NATL INST. OF STAND & TECH A11107 ESTbfiM

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

NBS Special Publication 672

Experimentation

and Measurement

Experimentation

and Measurement

'NATIONAL BUREAU

OF ST7V T f>AilDS-r NATIONAL BUREAU

, OF STANDARDS

LIBRARY W. J. Youden CArc

Statistical Consultant (1948-1965)

National Bureau of Standards

Washington, DC 20234 - ~

No. 6^

U.S. DEPARTMENT OF COMMERCE

Malcolm Baldrige, Secretary

National Bureau of Standards

Ernest Ambler, Director

Issued March 1984
Library of Congress Catalog Card Number: 84-601011

National Bureau of Standards Special Publication 672

Natl. Bur. Stand. (U.S.), Spec. Publ. 672, 127 pages (Mar. 1984)

CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1984

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

Experimentation and Measurement was written by Dr. W.J.

Youden, Applied Mathematics Division, National Bureau of

Standards in 196 1, and appeared as a VISTAS of SCIENCE book in

1 962. The VISTAS of SCIENCE series was developed and produced

by the National Science Teachers' Association.

Nearly a quarter of a century after its publication, Experimentation

and Measurement still enjoys wide popularity. Dr. Youden was

unsurpassed in his skill in communicating sophisticated ideas in

simple language. Moreover, he has created ingenious examples

based on common everyday measurements in this book. It provides an excellent introduction to the realistic consideration of errors of

measurement, and illustrates how statistics can contribute to the

design, analysis and interpretation of experiments involving meas- urement data.

The VISTAS of SCIENCE version has been out-of-print for a number of years. The original book has been reproduced in its entirety to preserve its authenticity, and to recognize the contributions of the National Science Teachers' Association.

H. H. Ku

CONTENTS

Preface 6

1. INTRODUCTION 8

2. WHY WE NEED MEASUREMENTS 15

3. MEASUREMENTS IN EXPERIMENTATION 23

4. TYPICAL COLLECTIONS OF MEASUREMENTS 41

5. MATHEMATICS OF MEASUREMENT 56

6. INSTRUMENTS FOR MAKING MEASUREMENTS 80

7. EXPERIMENT WITH WEIGHING MACHINES 96

8. SELECTION OF ITEMS FOR MEASUREMENT 104

9. MEASUREMENT OF THREAD STRENGTH 113

EpUogue 121

Table of Squares 122

Selected Readings 123

Glossary 124

Index 126
A WORD FROM THE AUTHOR

One approach to the topic "measurement" would be the

historical and factual. The curious early units of measurement

and their use would make an interesting beginning. The development of modern systems of measurement and some of the

spectacular examples of very precise measurement would also

make a good story.

I have chosen an entirely different approach. Most of those

who make measurements are almost completely absorbed in the

answer they are trying to get. Often this answer is needed to

throw light on some tentative hypothesis or idea. Thus the

interest of the experimenter is concentrated on his subject,

which may be some special field of chemistry or physics or other science.

Correspondingly, most research workers have little interest

in measurements except as they serve the purpose of supplying

needed information. The work of making measurements is all too often a tiresome and exacting task that stands between the

research worker and the verification or disproving of his thinking on some special problem. It would seem ideal to many

research workers if they had only to push a button to get the

needed data.

The experimenter soon learns, however, that measurements

are subject to errors. Errors in measurement tend to obscure

the truth or to mislead the experimenter. Accordingly, the

experimenter seeks methods to make the errors in his measure- ments so small that they will not lead him to incorrect answers

to scientific questions.

In the era of great battleships there used to be a continuous

struggle between the makers of armor plate and the gunmakers

who sought to construct guns that would send projectiles

through the latest effort of the armor plate manufacturers. There

is a somewhat similar contest in science. The instrument makers
continually devise improved instruments and the scientists continually undertake problems that require more and more accurate measurements. Today, the requirements for accuracy in

measurements often exceed our ability to meet them. One con- sequence of this obstacle to scientific research has been a growing interest in measurement as a special field of research in itself. Perhaps we are not getting all we can out of our measurements.

Indeed, there may be ways to use presently available instruments to make the improved measurements that might be

expected from better, but still unavailable, instruments.

We know now that there are "laws of measurement" just as fascinating as the laws of science. We are beginning to put these

laws to work for us. These laws help us understand the errors

in measurements, and they help us detect and remove sources

of error. They provide us with the means for drawing objective,

unbiased conclusions from data. They tell us how much data

will probably be needed. Today, many great research establishments have on their staffs several specialists in the theory of

measurements. There are not nearly enough of these specialists

to meet the demand for them.

Thus I have thought it more useful to make this book an

elementary introduction to the laws of measurements. But the

approach is not an abstract discussion of measurements, instead

it depends upon getting you to make measurements and, by

observing collections of measurements, to discover for yourself

some of the properties of measurements. The idea is to learn

something about measurement that will be useful — no matter

what is being measured. Some hint is given of the devices that

scientists and measurements specialists use to get more out of

the available equipment. If you understand something about the

laws of measurements, you may be able to get the answers to your own research problems with half the usual amount of work.

No young scientist can afford to pass up a topic that may double

his scientific achievements. —w. j, YOUDEN

1. INTRODUCTION

The plan of the book

Measurements are made to answer questions such as : How

long is this object? How heavy is it? How much chlorine

is there in this water?

In order to make measurements we need suitable units of

measurement. When we ask the length of an object, we expect an answer that will tell us how many inches, or how many

millimeters it is from one end of the object to the other end.

9
We need some way to find out how many times a unit quantity,

such as a millimeter, is contained in the length of a particular

object. Rarely will a unit of length go into the length of the

object a whole number of times. Almost always our answer will

be, "so many units plus some fractional part of a unit." If a

quantitative measurement of length is to be trusted, we must

take great care that the unit we choose is invariable and suitable

for the task. We must also have devised a method of measuring

the object with this unit.

