NBSIR 73-151

On Uncertainty in Mass Measurement

Janet R. Donaldson

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

March 1973

Interim

Prepared for Optical Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington. D. C. 20234

NBSIR 73-151

ON UNCERTAINTY IN MASS MEASUREMENT

Janet R. Donaldson

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

March 1973

Interim

Prepared for Optical Physics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

IVATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

0:i UIICERTAIIITY IN iIASS ME AST' RE' !ENT

INTRODUCTION

For every measurement there must be included a statement of uncertainty. This makes the measured

value usable, defining the expected consistency of the

value with the total measurement system. It provides

an interval about the measured value in v/hich the

limiting mean of a large number of measurements is

The uncertainty statement is based on

three factors: the standard, the

measurement process.

expected to be

a knowledge of

unknov/n, and the

THE STANDARD

It is assumed that the standard is a well defined

object and that its mass has been determined in a v:ell

characterized process which validly describes the

expected consistency of the mass value with reference

to the total measurement system. Barring any real

change, the actual mass of the standard is constant and

it is either greater than, ecual to, or less than its

assigned mass estimate. Its uncertainty, E, describes

the limits that the mass estimate will differ from the

actual mass of the. standard. Any error in the assigned

mass of this standard will be systematic, always the

same amount, in the same direction, although the

direction and magnitude is not known.

THE UNKNOW

It is possible that properties of the unknown, such as static, magnetic or hygroscopic characteristics, vrill

cause the total collection of data made over an

extended period of time to vary more than measurements

made over a short period of time reflecting essentially

the same conditions for each measurement.

The mass estimate of the unknown may also shovj

systematic shifts due to location, season, etc. These

shifts are often due to incorrect assumptions in the

treatment of the raw data and remain undetected until

the unknown is transferred to another facility which

has conditions significantly different than those of

the location where the original determinations were

made. Because these shifts cannot usually.' be estimated

by the measuring facility itself, they will not be

treatedinthispaper.

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THE liEASUnriFNT FTOCFSS

All msp riiC Inurements are measures of differences

betveen aa unkaov/n and a ."Standard. In its simplest

form, it is a determination of the difference between

tvo objects of the sam.e nominal r;ize. The measurement

process determines this difference. The balance, the

unknown, the standard, the weighing procedure, the

operator, the location and the treatment of the raw

data all define the measurement process. It is assumed

that for each v^eighinp, a set procedure will be used

for measuring the difference between the unknown and

the standard. For this discussion, it will also be

assumed that the difference is in mass units and that

corrections for buoyant effects have been made.

Par ame t e rs o f the Heasuremen t F ro ces

s

A collection of difference measurements can be thought

of as a set of random variables taken from some nrobabilitv distribution centered on a limiting mean value.

Each measure of the difference betTTpen the unknovn and

the standard will lie above or below this limiting mean

which Is estimated by the mean of the observed differences. For independent measurements the variabilltv of

these values about the mean is described bv t>ie

standard deviation of the nrocess, a. The mean of n observed differences has a standard deviation of a /n

.

Es t imat ion o f the Stand ard Devlat ion o f the Process

In some cases, a standard deviation can be assumed from

prior knowledge. However, in most cases, especially

where the unknown is not immune to environmental or

other factors, an estimate of the standard deviation of

the measurement process should be computed from

repeated measurements of the difference using the

equation:

s = JJJ^. - X)

Y

n - 1

where n is the number of determinations of the

difference made

1

X. is an individual determination of the

d i f f erence

is the mean of the collection of

di f f er ences

1 - 2 -
The estimate of the standard deviation for this y^rocess

is based on n-1 degrees of freedom. As more

determinations of this difference are made, the number

of degrees of freedom from vhich the standard deviation

has been computed will increase, producing a more

reliable estimate of the standard deviation.

Pooled Est imatos o f the S tandar d Deviat ion o f the

Process

It is not always feasible to measure any one object

enough times to establish a reliable estimate of the

standard deviation of the process, Hov^ever, knowledge

obtained from the measurements of a given unknown can

be applied to like measurements of similar objects.

For example, if the mass of several unknowns of the

same material and nominal size are to be measured using

the same procedure, an estimate of the standard

deviation of the process can be made from each group of

difference measurements. If one is careful to

that each groun of measurements reflects

conditions so that each estimate of the

deviation is a true estimate of the

variability of the process, then the

estimates

follows

of standard deviation can be

s =

be sure

varying

standard

long term

individual

pooled as

v;ue re

n

X

is the individual estimate of the standard

deviation

is the number of difference determinations

determining the estimate.

