide the interval between discrete settings.

Other instruments, such as the gage block comparator

used a linear relationship to subdivide a fixed

interval which, in turn, can only be used to determine

differences. Some elastic devices rely on a relationship which may or may not be linear over the

whole range of the instrument [18] . In these cases a linear indicating scale is often imposed on a nonlinear

response in such a way as to distribute the

nonlinearity over the range of the instrument.

Instrument "calibration" requires careful attention to

the nature of the instrument, the manner in which it is

used, and how the results are to be interpreted.

Assigning a value to a discrete instrument setting with

reference to a known "standard" is exactly similar to

the task of assigning a value to any other unknown with

reference to that "standard." The uncertainty of the

value assigned to the setting is the sum of the "unit

error" of the reference standard and of the instrument

itself. This summation becomes the "unit error" of the

instrument. If the instrument is sufficiently stable,

the magnitude of the "unit error" embodied in the

instrument can be reduced by the use of reference

standards in which the "unit error" or uncertainty is

small with respect to the instrument s.d., and by using

the average of a niimber of independent "calibration"

rims

.

The proportional part of the instrument indication can

be evaluated with two known "objects" which differ in

magnitude by an amount somewhat less than the smallest

incremental instrument setting. For example, in mass a small known "sensitivity" weight. A, is chosen in such

a manner that (S+A)>X>S. From the three observations:

- 50 -
(S-l-A) - 0^

one can foim the relation:

X = s + KCO^ - 03) - s + (r-f^)(o2 - O3)

where K " ( r— j

, the mass value per division of

^^1 ~ 3/ instrument reading scale.

Traditionally, for independence, the value of K as determined in a particular comparison has been used in

the computations. If, however, K is a stable property

of a particular instrument, the use of an average K

will result in a smaller s.d.^

With certain instruments, operator or servo, adjustment

of the instrument configuration to obtain a defined

"null" position is necessary before the indication can

be recorded. For the operator, this can be a tedious

and time consuming operation, particularly if the time

constant of the instrument is long. Either, or combinations of, reducing the instrument sensitivity,

and "rounding" the indication are used to alleviate

this condition in many practical measurement processes.

"Rounding" occurs when the operator is instructed to

"read to the closest marked interval", or to adjust to

within some defined limits around the desired "null"

positions, or by purposely dropping digits in the

instrument indicating system. The sensitivity may be

reduced to the point that all variability is masked.

For both of these cases, the "unit error" of the

instrument must be considered carefully.

If, with the instrument operating properly and with no intentional "rounding", a sequence of measurements on a "known", similar in all respects to other "unknowns"

which are to be measured, produces a sequence of

identical results, the instrument is not sufficiently

sensitive to detect normal process variability. In

this case, the random component of the uncertainty is

The same is true for the practice of resetting the zero of an instrument. If the instrument "drifts"

because of some environmental problem, the effects of

drift can be reduced somewhat by resetting zero immediately before each measurement. On the other

hand, if the change in zero is random in nature, the

practice of continually readjusting zero will result in

a larger s.d.

- 51 -
zero. That is, the instrument unit error is the

uncertainty of the measurement. The instrument unit

can be established by adjusting the reading scale so that the indication agrees "exactly" with the value

assigned to the "known." By adding, or removing small

increments from the "known" one can establish the

magnitude of change necessary to cause a change in the

instrument indication. This change, together with the

uncertainty of the value assigned to the "known", is

the instrximent "unit error." The second element of the

uncertainty of the result relates to the manner in

which the instrument is used. If the operator "rounds"

to the closest graduation, the value of the graduation

interval must also be included in the uncertainty

statement

.

It is not always possible to adjust the instrument

reading scale as suggested above. In this case, the

residual difference between the instrument reading and

the value assigned to the "known" is a systematic

error, or bias. Such a bias may exist at each

incremental setting of the instrument. The bias, or reading scale offset, for a particular setting applies

to all measurement data over the on-scale range of the

instrument for that particular setting. With suitable

tests, both the magnitude and the direction of the

offset can be determined and incorporated in the

measurement algorithm.

