Section H.3

FILE .COBCDo not remora NATIONAL BUREAU OF STANDARDS REPORT

NBS Project NBS Report

5011-60^0002 26 1.1arch 1952 15S9

STATISTICAL UHITS OP IaE-aSUREF*F.HT

hf

VJ o <J® Youden

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

This report is issuec

production in any foi writing is obtained i Washington 25, D. C.

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

ion, reprinting, or re- ted unless permission in

al Bureau of Standards,
FOREWORD

Ha© adoption of statistical procedures for the

interpretation of data will be encouraged, first, by

drawing the attention of scientists and engineers to

statistical procedures that, are the counterpart- of the

questions which inevitably arise in the examination

of data, and, second, by indicating that these pro®

cedures are valid for small sets of data*

The paper was prepared as the first of a series

of four lectures sponsored by the Philadelphia Chapter

of the American Society for* Metals®

J e Curtiss

Chief, National Applied

Mathematics Laboratories

A® Vo Astin

Acting Director

National Bureau of Standards

28 March 1952
STATISTICAL UNITS OF MEASUREMENT*

^7

W. J. Youden**

Engineers are accustomed to devise suitable measuring units

to express the magnitude of the physical properties of engineering

materials. In many cases these units are simple combinations of

the basic units employed in everyday life. The tensile strength

of an aluminum wire may be 35,000 pounds per square inch. A pound

per square inch ia a unit of measurement for tensile strength and

the number 35,000 expresses the engineer’s verdict regarding this

physical property of the metal. Such units are so familiar to us

that they are usually taken for granted, especially when the unit

can be visualized and found meaningful.

There are many cases in which the unit or scale has an arbitrary

basis. The scale of hardness for minerals is an example. The

values 1 to 10 are assigned to ten materials ranging from talc to

diamond. Other materials are rated by their response to scratching

with specimens chosen from the standard list. Obviously a longer

list of standard materials would give finer divisions. This scale

of hardness is something which men have agreed to use to convey

their knowledge of the properties of materials. Other hardness

scales are available for metals, such as the diameter of the indentation made by a hardened steel ball under specified circumstances.

Both these scales of hardness have a recognizable meaning in

terms of everyday experience with the concept known as hardness.

Sometimes the connection is more remote. If the measurement process

is destructive recourse may be had to some other form of measurement the results of which are closely correlated with the performance of the material. It is difficult to see any simple property

that is measured by the Charpy Notched-Bar Test yet such measurements are considered, at least by some, to furnish information on

certain performance characteristics of steel.

Making measurements of one kind or another is a primary

activity for most engineers and scientists. Attention is directed

to the answers, that is, the magnitudes which are obtained. Judgment

Talk given before the Philadelphia Chapter of the American Society

for Metals, March 4, 1952.

"National Bureau of Standards, Washington 25, D. C.
are then formed by appraising these magnitudes against the

individual's accumulated experience with such measurements.

Often the result or measurement is directly compared with some

specification to determine the fate of the material in question.

Men have learned also by experience that measurements vary and

are aware that the fate of border-line material may depend on

this variation. On one test the material may fall just short of

specification. On a repeat test the inherent variation in the

measuring process may turn up a result which meets the specification. The material has not changed; the uncertainty lies in the

limitations of the procedure for making the measurement or possibly

in the sample selected or in both.

In the early stages of the development of a craft attention

is properly directed to refining the measurement procedure.

Attempts are made to reduce the variation shown by repeated

measurements on the same’ material so that it can be neglected.

When this is achieved there is no uncertainty about whether or

not a product meets specification or about the performance of

the product. This happy state of affairs is not easy to attain.

Sometimes, by building extra quality Into the product, one can

be sure that all measurements meet specifications and there will

be no rejections. This can be expensive. Whenever we crowd the

specification we risk the chance that good material will be

rejected simply because measurements vary. Consequently great

efforts have been made to improve measurements and, in recent

times, to making the most of the measurements we have. It is at

this point that someone begins to look into the subject of statistics since it is rumored that statisticians can get information

out of data.

It will be my concern to give some indication of what it is

the statistician has to offer. The somewhat lengthy Introductory

remarks on units had a purpose. We shall need a unit to measure

variation. I shall mention three common units that are used to

give quantitative expression to our knowledge of the variation

that measurements may exhibit.