Some measurements require only a simple procedure and

little equipment. The apparatus may be no more than a scale

marked off in the desired units. It is easy to measure the width

of a table by using a meter stick marked off in centimeters and

millimeters. The air temperature of a room is found by looking

at a thermometer and reading the position of the mercury on

the scale. The pressure in an automobile tire is found by applying a tire gauge to the valve and looking at the scale to read the

pounds of air pressure per square inch of surface in the tire. When the proper instrument is available and used carefully,

many measurements require no more than a careful reading of a

scale. On the other hand, most scientific measurements involve

elaborate equipment $nd a complicated technique of using it. If a chemist wants to determine the amount of chlorine in a

material, he may perform a fairly lengthy sequence of operations. He must first weigh out a sample of the material and

record the weight. The sample must be treated with an acid

that will dissolve out all of the chlorine. Any insoluble residue

must be filtered off to obtain a clear solution, and the filter paper

must be washed carefully with excess acid to make sure that

none of the chlorine is left behind.

It then may be necessary to adjust either the acid concentration or the volume of the solution — or both — before adding

a second reagent to precipitate the chlorine. The usual reagent

is silver nitrate. Enough must be added to precipitate all the

chlorine as insoluble silver chloride. This precipitate of silver

10
chloride is separated from the acid by filtering the suspension

through a crucible with a porous bottom.

Before doing this, however, it will be necessary to weigh the

crucible, making sure that it is dry. The precipitate collected

in the crucible should be then washed with distilled water to

remove all traces of reagent and dried. The weight of the empty

crucible subtracted from the weight of the crucible and the

precipitate gives the weight of the silver chloride.

By using the atomic weights of silver and chlorine, the proportion of chlorine in the silver chloride molecule can be determined. The weight of silver chloride precipitate multiplied by

this proportion gives the weight of chlorine in the precipitate.

This, of course, is also the weight of the chlorine in the original sample. The weight of chlorine divided by the weight of

the sample and multiplied by 100 gives the per cent of chlorine

in the sample, thus completing the determination of chlorine.

The Errors in Measurements

If we consider that each weighing ( sample, empty crucible,

and crucible plus precipitate) is a measurement, we see that

three measurements are necessary to measure the amount of

chlorine in the material. This sketch of the analytical procedure

reveals that there are several steps, all of which must be taken

with great care. If the silver chloride precipitate is not carefully

washed, the silver chloride may be contaminated and appear

too heavy. If the precipitate is not transferred completely to

the crucible, some may be lost. None of these steps can be

carried out so that they are absolutely free of error. For example,

since the silver chloride is very slightly soluble, some of the

chloride will not be precipitated. This results in error. Evidently a measurement is subject to many sources of error, some of which may make the measurement too large, while

others may tend to make the measurement too small. It is the

aim of the experimenter to keep these sources of error as small

11
as possible. They cannot be reduced to zero. Thus in this or any

measurement procedure, the task remains to try to find out how

large an error there may be. For this reason, information about

the sources of errors in measurements is indispensable.

In order to decide which one of two materials contains the

larger amount of chlorine, we need accurate measurements. If

die difference in chlorine content between the materials is small

and the measurement is subject to large error, the wrong material may be selected as the one having the larger amount of

chlorine. There also may be an alternative procedure for determining chlorine content. How can we know which procedure

is the more accurate unless the errors in the measurements

have been carefully studied?

Making Measurements

The best way to find out about some of the difficulties in

making measurements is to make measurements. Much of this

book will be devoted to making measurements — to trying to

find out something about the sources of errors in measurements

and how they may be detected and minimized.

The second chapter is an easy one. It goes a little more into

detail about the importance of making good measurements and

tells us something about the role of measurements in our everyday life and in business and commerce. In the third chapter we

undertake a measurement that involves no more equipment

than a book and a millimeter scale. Everyone who reads this

book should try making several repetitions of the measurement

described there. We will examine 96 similar measurements made

by a class of girls. Such a large collection of measurements poses

the problem of finding some convenient and precise method of

describing the collection. Perhaps we can find some number

to represent the whole collection and some other number that

will represent the errors in the measurements.

When you have made the measurements described in Chap12
ter 3, you will have completed your first exploration of the world

of measurement. It will be natural for you to wonder if some

of the things that you have learned on the first exploration apply

to other parts of the measurement world.

Scientific measurements are often time consuming and require

special skills. In Chapter 4, we will examine the reports of other

explorers in the world of measurement. You may compare their

records with the results you found in order to see if there is anything in common. I hope you will be delighted to find that

the things you have observed about your own measurements

also apply to scientific and engineering data.

Mapping the Land of Measurement

One of the primary tasks of all explorers — and scientists are

explorers — is to prepare a map of an unknown region. Such a map will serve as a valuable guide to all subsequent travelers.

The measurements made by countless researchers have been

studied by mathematicians and much of the world of measure- ments has been mapped out. Not all of it, by any means, but

certainly enough so that young scientists will be greatly helped

by the existing maps. So Chapter 5 may be likened to a simplified map of regions already explored. Even this simplified map

may be something of a puzzle to you at first. Remember, Chapter 5r like a map, is something to be consulted and to serve as a guide. Yet people get lost even when

they have maps. Don't be surprised if you get lost. By the time

you have made some more measurements, which amounts to

exploration, you will begin to understand the map better and

will be able to use it more intelligently.

The rest of the book concerns some other journeys in the land

of measurement. Now that you have a map, you are a little better equipped. The next set of measurements that you can

undertake yourselves requires the construction of a small instrument. Most measurements do involve instruments. It is a good

13
idea to construct the instrument yourself.

Next we undertake an exploration that requires a team of

four working together. These easy measurements will reveal

how much can be learned from a very few measurements.

Then comes a chapter which discusses in a brief manner

another important problem confronting most investigators. We

cannot measure everything. We cannot put a rain gauge in

every square mile of the country. The location of the rain gauges

in use constitutes a sample of all possible sites. Similarly, we

cannot test all the steel bars produced by a steel mill. If the test involved loading each bar with weights until it broke, we would

have none left to use in construction. So a sample of bars must

be tested to supply the information about the strength of all the bars in that particular batch. There is an example in this

chapter that shows something about the sampling problem.

The final chapter describes a more complicated measurement

and the construction and testing of a piece of equipment. All

research involves some kind of new problem and the possibility

of requiring new apparatus. Once you have constructed a piece

of equipment and made some measurements with it, your re- spect for the achievements of the research worker will increase.

Making measurements that will be useful to scientists is an

exacting task. Many measurements are difficult to make. For

this reason we must make the very best interpretation of the

measurements that we do get. It is one of the primary purposes

of this book to increase your skill in interpretation of experiments.