"Rounding" by dropping digits also introduces a bias, or offset. In this case, for all X within the interval

I^ ^ X < I^, the recorded value for X is 1^, thus the

recorded value for X may be low by as much as the value

of the increment | I^ - I2 | . The consequences of this

type of rounding will be discussed later. In some

cases, it may be possible to establish an "unrounded"

value for X. If the instrument indication is I^ for X, one can add a small summation, , to obtain an indication I2. Continuing, add a second summation, A^,

to obtain an indication I^. Assuming the instrument

response to be linear over the range I, to I3, the

"unrounded" estimate of X Is then:

\ - A

1 2 h

In this case A2 is the amount necessary to change the

indication by one unit, I^ to l^^. For indication I^ , with I^<X<l2, Aj is the increment which must be added

to obtain I2 . There are numerous variations of this

procedure.

- 52 -
As it becomes necessary to announce smaller

uncertainties for the measurement results, more

sensitive, or precise measurement, processes must be

used in order to detect, with some assurance, small

changes in the property of interest. In these cases

the increment of the normally used rounding practices

may be smaller than the observable process variability.

Evidence that such is the case can again be obtained

from sequences of repeated measurements, as for

example, a sequence of 10 numbers, 8 of which are identical, and with one each plus one "increment" and

minus one "increment." A proper assessment of the

"unit error" and the process performance

characteristics cannot be derived from such a distribution. This is not to say that such procedures

will not produce adequate results, but rather that

uncertainty associated with the result must be based on

a more comprehensive study of the process.

One technique which is applicable to a variety of

measurement processes is that of changing the mode of

operation for the purpose of determining the total

process s.d. By changing from a direct reading to a comparative mode of operation, the rounding effect of

the instrument indicating system can effectively be

bypassed. With ingenuity, in addition to the total

process standard deviation, one can also determine the

instrument "unit error" for the incremental settings as well as the proportional subdivision of the increment

between settings.

For example, consider the case of the large multiple

lever weighing scale. With the Instrument settings in

a fixed position, a "known" weight can be placed on the

platform and, by adding summations of small "known"

weights, the position of the weigh beam can be brought

to some arbitrarily defined "null" position^ . The

large weight can be removed and replaced on the

platform, again adjusting the summation to obtain a "null" position. Reordering the changes in the

summation of small weights for a sequence of such

measurements provides a set of data which can be used

to determine an estimate of the total process standard

deviation. The method is directly extensible to the

comparison of two weights, and the "calibration" of one weight with respect to the other. Proper selection of

weights to be added or removed provides a means to

evaluate both the incremental instrument settings as well as the proportional subdivision. While this

method of testing is not identical to the manner in

which the instrument is normally used, the process

characteristics determined are appropriate for

determining the uncertainty for all modes of operation.

^ This may require the addition of a suitable pointer

and a small linear scale at the tip end of the weigh

beam as shown in reference [19].

IX3JJ.J J I I 1 1 1 1 I . I
7.4 Measurement Requirements

Production measurement requirements may be stated in

several ways, such as: (1) determine the magnitude of

the property within some specified uncertainty limits;

(2) determine that the magnitude of the property does

not deviate from some desired magnitude in excess of some specified limits; or (3) determine the magnitude

of the property with an uncertainty appropriate to the

requirement that the announced result will not disagree

with the result from another facility in excess of some specified limits. The goal of measurement assurance

efforts is to provide evidence that the performance of

a given process is adequate with respect to any one, or

all, of these requirements. Fundamental to this

assurance is a realistic estimate of the "unit" error,

and demonstratable evidence to support the total

process s.d. It should also be evident that it is of

primary importance to verify that the required limits

are valid with respect to the manner in which the

measurements are to be used.

A generalized uncertainty statement might be as follows

Uncertainty Instrument

"unit error"

Unaccounted

for S.E.