Even If but two measurements are available the difference

between them is a measure of the variation shown by such measurements. Once sufficient experience with a given procedure has

been' accumulated It is possible to pass' judgment on the agreement

shown by a pair of readings and determine whether or not confidence may be placed in the average of these two readings. If

more than two readings are available, a simple extension of the

rule of taking differences leads to using the range, i.e., the
3-

difference between the largest and smallest measurements in the

set, as a measure of the variation. One important drawback to

this simple unit is that it needs adjustment depending upon the

number of measurements in the set. Five measurements will, on

the average, have a range more than twice as large as the average

difference between two measurements. The average difference

between the largest and smallest of a group of 14 measurements

exceeds three times the average difference between a pair of

measurements. The intrinsic variation of the measurement operation

concerned is the same regardless of the number of measurements

taken, so a unit that depends upon the number in the group must

be adjusted by an appropriate factor.

/ Another unit for measuring variation is the average deviation — simply the average of all the differences (without regard to

sign) obtained by subtracting the average for the group from

every measurement. This unit is in widespread use by engineers

and scientists. It would not be used so much if the user knew

that this unit, like the range, depends upon the number of measurements in the group. Indeed the average deviation, if groups

of three measurements are taken, is, in the long run-, but seven

tenths the correct value. The correction factor depends on the

number in the group. Almost invariably this correction is ignored.

No engineer would use a unit of length that got larger for long

objects than for short ones.

Statisticians have their favorite unit for measuring variation

among measurements. It is called the standard deviation and is

computed by squaring the differences used in the average deviation

and dividing the sum of the squares by one less than the number

of measurements. The final step of taking the square root returns

the unit to the 'same scale as the original measurements. This

seems like a tedious process but so is the process of taking the

measurements in the first place. The square of this unit meets

the first requirement that, on the average its calculated size

is not influenced by the number of measurements used in its

computation. Equally important, this unit is particularly well

adapted to answering the questions that experimenters are bound to

ask about their data. The numerical operations with this unit

are simple.- Furthermore, the standard deviation extrac.ts more

information from the data than the range or average deviation.

A word of caution may be interjected at this point. We are

here considering measurements that are independent and of equal

precision. If the measurements are not independent the information

furnished by a measurement is, in part, a duplication of informatioi
4

already supplied by other measurements. This reduces the effective

size of the set of measurements. Also, if measurements have

varying precision it becom'es necessary to weigh them appropriately

otherwise a single imprecise reading may dominate the others.

Suppose wo enumerate some of the questions that experimenters

ask about data and give some examples to show how the standard

deviation is put to work to answer their questions. It will not

be possible to do a complete job in this paper but some examples

will show that a unit for the measurement of variation is indispensable for a really close examination of a set of measurements.

In the first place it is often desirable to be able to state

the variability of a measurement procedure.

TABLE I

Measur ements Diff. from Average Diff. Squ;

2.59 .06 .0036

2.31 -.02 ,0004

2.38 .05 .0025

2.32 -.01 .0001

2.27 -.06 .0036

Average 2.33 Sum 0.0102

Standard Deviation = /0.0102

j _->/0.0025 = 0.05

V 5-1

Table I shows the computation or 'estimation of the standard

deviation from a group of five measurements. The result, 0.05,

is called an estima te of the standard deviation because, obviously

enough, another set of five similar measurements will lead to

another result differing more or less from the first. The original

readings, or estimates of the property, show variation and consequently we may anticipate variation among a series of estimates

of the variation.

Further questions immediately come to mind. Suppose these

measurements have been obtained by a modification of a test

procedure which long experience has shown to have a standard
5

deviation of 0.12. Are those data sufficient to warrant the

conclusion that a real improvement has "been achieved in the test

procedure? The statistician will divide the sum of squares,

0.0102, from above by the square of 0.12 obtaining

0.0102

0.0144

0.71

This ratio, known as Chi square, is looked up in a statistical

table (see Appendix 1) which informs the statistician that only

once in 20 times would he get five measurements to agree as well

as these do if the procedure has not been improved and it still

has the original standard deviation of 0.12. Perhaps this is the

lucky one in twenty shot. The usual conclusion, however, is that

an improvement has been effected.