This is

based on

a pooled standard deviation of the process

n n =i|fn.-l)

degrees of freedom. The standard deviation of the mean

of the difference beti:een one object and the standard

would be s//n^ where s is the pooled estimate of the

standard deviation of the process and nj^ is the number

of measurements made of the difference between the

unknov/n in question and the standard.

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Sinc'^- thr standard deviation thn r>rocess may

reflect, in part, the variability^ of tlie unl'.nowu, one

must verify tliat tl;e procer.foes can be ansumed to be the

sane no tiiat estxr. ater of standard deviation obtained

fron all groups of similar measurements can be pooled

le pit imate ly

.

A limit to the poEs:'*ble effect of the random, variation

can be obtained using normal distribution theory by

taking a multiple, t, of the standard deviation of the

distribution. This multiplier denends on the state of

our knowledge of the underlying distribution: for most

cases \7hen the standard deviation, a, is well known the

multiplier, 3.0, is used. For other cases one may use

the factor from the standard t distribution V7hich

corrects for the smallness of the number of

observations in the estimation of o,

COIIPUTATTON OF UMCERTAINTY

The total uncertainty of the mass estimate of the

unknovn is the sum of the uncertainty of the difference

measurement (the random component) and the uncertainty

of the mass estimate of the standard (E, the systematic

component) just as the mass of the un]:nown is the sum

of the mass estimate of the difference between the

unknnvm and the standard, and the mass estimate of the

standard. Repeated measurements of the difference can

decrease the random component of tlie uncertainty,

iiov^ever only a redetermination of the mass of the

standard can decrease the systematic component of

uncertainty,

Propapat ion o f Error

In many measurement processes, the mass of the unknown

is not determined directly against a standard, but is

measured indirectly through a chain of difference

measurements. The follox/ing example is offered as an

explanation of the logic in the propagation of error

through a chain of measurements. It is not expected

tb.at the uncertainty of a complex chain of measurements

can be computed from this example particularly if

complicated least snuare solutions are involved in

determining the uncertainty of the values obtained.*

* Solutions for all common weighing designs are available free of charge from the Office of Measurement Services at the National Bureau of Standards, Washington,

D. C. 20234.

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In the s d i f f e r en

a start

100 g r a r unkn ovn

dif f eren

standard

b e tve en

values

v/itb the

Implest form, this chain is a series

ce measurements. For example, if V7e ing f^tandard, T (with assigned value

valu

me a s then one could determine the5 0 pr am v^e igl; t s , first by

ce , X , be tt;een

T,

the tv;o 50 gram weights. The table

along v^ith the standard deviations

me asu remen t

.

the Z100=(50) +(50)

and then by measuring the difr

o f sum and

begin t 7 i t h

K) , s ay at

e of t V7 o uring the

and the

e ren ce , Y

,

shows the

associated

Observation Quantity Measured Standard Deviation

X [(50)j+ (50)2] - T a

Y (50)j- (50)2

The value for (50)^ and (50) ^ are given by

Quantity Mass Value

(50) 1 A = (^X + i^Y) +

(50) B = (^X - ijY) +

The usual formula for the standard deviation of a linear combination, (ax+by) , of independent random

variables is

(ax+by) X y

Because the value K is provided by an outside

laboratory and is not determined in the measurement

process described above, it is not a random variable of

that process. Therefore, the standard deviation of the

value A is

(s.d. of A) = Ah)^(o^)^ + (^5) 2 (0^)2

The formulas for the computation of the variance of

other functional forms of random variables can be found

in Table 2-3, Statistical Concepts in Hetrology by

Harry H. Ku , Precision Measurement and Calibration, NBS

Special Publication 300, Vol. 1, 5.2, pp. 315-339,

1969.

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The uncertainty^ in K will be the same for all uses of

the standard and cannot be reduced by making more

measurements in the home laboratory, A proportional

part of the uncertainty in K will be the systematic

component of the uncertainty in A, The proportionality

factor is computed from the ratio of the nominal size

of the unknov.-n, in this case 50g, to the nominal size

of the starting standard, in this case lOOg. The

uncertainty in A is the sum of the random and

systematic components of uncertainty:

Uncertainty in A = 3(s.d. of A) + (50/100) (Uncertainty in K)

Similarly for B

(s.d. of B) = /(J5)2(a^)2 + i-h)^(o^)^ = /(J5)2(a^)2 + (h)^(a^)^

Uncertainty in B = 3 (s.d. of B) + (50/100) (Uncertainty in K)