14
2. Why we need

measurements

Ameasurement is always expressed as a multiple of some unit

quantity. Most of us take for granted the existence of the

units we use; their names form an indispensable part of our vocabulary. Recall how often you hear or use the following

words: degree Fahrenheit; inch, foot, and mile; ounce, pound,

and ton; pint, quart, and gallon; volt, ampere, and kilowatt

hours; second, minute, and day. Manufacturing and many other
commercial activities are immensely helped by the general ac- ceptance of standard units of measurement. In 1875, there was

a conference in Paris at which the United States and eighteen

other countries signed a treaty and established an International

Bureau of Weights and Measures. Figure 1 shows a picture of

the International Bureau in France.

Numbers and Units

The system of units set up by the International Bureau is based on the meter and kilogram instead of the yard and pound.

The metric system is used in almost all scientific work. Without

a system of standard units, scientists from different countries

would be greatly handicapped in the exchange of scientific

information. The task of defining units still goes on. The problem is not as easy as it might seem. Certain units may be chosen

arbitrarily; for example, length and mass. After four or five units are established in this way, it turns out that scientific laws set up certain mathematical relations so that other units — density,

for example — are derived from the initial set of units.

Obvious also is the need of a number system. Very likely the

evolution of number systems and the making of measurements

were closely related. Long ago even very primitive men must

have made observations that were counts of a number of objects,

such as the day's catch of fish, or the numerical strength of an

army. Because they involve whole numbers, counts are unique.

With care, they can be made without error; whereas measure- ments cannot be made exactly.

Air temperature or room temperature, although reported as 52°F., does not change by steps of one degree. Since a report

of 51° or 53° probably would not alter our choice of clothing, a

report to the nearest whole degree is satisfactory for weather

reports. However, a moment's thought reveals that the temperature scale is continuous; any decimal fraction of a degree is possible. When two thermometers are placed side by side, care- 16
ful inspection nearly always shows a difference between the

readings. This opens up the whole problem of making accurate

measurements.

We need measurements to buy clothes, yard goods, and carpets. The heights of people, tides, floods, airplanes, mountains,

and satellites are important, but involve quite different procedures of measurements and the choice of appropriate units.

For one or another reason we are interested in the weights of

babies (and adults), drugs, jewels, automobiles, rockets, ships,

coins, astronomical bodies, and atoms, to mention only a few.

Here, too, quite different methods of measurements — and units — are needed, depending on the magnitude of the weight and

on the accessibility of the object.

Significance of Small Differences

The measurement of the age of objects taken from excavations of bygone civilizations requires painstaking measurements

of the relative abundance of certain stable and radioactive

isotopes of carbon; C-12 and C-14 are most commonly used.

Estimates of age obtained by carbon dating have a known

probable error of several decades.

Another method of measuring the age of burial mounds

makes use of pieces of obsidian tools or ornaments found in

them. Over the centuries a very thin skin of material — thinner

than most paper — forms on the surface. The thickness of this

skin, which accumulates at a known rate, increases with age

and provides an entirely independent measure of the age to

compare with the carbon-14 estimate. Here time is estimated

by measuring a very small distance.

Suppose we wish to arrange some ancient objects in a series

of ever-increasing age. Our success in getting the objects in the

correct order depends on two things: the difference in ages

between objects and the uncertainty in the estimate of the

ages. Both are involved in attaining the correct order of age.

17
If the least difference in age between two objects is a century

and the estimate of the age of any object is not in error by

more than forty years, we will get the objects in the correct

order. Inevitably the study of ancient civilizations leads to an

effort to get the correct order of age even when the differences

in age are quite small. The uncertainty in the measurement

of the age places a definite limitation on the dating of archeological materials.

The detection of small differences in respect to some property is a major problem in science and industry. Two or more

companies may submit samples of a material to a prospective

purchaser. Naturally the purchaser will want first of all to

make sure that the quality of the material he buys meets his

requirements. Secondly, he will want to select the best mateFigure 1. Measurement standards for the world are maintainec if
rial, other things, such as cost, being equal.

When we buy a gold object that is stated to be 14 carats

fine, this means that the gold should constitute 14/24 of the

weight. We accept this claim because we know that various

official agencies occasionally take specimens for chemical analysis to verify the gold content.

An inaccurate method of analysis may lead to an erroneous

conclusion. Assuming that the error is in technique and not

some constant error in the scales or chemicals used, the chemical analysis is equally likely to be too high as it is to be too

low. If all the items were exactly 14 carats, then chemical

analysis would show half of them to be below the specified

gold content. Thus an article that is actually 14 carats fine

might be unjustly rejected, or an article below the required

in Paris at the International Bureau of Weights and Measurements.
content may be mistakenly accepted. A little thought will show

that if the error in the analysis is large, the manufacturer of

the article must make the gold content considerably more than

14/24 if he wishes to insure acceptance of nearly all the items

tested.

There are two ways around this dilemma. The manufacturer

may purposely increase the gold content above the specified

leveL This is an expensive solution and the manufacturer must

pass on this increased cost. Alternatively, the parties concerned

may agree upon a certain permissible tolerance or departure

from the specified gold content. Inasmuch as the gold content

cannot be determined without some uncertainty, it appears

reasonable to make allowance for this uncertainty. How large a tolerance should be set? This will depend primarily on the

accuracy of the chemical analysis. The point is that, besides

the problem of devising a method for the analysis of gold

articles, there is the equally important problem of determining

the sources of error and size of error of the method of analysis.

This is a recurrent problem of measurement, regardless of the

material or phenomenon being measured.

There may be some who feel that small differences are un- important because, for example, the gold article will give

acceptable service even if it is slightly below 14 carats. But

small differences may be important for a number of reasons.

If one variety of wheat yields just one per cent more grain

than another variety, the difference may be unimportant to a

small farmer. But added up for the whole of the United States

this small difference would mean at least ten million more

bushels of wheat to feed a hungry world.

Sometimes a small difference has tremendous scientific con- sequences. Our atmosphere is about 80 per cent nitrogen.

Chemists can remove the oxygen, carbon dioxide, and moisture.

At one time the residual gas was believed to consist solely of

nitrogen. There is an interesting chemical, ammonium nitrite,

NH4NO2. This chemical can be prepared in a very pure form.