There is another question we might consider even before the

measurements are taken. We may set our sights a bit lower and

decide, in advance, that we would be pleased if the modification

in the method cut the actual standard deviation (not merely its

estimate) to one half the former figure of 0.12. 'If the modification has really brought about this much improvement we would

like very much to establish that fact. It is necessary to face

the situation that, in any given group of measurements, the estimate

of the standard deviation may, by chance, be low and we could be

misled. We shall insist, therefore, that the estimate obtained

from the measurements, when they become available, be below such

chance values or we will not be impressed by it. On the other

hand, the modification may really cut the standard deviation down

to one half its former value and nevertheless the set of measurements obtained give, by chance, an estimate somewhat above the

value 0.06. How many measurements should we take to prevent our

concluding an improvement has been achieved when in fact it has

not and also give us a good liklihood of catching this improvement if it really has been achieved. Tables have been prepared

to guide the experimenter in this matter. If the experimenter

decided he wants to risk only one chance in twenty of believing

in an improvement, when none has taken place, and to have nine

chances in ten of picking it up if this 50 percent reduction has

been achieved, such tables show that 12 measurements should be

collected. If this seems a largo number, it means that some

people underestimate the amount of work that it takes to demonstrate that such an improvement has been made. A short table

Is given In Appendix II.
- 6 -

These techniques for assessing the variation among measurements are, of course, equally applicable to the assossment of the

variation shown by units of a product that is -being manufactured.

Almost always the objective is to achieve greater uniformity in

a product and various steps are taken with this in mind. Sooner

or later the question arises? Has the variation been reduced by

the steps taken? It may be that there is no limit on the number

of units available. On the other hand, if the tests are destructive of a valuable item, or if the measurement . of quality or

performance of the product is expensive, then it is useful to be

able to make an estimate, in advance, of the amount of work

required in order to arrive at a sound decision. .Not the least

benefit accruing from the statistical approach is that it removes

the matter from the field of personal opinion and submits -it to

an objective criterion.

Another variation of this question is often encountered. A

purchaser may be offered items from two suppliers. Naturally he

will wish to choose the more uniform material; the^one showing

least variation. Presumably this will assist him in producing a

uniform product. Here the variability of neither material is

known In advance. The purchaser must first estimate the standard

deviations. These are compared by taking the ratio of the squares

of the standard deviations, setting up the fraction so that the

ratio is greater than unity. This ratio, called F, must attain

a certain critical value before it can be concluded that a difference In variability between the products really exists (see

Appendix III). Furthermore, statistical tables are available

to guide the purchaser in the amount of data required,^ to be

reasonably sure of detecting a difference in variability that

would be Important in his process. (See Appendix IV).

These examples by no means exhaust the list of ways of making

good use of knowledge of the variation exhibited by a group of

numerical quantities. This variation is one of the reasons for

making repeat tests. It is well known that the average of a serie

is less prone to variation than the individual measurements enter!:

into the average. In fact, the standard deviation of the average

of n measurement’s is estimated by dividing the standard deviation

(as”"calculated in Table I) by the V?T. This Is one step toward

answering a question of greater interest to the man with the data,

He would like to know whether an interval can be specified such

that, when centered on the observed average, he can have a certain

confidence that it will bracket an average based upon a great

many measurements. If this interval does not overlap some

rejection value the data do not contradict the claim that
7

the product meets the specification. Of course more data might

show that the product does not meet the specification.

Here again there are two distinct situations — first, the

case where the standard deviation has been established through

an accumulation of earlier records and, second, the case in which

the limited set under examination is the sole source of information on the variation of the data. It is reasonable to expect

that a narrower interval can be set if there is available a well

established figure for the standard deviation. Indeed, for 95

percent confidence it is customary to make the interval extend

two (more accurately 1,96) standard deviation'* on either side

of the average. The factor becomes larger as the number of

measurements available for estimating the standard, deviation

decreases. With 20 measurements the factor is 2,09, for 10

measurements 2,26, and for 5 measurements 2.78, These are the

well known jb values taken from widely available Tables of t, (See

Appendix V), The larger factors reflect the additional uncertainty introduced by the variation In the estimates of the

standard deviation which naturally enough gets worse as the

number of measurements becomes smaller.

When earlier records provide a reliable estimate of the

standard deviation it is possible to specify the number of measurements required to have a given chance of picking up a shift in

the average. The shift Is expressed in units of the standard

deviation and a table may be consulted (see Appendix VI) for

the required number of measurements* The table vividly shows the

large number of measurements necessary to be reasonably sure of

catching a small shift in the average.

There are still some workers who hold that statistical

procedures can only be used when large masses of data are involved, This simply is not true. Often these same workers do

not hesitate to express their intuitive judgments upon rather

small sets of data. But statistical procedures are nothing more

than the translation of such intuitive judgments into more exact

appraisals by making use of an appropriate unit of measurement

for the evaluation of the variation in the observed results.