If the (50)2 weight x^ere really a summation

50= (25 ) j+(25 ) 2 » then one would need an additional

measurement Z of the difference between the two 25 gram

weights. Vie \7±11 presume the standard deviation of

this measurement to be a^. The weighing design is

quite simple:

Quantity Measured Value

(25)i + (25)2 B

(25)^ - (25)2 Z

Unknown Estimate of Unknown

(25) 1 c = h(B+z) = h[(hx-hY)+h^] + hz

(25)2 D = %(B-Z) = i2[(^X-J2Y)+i5K] - ^Z

If one does not ever repeat the original measurements X

and Y when redetermining values for the 25 gram

weights, the situation is just like the case when the

value B has been assigned by an outside laboratory. In

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this case, Z is the only ran don variable and the

standard deviation of C is

and of D is

The systematic comnonent of the uncertainty in C and D

is a proportional part of the uncertainty in B. The

uncertainty in C and D would be computed as follows:

Uncertainty in C = 3(^2)^ + (25/50) (Uncertainty in B)

La Uncertainty in D = 'hih)^^ + (25/50) (Uncertainty in B)

Hov7ever, if the whole series of measurements is

repeated to redetermine the values for the 25 gram

weights, then X, Y and Z are all random variables and

the standard deviation of C is

(s.d. of C)

and of D is

(s.d. of D

)

= /(J^) 2 [ {^^^'^ ih) 2 ia^VVHh) 2 (a^) 2

The assigned value of T is still constant and a proportional part of its uncertainty, now

( 25 /I GO ) (Uncer t ain t V in K) , is the s^'stematic component

of the uncertainty in C and D. In this case the

uncertainty of C would be

Uncertainty in C =

3/(^2) 2 [ ih) H^^) ^'^(^^(0Y^

~^ ]+(^)

^"(^z)

^ + (25/100) (Uncertainty in K)

and the uncertainty in D would be

Uncertainty; in D = 3/(?)^ [ ih) 2 (a^) 2+ ih) 2 (a^) 2 ] + (J^) 2 (a^) ^ + (25/100) (Uncertainty in K)

The above example is analagous to the situation where

one is measuring a set of weights ina5 3 2 1 series

= /(J5)2(s.d.of B)2+(^)2(<;,^)2

= Ah) 2

[ (^) 2 (a^) 2+ ih) 2 ia^) 2 ] + (3^) 2 (a^) 2

= /(55)2(s.d.of B)2+(-J5)2(a )2

- 7 -
v/here a value, for the Zin = 5 + 3+2 has been determined in

a 50 30 20 10 ZIO series. In the general form, the

uncertainty of a given object is computed as follows:

s/d^G 2+(k)2(a^)2 + KE w E

vrhere d is a design factor obtained from the

least souare solution of the last series

of measurements

o is the standard deviation of the

measurement process used for the last

series

k is the ratio of the nominal size of the

object in question to tlie nominal size of

the summation measured in the preceding

series

is the standard deviation of the

summation

K is the ratio of the nominal size of the

object in question to the nominal size of

the vStarting standard and

E is tlie uncertainty in the assifrned value

of the starting standard.

The above article was prepared by Janet R. Donaldson,

Mass, Length and Volume Section, Room A211 MET,

National Bureau of Standards, Washington, D. C. 20234.

Telephone: Area Code 301 921-2511.

USCOMII-NBS-DC

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FoRv NBS-114A (t-7ti —**rs. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA NBSIR 7^-1=^1

SHEET - '-^

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

ON UNCERTAINTY IN IIASS TCASURETCNT

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

Tanpt R. TVinpl(i=;nn

8. Performing Organization

NBSIR 73-151

9. PERFORMING ORGANIZATION NANE AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTNIENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. 2320190

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

National Bureau of Standards

Department of Camerce

Vfeshington, D. C. 20234

13. Type of Report & Period

Covered 1967 - 1973

Interir?

1 A f\n c r\ n n o Ao^rt^rj i rt/n**

15. SUPPLEMENTARY NOTES

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

bibliography or literature survey, mention it here.)

This paper Is a review of the factors which determine the

total uncertainty of a measured mass value. It explains

IIUW UU Lit; UciiILLilc l/Ilc b UcLriU-cilAj. U.cvJ.d.L±On Ox l/Rc IiKdoU-LxJilKrilu process 3 how to combine error terms to provide a realistic

estimate of the "uncertainty , and how these error terms are propagated through a chain of mass measurements

.

17, KEY WORDS (Alphabetical order, separated by semicolons) MaSS ITEaSUreiTeJlt

J

random variability; standarxi deviation; systematic error; uncertainty

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