20
When heated, ammonium nitrite decomposes to give nitrogen

(N2 ) and water (H2O). Now pure nitrogen, whether obtained

from air or by the decomposition of NH4NO2, should have

identical chemical and physical properties. In 1890, a British

scientist, Lord Rayleigh, undertook a study in which he compared

nitrogen obtained from the air with nitrogen released by heating

ammonium nitrite. He wanted to compare the densities of the

two gases; that is, their weights per unit of volume. He did this

by filling a bulb of carefully determined volume with each gas

in turn under standard conditions: sea level pressure at 0°

centigrade. The weight of the bulb when full minus its weight

when the nitrogen was exhausted gave the weight of the

nitrogen. One measurement of the weight of atmospheric nitrogen gave 2.31001 grams. Another measurement on nitrogen

from ammonium nitrite gave 2.29849 grams. The difference,

0.01152, is small. Lord Rayleigh was faced with a problem: was

the difference a measurement error or was there a real difference

in the densities? On the basis of existing chemical knowledge

there should have been no difference in densities. Several additional measurements were made with each gas, and Lord Rayleigh

concluded that his data were convincing evidence that the observed small difference in densities was in excess of the experimental errors of measurement and therefore actually existed.

There now arose the intriguing scientific problem of finding a reason for the observed difference in density. Further study

finally led Lord Rayleigh to believe that the nitrogen from the

air contained some hitherto unknown gas or gases that were

heavier than nitrogen, and which had not been removed by

the means to remove the other known gases. Proceeding on this

assumption, he soon isolated the gaseous element argon. Then

followed the discovery of the whole family of the rare gases,

the existence of which had not even been suspected. The small

difference in densities, carefully evaluated as not accidental,

led to a scientific discovery of major importance.

Tremendous efforts are made to improve our techniques of

21
making measurements, for who knows what other exciting discoveries still lie hidden behind small differences. Only when

we know what are the sources of error in our measurements

can we set proper tolerances, evaluate small differences, and

estimate the accuracy of our measurements of physical constants. The study of measurements has shown that there are

certain properties common to all measurements; thus certain

mathematical laws apply to all measurements regardless of what

it is that is measured. In the following chapters we will find out

some of these properties and how to use them in the interpretation of experimental data. First,we must make some measure- ments so we can experience first hand what a measurement is.

22
3. Measurements in

experimentation

The object of every scientific experiment is to answer some

question of interest to a scientist. Usually the answer comes

out in units of a system of measurement. When a measurement

has been made the scientist trusts the numerical result and uses

it in his work, if the measurement apparatus and technique are adequate. An important question occurs to us right away. How

do we know that the measurement apparatus and technique
are adequate? We need rules of some kind that will help us to

pass judgment on our measurements. Later on we will become

acquainted with some of these checks on measurements.

Our immediate task is to make some measurements. The

common measurements made every day, such as reading a thermometer, differ in a very important respect from scientific

measurements. Generally we read the thermometer to the nearest whole degree, and that is quite good enough for our purposes. If the marks are one degree apart, a glance is enough

to choose the nearest graduation. If the interval between

adjacent marks is two degrees, we are likely to be satisfied

with a reading to the nearest even number. If the end of the

mercury is approximately midway between two marks, we

may report to the nearest degree, and that will be an oddnumbered degree.

The Knack of Estimating

Fever thermometers cover only a small range of temperature.

Each whole degree is divided into fifths by four smaller marks

between the whole degree graduation marks. The fever thermometer is generally read to the nearest mark. We get readings

like 98.6°, 99.8°, or 100.2° F. As the fever rises, readings are taken more carefully and the readings may be estimated between marks, so that you may record 102.3°. Notice that body

temperature can easily be read to an extra decimal place over

the readings made for room temperatures. This is possible

because the scale has been expanded.

Examine a room thermometer. The graduation marks are approximately one sixteenth of an inch apart. The mercury may

end at a mark or anywhere in between two adjacent marks. It

is easy to select a position midway between two marks. Positions one quarter and three quarters of the way from one mark

to the next mark are also fairly easy to locate. Usually we do

not make use of such fine subdivisions because our daily needs

24
do not require them. In making scientific measurements, it is standard practice to estimate positions in steps of one tenth

of the interval. Suppose the end of the mercury is between the

70 and 71 degree mark. You may feel a little uncertain whether

the mercury ends at 70.7° or at 70.8°. Never mind, put down

one or the other. Practice will give you confidence. Experts

may estimate the position on a scale to one twentieth of a scale

interval. Very often a small magnifying glass is used as an aid

in making these readings.

Here is an example of a scientific problem that requires precise temperature readings. Suppose that you collect some

rain water and determine that its freezing point is 32 °F.

Now measure out a quart of the rain water sample and add one ounce of table sugar. Place a portion of this solution in a freezing-brine mixture of ice and table salt, stirring it all the while.

Ice will not begin to appear until the temperature has dropped

to a little more than 0.29 °F., below the temperature at which

your original sample begins to turn to ice.

From this simple experiment you can see that freezing points can be used to determine whether or not a solvent is pure. The

depressions of the freezing point produced by dissolving substances in solvents have long been a subject of study. In these

studies temperatures are usually read to at least one thousandth

of a degree, by means of special thermometers. The position of

the mercury is estimated to a tenth of the interval between

marks which are 0.01 of a degree apart. Very exact temperature

measurements taken just as the liquid begins to freeze can be

used to detect the presence of minute amounts of impurity.

In scientific work the knack of estimating tenths of a division

on scales and instrument dials becomes almost automatic. The

way to acquire this ability is to get some practice. We will now

undertake an experiment that will quickly reveal your skill in

reading subdivisions of a scale interval. The inquiry that we

are to undertake is to measure the thickness of the paper used

in one of your textbooks. Although a single sheet of paper is 25
much thinner than the smallest scale divisions on a ruler, a simple procedure will make it possible for you to determine the

thickness of the paper quite accurately. The procedure consists

of reading the thickness of a stack of sheets of the paper and

dividing the reading by the number of sheets in the stack.

Simple as this procedure appears, we will find that it reveals

quite a lot about measurements.

Prepare a form for recording the measurements. Strangely

enough, careful measurements are sometimes so poorly recorded even by professional scientists that not even the experimenter can interpret them sometime later. A form suitable for this

experiment with spaces to enter four separate observations is shown in Table 1. Note that the form identifies the observer,

the source of the data, and the date of the experiment

or observation. These are characteristics of useful forms.