Random

variability

For any given situation, any one of the three elements

may be predominant. The instrument unit error is fixed

by the mode of operation. While the magnitude can be

established, the direction cannot. Generally, this

term can be reduced by changing to a comparative mode

of operation in which the "unit error" is the

uncertainty of the reference standard used. The

"unaccounted for S.E." includes "rounding",

uncorrected bias, and using inappropriate algorithms.

Some of these may be known both in magnitude and

direction. It is generally possible to reduce the term

to the point that such effects are no longer

identifiable in the end results. The "random

variability" is a function of the total process s.d.,

Oj^. The magnitude depends upon the amount of effort

^ Operator skills are included in all nonautomated

measurement processes. The least important are those

associated with "setting zero", estimating tenths of

divisions, and the like. It is reasonably easy to

teach operators to make unbiased estimates of pointer

positions between two numbered intervals. In many

cases devices such as digital "readouts" and card

printers replace this function. It is most important

for the operator to clearly understand the principles

of the instrument as they relate to the measurement

task. _ 54 _
expended in a given measurement. The direction cannot

be determined. Considering the economy of measurement

effort, for conditions in which relatively large uncertainties are acceptable, the magnitude of the second

term in the above relationship should predominate.

Practices which are adequate for the requirements may

mask completely the last term. In the limit, with

sufficiently well known standards available, the last

term is the predominate one. In this case, the

residual systematic effects after correction for known

sources of variability and the effects from unknown

sources are the components of the random variability.

The problems associated with testing for compliance

with specified tolerances are somewhat more complex.

Tolerance may be associated with process control, as for example, the exceeding of specified tolerance

limits may be a signal for change of process

parameters, e.g. change in part size as related to

cutting tool wear. Tolerances may also specify a band

within which one assigns a common number to indicate

the magnitude of a particular property. In the latter

instance, if the variability between items from the

production process is small relative to the tolerance

band, the process could be adjusted so that the average

value is near the tolerance limit. This introduces an unexpected bias. One normally expects the average

product value to be in reasonable agreement with some specified nominal value. Such a condition could go

undetected if the procedure used for testing for

compliance is not sufficiently precise, and the tests

occur at random over long time intervals. On the other

hand, if the production process is not capable of

meeting the tolerance specifications, there may be

endless haggling over the question of compliance or noncompliance irrespective of the suitability of the

product in its intended usage.

If the acceptance of a tolerance structure is

appropriate to a certain requirement, the same philosophy must be extended to the problem of

detemnining compliance with the structure for the

objects with values at or near the tolerance limits.

As an example, for some classes of weights, two

tolerance limits are specified, one for adjustment and one for maintenance. Wliile it is normally assumed that

the maintenance tolerance is associated with a certain

allowance for wear, it also tends to prevent rejection

for noncompliance, or re-adjustment based on the

assumption of change, when tests are made by various

processes with unknown performance characteristics.

- 55 -
For tolerance tests with processes in which the

predominant error is systematic, that is, the sum of

the "unit error" and "unaccounted S.E.'s", the

tolerance limits should be narrowed by the magnitude of

the systematic error of the process. Any value which

lies within the narrowed limits could safely be

considered to be indicative of an in-tolerance

situation. If the random variability is the major

component of the process variability two actions are

required. The tolerance limits must be narrowed by the

"unit error" of the instrxment, or the local accessible

unit, and a "rule" must be established to define

compliance relative to the narrowed limits. A simple

rule for judging compliance is the same as above, any

value which is within the narrowed limits could be

considered as indicative of a within tolerance

situation. If the risk associated with an out-oftolerance condition is large, the total s.d. of the

process should be some fraction of the narrowed

tolerance limits. In most cases, a requirement that

the s.d. of the process should be on the order of onefifth of the narrowed limits should be satisfactory.

In any event, rejection on the basis of an out-oftolerance condition on the order of a fifth of the

tolerance band negates the whole philosophy of the

convenience of a tolerance structure.

Any meaningful comparison of measurement results from

different processes must be predicated on the fact that

both measurement processes are well characterized.