Ever since men began to make measurements they have focused

their attention on the quantity under measurement. There is

always a second quantity which requires measurement. This

Standard deviation of the aver a ge ,
8

second quantity is the variation in the original measurements —

the scatter they exhibit about their average. The examples given

above show emphatically that it is desirable to have a unit to

measure this variation. Once this variation has been measured

it is relatively easy to face the questions about, the data that

sooner or later confront every experimenter.
APPENDIX I

Critical Values for Chi Square

One in 20 Level, n Measurements

n Lower Upper

2 0.00393 3.841

3 0.103 5.991

4 0 . 352 7.815

5 0.711 9 c 488

10 3.325 16.919

15 6.571 23.685

20 10.117 30,144

This table is used to determine whether a group of n

measurements shows (a) less variation than some standard value

or (b) more variation than some standard value for the Standard

Deviation.

Chi square Is computed by taking the ratio of the sum of

the squares of the deviations from the average to the square

of the known standard deviation.

( x - x) 2

Chi Square = 77 rrr“l, _ (known S.D .

)

c

One in 20 sets of measurements will yield values below the

lower critical values, and one in 20 above the critical values

even if the procedure really has the Standard Deviation entered

in the denominator.

This table is extracted from larger tables available in

most statistical texts. In these texts the entries will be

found opposite (n - 1).
APPENDIX II

Number of measurements required to reveal

whether a given procedure has a standard deviation

that is a given fraction of an assigned value

Given fraction

of assigned

value

Chance that tho indicated number of

measurements will reveal the improvement

• Pour out of five Nine out of ten

.70 28 38

.65 20. 26

.60 15 19

.55 12 14

o 50 9 12

.45 8 9

.40 7 8

The value of Chi square for the set of measurements, when

they are obtained, is calculated as in Appendix I and is judged

by the lower critical limit values. The numbers of measurements that are entered in the above table arc sufficient to

reduce the chance' of claiming an improvement, when in reality

there has been no improvement, to one in 20. The two tolunns

refer to the chance of detecting the improvement, if actually

there to the extent indicated in the first column.

For larger tables see ’’Selected Techniques of Statistical

Analysis”, edited by Eiscnhart, Hastay and Wallis. McGraw-Hill

Book Company, Inc., 1947.
APPENDIX III

Critical Values for the F Ratio

Each Group with n Measurements

n F

2 161 One chance

3 19 in ten of

4 9.28 attaining F

5 6.39 if both sets

10 3.18 come ffom

15 2.48 sources with

20 2.13 same standard

25 1.98 deviation.

Compute the squared standard deviation for each set and

divide the larger result by the smaller to get the F ratio.

If circumstances dictate that one of the sources must

have a smaller standard deviation than the other in order to

be of interest and the squared standard deviation from this

source made the denominator of the ratio, there is but one

chance in 20 of attaining the F ratio shown above if both

sources In fact have the same standard deviation.
APPENDIX IV

Number* of measurements required in each of two groups

to have a four out of five chance of detecting

whether the standard deviation of one source

is a given multiple of the other.

Multiple No.

1.5

2.0

2.5

3.0

3.5

4.0

of Measurements

39

15

9

7

6

5

The ratio of squared standard deviations is obtained by

putting the value from the source expected to have the smaller

standard deviation in the denominator. The ratio is judged by

the critical values in Appendix III.

APPENDIX V

Factors to multiply the standard deviation of the average

estimated from n measurements

in order to set up a 95 percent confidence interval

for the average.

n t

2 12.706

3 4.303

4 3.182

5 2.776

10 2.262

15 2.145

20 2.093

25 2.064

00 1.960
APPENDIX VI

Given that past experience- with a process has provided

a good estimate of the standard deviation of the output,

the table shows the number of measurements necessary

to detect whether the process average has shifted

from a standard value.

Shift in

Average

Measurements needed

4 out of 5 9 out of 10

0.5 S.D. 30 42

1.0 S.D. 8 11

1.5 S.D. 4 5

2.0 S.D. 2 3

2.5 S.D. 2 2

3.0 S.D. 1 2

Compute

standard value - average

S jD* /

r -jy/

n

Accept shift if t> 1.96. Then there is one chance in 20 that

there has been no shift in the process average.

The S.D. is taken from prior work.

SEE: Ferris, Grubbs, and Weaver: Annals of Mathematical

Statistics, XVII, 178, 1946.