Choose a book without inserts of special paper. First open

the book near the beginning and also near the end. Hold the

stack of pages together. This is the stack whose thickness we

will measure with a ruler marked off in centimeters and millimeters. We will estimate tenths of a millimeter.

There are a number of details to observe even in this simple

Table L Example of a form for keeping a record of measurements.

Experiment to measure thickness of sheets of paper

Observer: B. G. Date: March 16, 1961

Book title and author: MODERN CHEMISTRY, Dull, Metcalfe and Williams

obser- page number pages sheets stack thickness

vation at at in in thickness per sheet number front back stack stack mm. mm.

1 67 387 320 160 12.4 0.0775

2 41 459 418 209 16.4 0.0785

3 23 549 526 263 20.0 0.0760

4 35 521 486 243 18.5 0.0761

Total 0.3081

Average 0.0770

[D»U taken by t student it Immtculatt High School, Washington, D C ]
experiment. Read the numbers on the first page of the stack

and that on the page facing the last page of the stack. Both

numbers should be odd. The difference between these two numbers is equal to the number of pages in the stack. This difference must always be an even number. Each separate sheet

accounts for two pages, so divide the number of pages by two to get the number of sheets. Enter these data on the record form.

Pinch the stack firmly between thumb and fingers and lay

the scale across the edge of the stack. Measure the thickness of

the stack and record the reading. The stack will usually be

between one and two centimeters thick; i.e., between 10 and

20 mm. (millimeters). Try to estimate tenths of a millimeter.

If this seems too hard at first, at least estimate to the nearest one fourth (0.25) of a millimeter. Record these readings as decimals. For example, record 14 and an estimated one fourth

of a division as 14.25.

Measurements Do Not Always Agree

After you have made the first measurement, close the book.

Then reopen it at a new place and record the new data. Make

at least four measurements. Now divide the reading of the thick- ness by the number of sheets in the stack. The quotient gives

the thickness of one sheet of paper as a> decimal part of a millimeter. When this division is made for each measurement, you

will certainly find small differences among the quotients. You

have discovered for yourself that measurements made on the

same thing do not agree perfectly. To be sure, the number of

sheets was changed from one measurement to the next. But that

does not explain the disagreement in the answers. Certainly a

stack of 200 sheets should be just twice as thick as a stack of

100 sheets. When the stack thickness is divided by the number

of sheets we should always get the thickness of a single sheet.

There are two major reasons for the disagreement among the

answers. First, you may pinch some of the stacks more tightly

27
than others. You could arrange to place the stack on a table top

and place a flatiron on the stack. This would provide a uniform

pressure for every measurement. The second major reason is your inexpertness in reading the scale and estimating between

the scale marks. Undoubtedly this is the more important of the

two reasons for getting different answers. The whole reason for

insisting on closing the book was to make sure the number of

sheets was changed each time. You knew the thickness would

change and expected to get a change in your scale reading.

Unless you are very good in mental arithmetic you could not

predict your second scale reading.

Suppose, however, that you were able to do the necessary

proportion in your head. If you knew in advance what the

second scale reading should be to make it check with your first result, this would inevitably influence your second reading.

Such influence would rob the second reading of the indispensable independence that would make it worthy of being called a measurement.

It may be argued that all four answers listed in Table 1 agree

in the first decimal place. Clearly, all the answers are a little less than 0.08 mm. Thus any one of the results rounded off would

give us this answer. Why take four readings?

Just to take a practical everyday reason, consider the paper

business. Although paper in bulk is sold by weight, most users are also concerned with the thickness of paper and size of the

sheet. Thick paper will mean fewer sheets of a given size per

unit weight paid for. A variation of as little as 0.01 mm. — the

difference between 0.08 mm. andO.09 mm. — would reduce the

number of sheets by more than ten per cent. We need to know

the answer to one or two more decimal places. In a situation

like this, it is usual to obtain the average of several readings.

You should note, however, that although repetition alone doesn't

insure accuracy, it does help us locate errors. Many people seem to feel that there is some magic in the

repetition of measurements and that if a measurement is re- 28
peated frequently enough the final result will approach a "true"

value. This is what scientists mean by accuracy.

Suppose that you were in science class and that the next two

people to come into your classroom were a girl five feet ten

inches tall and a boy five feet nine inches tall. Let each of the

30 students already in the class measure the two new arrivals

to the nearest foot. The answer for both is six feet. Has repeated

measurement improved the accuracy?

Suppose that the hypothesis to be tested was that girls are

taller than boys. This time the boy and the girl were each

measured 30 times with a ruler that read to 1/100 of an inch.

Could we conclude that the repeated measurements really supported the hypothesis? The point is that repeated measurements

alone do not insure accuracy. However, if a set of measurements

on the same thing vary widely among themselves we begin to

suspect our instruments and procedures. This is of value if we

are ever to achieve a reasonable accuracy.

Paper thickness is so important in commerce that the Ameri- can Society for Testing Materials has a recommended procedure

for measuring the thickness of paper. A standard pressure is put

on the stack and a highly precise instrument called a micrometer

is used to measure the thickness. Even then the results show a

scatter, but farther out in the decimal places. Improved instruments do not remove the disagreement among answers. In fact

the more sensitive the apparatus, the more surely is the variation among repeat measurements revealed. Only if a very coarse

unit of measurement is used does the disagreement disappear.

For example, if you report height only to the nearest whole

meter practically all adult men will be two meters tall. It used to be a common practice among experimenters to pick

out the largest and smallest among the measurements and report

these along with th§ average. More often today the difference

between the largest and smallest measurement is reported to- gether with the average from all the measurements. This differ- ence between the maximum and minimum values is called the

29
range of the group. The range gives some notion of the variation

among the measurements. A small range, or a range which is a

small percentage of the average, gives us more confidence in

the average. Although the range does reveal the skill of the

operator, it has the disadvantage of depending on the number

of measurements in the group. The same operator usually will

find his average range for groups of ten measurements to be

about 1.5 times as large as the range he gets for groups of four

measurements. So the number of measurements in the group

must always be kept in mind.