That is, significant systematic effects must be

accounted for and the uncertainty associated with each

process must be realistic. For two such processes with

very nearly the same total s.d., the expected limit for

disagreement between results from a single measurement

would be (3/2) a . Thus for some specified limit on the

T agreement , L

(3/2)0^ - L or -

from which it follows that the individual process s.d.

should be on the order of 1/5 of the specified limit.

If more than two processes are involved in measurements

of the same item, and the requirement is that only one

time in a hundred shall any two results disagree in

excess of L, the individual process s.d.'s should not

exceed those shown in table 3 [21],

- 56 -
Table 3

No. of

Processes

Desired

Process s.d.*

5 (L/4.6)

10 (L/5.16)

20 (L/5.65)

30 (L/5.91)

* The

the

the

denominator is the value for

upper one percent point in

distribution of the range.

To illustrate the manner in which the random and

systematic components of uncertainty are combined,

consider the task of assigning a value to a 10 000 lb

artifact, and the problem of assigning an uncertainty

for the announced value. Such an artifact might be

used as the "accessible unit" in a particular

measurement process. For this example, all of the

measurement processes are well characterized, and the

announced value for any measurement is defined as the

average of n "single measurements" with total s.d. a. The accessible unit for the operation is a 50 lb

reference artifact with known value and uncertainty,

Ug. The measurement operations are, in sequence:

(1) Each of ten 50 lb artifacts with reference

to the "standard"

(2) Each of five 500 lb artifacts with

reference to the summation 500 lb

established in (1)

(3) Each of four 2 500 lb artifacts with reference

to the summation 2 500 established in (2)

(4) The 10 000 lb artifact with reference to the

summation 10 000 lb established in (3).

- 57 -
The uncertainties for each sequence are as follows

(1) For a 50 lb artifact:

For the summation 500 lb:

(2) For the 500 lb artifact:

For the 2500 lb summation:

"E2500

- '"500 * <5)(3)(o,„„/^-)

(3) For a 2 500 lb artifact:

U " + 3(o //n)

2500 Z5G0 2500

For the summation 10 000 lb

UjlOOOO = ^^2 500 + (4)(3)(a25oo/^)

(A) For the 10 000 lb artifact

"loooo = "^noooo ^^^oooo/*^^

Since the value for the 10 000 lb artifact will be used

as a constant in the next process, the uncertainty of

(4) above is the "unit error" of that process. It is a

± systematic error which must be added to the results

of comparisons with the "calibrated" 10 000 lb

artifact.

The above procedures can be extended establishing the

uncertainty of an inventory, provided that the

measurements are made with a well characterized

process, and that the instrument "unit error" is very

nearly the same over the range of objects which must be

weighed. For an inventory of m objects, the

uncertainty of Zm is:

U_ " m(instrument unit error) + 3/m (a^) Lm T

In the above relation, the value for each m is a single

"direct reading" with standard deviation a^.

- 58 -
In cases where the "direct reading" mode of operation

is used, and the stonnnation (Instrument "unit error" +

Uncorrected S.E.'s) completely mask the process

variability, the "inventory" uncertainty is:

Both terms in this relation must be considered

carefully. In some cases, it may be possible to adjust

the inventory value for bias associated with the

instrxjment "unit error." If digital rounding of the

type mentioned earlier occurs, with a large m and with

the objects distributed over several rounding

increments, the indicated inventory will be less than

the actual inventory by the amount m(rounding

increment/2). On the other hand, if the inventory

consists of "net" values, computed from "gross" and

"tare" values which have been obtained using the same digital rounding increment, the computed inventory

summation is not biased. In some cases, it may be

possible to reduce bias from digital rounding by

relocating the rounding increment relative to the

indicating scale. If the increment can be set so that,

for some objects, the indicated value is biased in one direction, and for others, the bias is reversed, the

effect on the total inventory will be reduced.