Averages, Ranges, and Scatter

The data for operator B. G. have been given in complete

detail in Table 1. This operator was one of a class of 24 girls, all of whom made four measurements on the same book. This

larger collection of data will reveal still more about measurements. The measurements made by these girls are tabulated in

Table 2, which shows the computed thickness for each trial. The details of pages and millimeters have been omitted. Most of

the girls did not estimate tenths of a millimeter but did read to

the nearest quarter millimeter. Two or three had readings only

to the nearest whole millimeter. A gross misreading of the scale

was evidently made by girl U on her last trial. This value has

been excluded from the average and no range entered for this

student.

The remaining 23 ranges vary widely. This does not necessarily mean that some girls were better than others in reading

the scale. Even if all girls had the same skill, the range may vary

severalfold when it is based on just four measurements. Of

course, if this class of girls repeated the measurements and the

very small ranges and very large ranges were produced by the

same girls as before, this would indicate that some girls can

repeat their measurements better than other girls. One way to

summarize the results is to give the average thickness, 0.07709,

30
and the average of the 23 ranges, 0.00483. The last two places of

decimals are quite uncertain for both these averages. Indeed, a mathematician would say we cannot be sure that the second

seven in 0.07709 is correct. Additional decimal places are always

carried when making computations and these should be entered

in the data record.

Table 2. Tabulation of 96 measurements of paper thickness made by

24 girls.

girl thickness in mm. average range A .0757 .0762 .0769 .0746 .0758 .0023 0 4

B .0808 .0793 .0781 .0821 .0801 .0040 4 0 c .0811 .0772 .0770 .0756 .0777 .0055 2 2 D .0655 .0683 .0714 .0746 .0700 .0091 0 4

E .0741 .0710 .0748 .0711 .0728 .0038 0 4

F .0756 .0772 .0776 .0759 .0766 .0020 2 2

G* .0775 .0785 .0760 .0761 .0770 .0025 2 2

H .0747 .0765 .0735 .0776 .0756 .0041 1 3

I .0719 .0762 .0802 .0713 .0749 .0089 1 3

J .0734 .0833 .0833 .0783 .0796 .0099 3 1 K .0755 .0740 .0714 .0743 .0738 .0041 0 4

L .0788 .0817 .0794 .0766 .0791 .0051 3 1 M .0731 .0716 .0726 .0714 .0722 .0017 0 4

N .0833 .0794 .0783 .0788 .0800 .0050 4 0

0 .0767 .0775 .0765 .0793 .0775 .0028 2 2 P .0787 .0798 .0864 .0817 .0816 .0077 4 0

Q .0784 .0799 .0789 .0802 .0794 .0018 4 0

R .0784 .0820 .0796 .0818 .0804 .0036 4 0

S .0830 .0796 .0778 .0767 .0793 .0063 3 1 T .0741 .0680 .0733 .0723 .0719 .0061 0 4

U .0759 .0766 .0772 .0466** .0766 1 3

V .0810 .0812 .0789 .0776 .0797 .0036 4 0

w .0777 .0759 .0795 .0790 .0780 .0036 3 1 X .0784 .0786 .0797 .0859 .0806 .0075 4 0

Total for 95 measurements = 7.3239

Average for 95 measurements == 0.07709

Average for 23 ranges = 0.00483

*G is the same student, B.G., reported in Table 1. **The last measurement made by student U is .0466. This appears to be a mistake as it is little more than half as large as the other measurements. This measuremerit is omitted from the collection and the total and average computed from

the remaining 95 measurements.
Notice that the number of sheets in the stack involves three

significant figures. The quotient or thickness per sheet is therefore carried out to three significant figures. To use only two

figures would be equivalent to rounding off the number of sheets

before dividing. Failure to carry enough figures tends to conceal

the variation in the data.

Condensing Our Data

Tabulated results, as shown in Table 2,look like a sea of numbers. There is a way to bring out the essential characteristics of

such collections of many measurements on the same thing.

Paradoxically we may condense the data into more compact

form and at the same time get a better picture of the collection.

The smallest of the 95 results is 0.0655 mm. and the largest is 0.0864 mm. The range for these 95 results is therefore 0.0209.

Suppose we form a series of intervals to cover this range. We

may start out at 0.0650 mm. and make each interval equal to

0.0020 mm. The size of the interval should be small enough so that at least six intervals will be needed. If there are many

measurements there should be more intervals than with few

measurements. Table 3 shows eleven intervals that completely

cover the whole range of values.

The intervals are written down in a column. Then each of the

values in Table 2 (except the apparent mistake) is placed in its proper interval by making a pen stroke opposite the interval

class. The actual values are, in effect, replaced by the mid-values

of the interval class to which they have been assigned. The

slight change made by using the mid-values of the intervals is of no consequence. Indeed, some values are slightly increased

and others decreased. Much of the effect therefore cancels out. Now we are beginning to get some order in our sea of numbers. The mass of individual items in the data now have been

replaced by the eleven different mid-values along with the number of measurements assigned to each mid-value. The last

32
column shows the product of each mid-value by the number of

measurements that go with it. The total for this column, 7.32925,

is close to the total, 7.3239 (Table 2), of the actual measurements. The averages obtained by dividing each total by 95 are 0.07715 and 0.07709. The difference is quite unimportant.

Table 3. Retabulation of data in Table 2 measurement number of measurements in this mid-value of no. times

interval interval interval mid-value .Ubou - .Ubby / 1 . \J\J U Z/ >J .0670 -.0689 // 2 .06795 .13590

.0690 -.0709 0 .06995

.0710 -.0729 /// /// /// / 10 .07195 .71950

.0730 -.0749 /// /// /// /// 12 .07395 .88740

.0750 -.0769 /// /// /// /// /// /// 18 .07595 1.36710

.0770 -.0789 /// /// /// /// /// /// /// /// 24 .07795 1.87080

.0790 -.0809 /// /// /// /// // 14 .07995 1.11930

.0810 -.0829 /// /// // 8 .08195 .65560

.0830 -.0849 /// / 4 .08395 .33580

.0850 -.0869 // 2 .08595 .17190

Total 95 7.32925

Average 0.07715

Grouping measurements into intervals is standard practice.

The presentation of the data is more concise. Furthermore a glance at the check marks made opposite the intervals in Table 3

tells us something about the data. We see that the interval

containing the average contains more measurements than any

other — it is called the modal interval. Intervals on either side of

the modal interval have fewer measurements in them. The number in each interval falls off sharply near the end intervals.

Apparently measurements that differ considerably from the

average are relatively scarce. This is an encouraging thought

for experimenters. Obviously, however, there is a chance of

getting one of these scarce measurements. Experimenters are 33
naturally much interested in knowing what the risk is of getting a measurement quite distant from the average.

Often the counts of the measurements in the intervals are shown graphically. One way to do this is by means of a histogram as shown in Figure 2. To make this histogram, the 11

intervals were marked off as equal segments on a horizontal line.

A suitable scale is laid off on a vertical line to designate the

number of measurements in each interval. Horizontal bars are drawn at the proper heights and connected as shown. The general form of this histogram, the intervals, and the number in

each interval, tell the expert just about evervthing that the

actual measurements would.

We have seen one histogram and obtained some idea of the

wav this collection of measurements is distributed around an

Figure 2. Histogram for 95 measurements of paper thickness

25 —

20 —

15 —

10

5 —

.0650- .0670- .0690- .0710- .0730- .0750- .0770- .0790- .0810- .0830- .0850- .0669 .0689 .0709 .0729 .0749 .0769 .0789 .0809 .0829 .0849 .0869 Thickness per sheet in millimeters 34
average. In Chapter 4 several different collections of measure- ments are represented by histograms. You will then be able to

observe that in many collections of measurements there are

similarities in the distributions regardless of the objects being

measured. This fact has been of crucial importance in the development of the laws of measurement.

Let's return to our measurements of paper thicknesses and

investigate some of the properties of this collection. The measurements in the collection should meet certain requirements.

One of these requirements is that each of the four measure- ments made by a student should be a really independent measurement. By that we mean that no measurement is influenced by

any following measurement. Another requirement is that all participants should be equally skillful. If some measurements

were made by a skilled person and some by a novice, we should

hesitate to combine both collections. Rather we should make a separate histogram for each individual. We would expect the

measurements made by the skillful one to stay closer to the

average. His histogram might be narrow and tall when compared with the histogram for the novice. The readings made by

the novice might be expected to show a greater scatter. Histograms can vide a quick appraisal of the data and the technique of thv iasurer.

Four measurements are too few to rate any individual. Nevertheless, the availability of 24 individuals makes it possible to explore still another property of these data. If we think about

the measurement procedure, we see that it is reasonable to

assume that any given measurement had an equal chance of

being either larger or smaller than the average. In any particular

measurement the pressure on the stack could equally well have

been either more or less than the average pressure. The scale

reading may have erred on the generous side or on the skimpy

side. If these considerations apply, we would expect a symmetrical histogram. Our histogram does show a fair degree of

symmetry.

35
Insights From the Laws of Chance

Before we conclude that the requirements for putting all the

measurements into one collection have been fully satisfied, we

must carefully examine the data. The reason we changed the

number of pages for each measurement was to avoid influencing

later readings by preceding readings. If we happened to get too

large a reading on the first measurement, this should not have

had the effect of making subsequent readings too large. We are assuming, of course, that the pressure applied to the stack varied

with each measurement, and that the reading of the scale was

sometimes too large and sometimes too small. It also seems

reasonable to assume that there is a 50-50 chance of any one

measurement being above or below the average value. Is this

true of the measurements made by the girls in this science class?

It is conceivable, of course, that a particular individual always

squeezes the paper very tightly and in consequence always gets

lower readings than the average for the class. Another person

might always tend to read the scale in a way to get high readings. If this state of affairs exists, then we might expect that all readings made by a particular individual would tend to be either

higher or lower than the average, rather than splitting 50-50.

Let us think about a set of four measurements in which each

measurement is independent and has the same chance to be

more than the average as it has to be less than the average.

What kind of results could be expected by anyone making the

four measurements? One of five things must happen: All four

will be above the average, three above and one below, two above

and two below, one above and three below, all four below.

Our first impulse is to regard a result in which all four meas- urements are above ( or below ) the average as an unlikely event.

The chance that a single measurement will be either high or low

is 50-50, just as it is to get heads or tails with a single coin toss. As an illustration, suppose a cent, a nickel, a dime, and a quarter

are tossed together. The probabilities of four heads, three heads,

two heads, one head, or no heads are easily obtained. The pos36
Table 4. Possible ways four coins may fall when they are tossed together.

One way to get no heads

Four ways

to get only one head

Six ways

to get only two heads

Four ways

to get only three heads

One way to get four heads

C6nt nickel dime nuartpr T T T T

H T T T T H T T

T T H T

T T T H

H H T T

LJn

T1 MT1 n

T1 T1 u

T1 un

un

T1 T H T H

T T H H

H H H T

H H T H

H T H H

T H H H

H H H H

sible ways the four coins might fall are enumerated in Table 4. There are just sixteen different ways in which the coins may

fall. We may easily calculate our chances of getting no heads, one head, two, three, or four heads. For example, we find there

is only one way to get four heads — the chance is 1 in 16. Remember that this calculation assumes that a tossed coin is equally likely to fall heads as it is tails. Incidentally, the

chances are not altered if four cents are used, as you can determine for yourselves by trying it out. The mathematical

experu among the readers will know that (H + T) 4 = H 4 +

4H 3 T + 6H 2 T

2 + 4HT3 + T4

. Observe that the coefficients

1, 4, 6, 4, 1 correspond to the counts shown in Table 4. Some of

you may be inclined to find out whether or not this relationship

holds if three, five, or n coins are tossed instead of four coins. Let's now see how the results from tossing four coins can serve

as a useful model in examining the collection of measurements

37
made on the thickness of. paper. If — as in the case of heads or

tails, when coins are tossed — high or low readings are equally

likely, we conclude that there is 1 chance in 16 of getting four

high readings and 1 chance in 16 of getting four low readings.

There are 4 chances in 16 of getting just one high reading and

an equal chance of getting just three high readings. Finally there are 6 chances in 16 of getting two high readings and two low

readings.

Now to apply this model to the entire collection of 24 sets of

four measurements each, we can multiply each of the coefficients on the preceding page by 1.5 (24/16 — 1.5). This will

give us the expected frequencies of highs and lows for 24 sets of four measurements as shown in the third line of Table 5. We must not expect that these theoretical frequencies are going to turn up exactly every time. You can try tossing four

coins 24 times and recording what you get. There will be small

departures from theory, but you may confidently expect that in

most of the 24 trials you will get a mixture of heads and tails

showing on the four coins.