For those who interpret measurement results, it must be

emphasized that the most one can expect of a sequence

of repeated measurements from a typical measurement

process, operating in a reasonable state of control, is

a reasonably symmetric distribution about some limiting

mean. While the results of repeated measurements from

a few select processes which are very well

characterized, which are generally located in a controlled environment, and which have been in

operation for long periods of time may be nearly

normally distributed about the mean, such a distribution is not necessarily a characteristic of all

measurement processes. It is suspected that the basis

for the assumption of normal distribution extends far

back in history, where the variability associated with

the operator and his ability to estimate the proper

instriiment reading was the largest source of

variability in the processes under study. With

appropriate training variability associated with

operator "reading" error is generally of no consequence. In a well characterized mass measurement

in which comparison designs are used, it can be

demonstrated that attempts by the operator to

"override" the normal process variability in order to

obtain a smaller within-group standard deviation are evident in a loss of control on the value obtained for

the "check standard."
8.0 Summary

The concept of a measurement process as a production

process is relatively new, having evolved in the last

ten years. There have been significant contributions

from many sources which have served to refine the

initial ideas. While some of the techniques may not be

appropriate for certain highly specialized measurement

processes, it is felt that the concepts are applicable

to practically all measurement processes. For certain

types of general measurement processes, which must

operate in a variety of environments, and which must

accommodate a variety of materials and properties, the

techniques have been invaluable in understanding the

manner in which measurement processes operate in a "real" world.

The Measurement Assurance Programs, associated with

this concept, are only new words and minor refinements

of something that has been going on for a long time —

doing whatever is necessary to provide assurance that

the measurements are adequate for the intended usage.

While these programs emphasize the importance of

verifiable evidence, such evidence is not always

expensive to obtain. Suitable "check standards" and some form of redundancy can be incorporated in almost

all measurement processes. Simple analytical

techniques will either verify that the process is

performing as expected, or that it is not. Fundamental

to the whole approach is the need to understand in the

beginning just what the measurement operation is

supposed to accomplish.

Any one measurement process is a particular realization

of an accepted unit-algorithm-model concept.

Meaningful quantitative evaluation of the product, that

is, the measurement results, must be based on consistency, both with itself and with other

measurement processes. The important questions are:

"If I did it over again, right away, next week or next

year, would the new result agree with my current

result?" and "If someone else repeated the measurement

with his process, would his result agree with mine?"

The answers to both questions are predictable and

verifiable for most measurement processes.

- 60 -
) References

[ 1] Carnap, R. , Philosophical Foundations of Physics, Chapters

6 and 7, (Basic Books, Inc., New York, 1966).

[ 2] Campbell, Norman R. , An Account of the Principles of Measurement and Calculation (Longmans, Green and Co., Ltd.,

New York, 1928).

[ 3] Eisenhart, Churchill, Realistic Evaluation of Precision and

Accuracy of Instrument Calibration Systems, J. Res. Nat.

Bur. Stand. (U.S.), 67C (Eng. and Instr.), No. 2, 161-187

(Apr. -June 1962)

.

[ 4] Pontius, P.E. and Cameron, J.M., Realistic Uncertainties

and the Mass Measurement Process, Nat. Bur. Stand. (U.S.),

Monogr. 103, 17 pages (1967).

[ 5] Pontius, P.E., The Measurement Assurance Program — A Case

Study: Length Measurements, Part I, Nat. Bur. Stand.,

(U.S.), Monogr. to be published 1974.

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

of Small Groups of Standards in the Presence of Drift, Nat.

Bur. Stand. (U.S.), Tech. Note 844, 32 pages (1974).

[ 7] Aimer, H.E., National Bureau of Standards One Kilogram

Balance NBS No. 2, J. Res. Nat. Bur. Stand. (U.S.), 76C

(Eng. and Instr.), Nos. 1 and 2, 1-10 (Jan. -June 1972).

[ 8] Schweitzer, W.G.Jr., Kessler, E.G. Jr., Deslattes, R.D.,

Layer, H.P., and Whetstone, J.R., Description, Performance,

and Wavelengths of Iodine Stabilized Lasers, Applied

Optics, Vol. 12, p 2927, December 1974.

[ 9] Calibration and Test Services of the National Bureau of

Standards, Nat. Bu. Stand. (U.S.), Spec. Publ. 250 (Dec.

1970)

.

[10] Catalog of Standard Reference Materials, Nat. Bu. Stand.

(U.S.), Spec. Publ. 260 (Apr. 1973).

[11] Page, C.H. and Vigoureux, P., The International System of

Units (SI), Nat. Bu. Stand. (U.S.), Spec. Publ. 330 (Apr.

1972).

[12] Youden, W.J., Graphical Diagnosis of Interlaboratory

Results, Industrial Quality Control, Vol. XV, No. 11 (May

1959)

.

- 61 -
[13] Progress Report - TAPE I, "Overlap" No. 2 (July 1969)*.

[14] Progress Report - TAPE II, "Overlap" No. 6 (March 1974)*.

[15] Puttock, M.J. and Thwaite, E.G., Elastic Compression of

Spheres and Cylinders at Point and Line Contact, National

Standards Laboratory Technical Paper No. 25, Commonwealth

Scientific and Industrial Research Organization, Australia

(1969).

[16] Natrella, M.G., Experimental Statistics, Nat. Bur. Stand.

(U.S.), Handbook 91, pp. 2-12, 2-13, U.S. Government

Printing Office (1963).

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

for Mass Calibration, Nat. Bur. Stand. (U.S.), Tech. Note

288, 39 pages (1968).

[18] Anderson, G.B. and Raybold, R.C., Studies of Calibration

Procedures for Load Cells and Proving Rings as Weighing

Devices, Nat. Bur. Stand. (U.S.), Tech. Note 436, 22 pages

(1969).

[19] Briggs, C.A. and Gordon, E.D., Weighing by Substitution,

Nat. Bur. Stand. (U.S.), Technologic Paper No. 208 (Feb.

1922) .

[20] Volodarskii, V.Ya, Rorenberg, V.Ya and Rubichev, N.A.,

Effect on the Precision of Measurements of the Discrepancy

Between the Investigated Object and its Model, (translated

from Izmeritel'naya Tekhnika, No. 7, pp. 18-20, July 1969)

Measurement Techniques, No. 7 page 907 (1969).

[21] Pearson, E.S., Percentage Limits for the Distribution of

Range in Samples from a Normal Distribution, Biometrika,

Vol. 24, pp. 404-417 (1932)

.

[22] Blair, B.E., Editor, Time and Frequency: Theory and Fundamentals, Nat. Bur. Stand. (U.S.), Monogr. 140, Ch. 8,

pp. 153-204 (1974).

[23] See reference [16], Ch. 5.

* Copies of Overlap may be obtained from P. E. Pontius, Mass and

Volume Section, Room A123 MET, National Bureau of Standards,

Washington, D. C. 20234.

USCOMM-NBS OC - 62 -
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4. Ti l l.!-: AND sbunr

Notes on the Fundamentals of Measurement

and Measurement as a Production Process

5. Publication Date Sept. 19 74

6. Performing Organization Code

7. AUTHOP.(S)

Paul E. Pontius

8. Performing Organ. Report No. NBSIR 74-545

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16. ABSTRAC T (A 200-word or less factual summary of most sigpificant information. If document includes a significant

bibliography or I iterature survey, mention it here.)

The concept of a measurement process as a production process is

relatively new, having evolved in the last ten years. There

have been significant contributions from many sources which have

served to refine the initial ideas. The generalized concept of

a measurement process is discussed together with techniques and

examples for verifying the validity of the result. While some

of the techniques may not be appropriate for certain highly

specialized measurement processes, it is felt that the concepts

are applicable to practically all measurement processes. For

certain types of general measurement processes, which must

operate in a variety of environments, and which must accommodate

a variety of materials and properties, the techniques have been

invaluable in understanding the manner in which measurement

processes operate in a "real" world.

17. KEY WORDS (six to twelve entries; alphatietical order; capitalize only Che first letter of Che first key word unless a proper name; separated by semicolons

)

Calibration; comparator; Interferometer; length; long gage blocks;

measurement process; uncertainty.

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