The last two columns in Table 2 are headed by a plus and by

a minus sign. In those columns the individual readings are compared with the average of all the readings, 0.07709, to determine

whether they are above (plus) or below (minus) the average.

Note that girl A had four readings all below the average, so four

is entered in the minus column and zero in the plus column.

Girl B's readings are just the reverse, all four are above the

average. Girl C had two above and two below. We next count

up the frequencies for the various combinations, and find them

to be 6, 3, 4, 4, and 7 respectively. These numbers are entered in

the fourth line of Table 5. When we examine these frequencies a surprising thing confronts us. We find far too many girls with measurements either

all above or all below the average. In fact there are 13 of these

against an expected three. This disparity is far too great to be

accidental. Evidently our assumed model does not fit the facts.

38
Table 5. Five ways to place 24 sets of four measurements with reference

to the average.

High readings

(Above the average)

4 3 2 1 0

LOW rcauingb

(Below the average) nu

ii £ 3

Expected frequency

for 24 measurements

1.5 6 9 6 1.5 Observed frequency 7 4 4 3 6

(from our data)

The hope of complete independence for the readings has not

been realized. It seems that if the first reading was high, subsequent readings also tended to be high. The same holds true if the first reading happened to be low. Evidently many of these

girls had a particular way of measuring that persisted throughout all four measurements. We see that for many of these girls

agreement of the four measurements with each other does not

tell the whole story. All four measurements may be quite high

or quite low. We sometimes say that such individuals are subject

to biases.

Bias—a Major Consideration

Once a scientist or measurement specialist detects or even

suspects that his readings are subject to a bias, he tries to take

steps to locate the bias and to correct his measurement procedure. The goal is to reduce bias as far as possible. Experience

shows that only rarely can biases be completely eliminated. We

can be quite sure in this case that some of the girls have rather

marked biases and this complicates the interpretation of the

data. Nevertheless, since there are nearly as many girls with

plus biases as those with negative biases, the histogram is still reasonably symmetrical.

One way to think about these measurements is to regard the

39
set of four measurements made by any one girl as having a

certain scatter about her own average. Her average may be

higher or lower than the class average; so we may think of the

individual averages for all the girls as having a certain scatter

about the class average. Even this simple measurement of the

paper thickness reveals the complexity and problems of making

useful measurements. A measurement that started out to be

quite simple has, all of a sudden, become quite a complicated

matter, indeed.

One more property of these data should be noted. Table 2

lists the average of the four measurements made by each girl.

There are 23 of these averages (one girl's measurements were

excluded). The largest average is 0.0816 and the smallest is 0.0700. The largest of the measurements, however, was 0.0864

and the smallest was 0.0655. Observe that the averages are not

scattered over as wide a range as the individual measurements.

This is a very important property for averages.

In this chapter we have used data collected in only a few

minutes by a class of girls. Just by looking at the tabulation of

96 values in Table 2 we found that the measurements differed

among themselves. A careful study of the measurements told us quite a lot more. We have learned a concise and convenient way to present the

data, and that a histogram based on the measurements gives a good picture of some of their properties. We also observed that

averages show less scatter than individual measurements. And

most interesting of all, perhaps, we were able to extract from

these data evidence that many of the students had highly personal ways of making the measurement. This is important, for

when we have located shortcoming in our ways of making

measurement we are more likely to be successful in our attempts

to improve our measurement techniques.

40
4. Typical collections

of measurements

In the preceding chapter a careful study was made of 96

measurements of the thickness of paper used in a textbook.

We learned how to condense the large number of measurements

into a few classes with given mid-values. The mid-values to- gether with the number in each class provided a concise sum- mary of the measurements. This information was used to construct a histogram, which is a graphical picture of how the
measurements are distributed around the average value of the

measurements. In this chapter a number of collections of data

will be given together with their histograms. We are going to

look for some common pattern in the way measurements are distributed about the average.

The first example concerns 100 measurements to determine

the amount of magnesium in different parts of a long rod of

magnesium alloy. Chemists find it convenient to have on hand

specimens of an alloy of known composition. Such specimens

make it easy for the chemist to calibrate the equipment used in

the analysis of metals and make sure that it is working properly.

In this example, an ingot of magnesium alloy was drawn into a rod about 100 meters long and with a square cross section

about 4.5 centimeters on a side. The long rod was cut into 100

bars, each a meter long. Five of the bars were selected at random and a flat test piece about 1.2 centimeters thick was

cut from each. These served as test specimens.

Table 6. Duplicate determinations of magnesium at 50 test points

position bar 1 bar 5 bar 20 bar 50 bar 85

12

34

5

6

7

8

910

0.076

0.071 0.070

0.067

0.071 0.065 0.067

0.071 0.066

0.068

0.067

0.066 0.065

0.066

0.065

0.067

0.067

0.067

0.063 0.068

0.069

0.071 0.068

0.071 0.066 0.068

0.071 0.069

0.070

0.068

0.066

0.062

0.066

0.066 0.065

0.066

0.067 0.067

0.065

0.068

0.073

0.069 0.068

0.069

0.070

0.070 0.065

0.067

0.067 0.064

0.070

0.068

0.066

0.067 0.068

0.065 0.068

0.069

0.073 0.068

0.073 0.075

0.069

0.072

0.069 0.069

0.072

0.063 0.069

0.069

0.063 0.069

0.067 0.068

0.066

0.065

0.082

0.063 0.066

0.066

0.070

0.066

0.068

0.068 0.064

0.070

0.069

0.067

0.069

0.067

0.069 0.064

0.063 0.064

0.067

0.065 0.064

0.064

0.068

0.069 On each of the five specimens ten test points were located in

the pattern shown in Figure 3. This gave 50 spots in all. Two

determinations of the magnesium content were made at each

spot.

The total collection of 100 determinations is shown in Table 6. The determinations range from 0.062 to 0.082 per cent of mag42
(D / ©

) >9l Figure 3. Pattern for locating

test points on cross section of

alloy bar.

nesium. One unit in the last place was used as the interval for drawing the histogram in Figure 4. To avoid

crowding the scale the intervals are labeled 62, 63, . . . instead of 0.062, 0.063. . . . On the histogram the lone

high reading of 0.082 shows

up like a sort thumb. How

can we account for it? Perhaps the analyst misread his instrument. That

seems more likely than to assume the existence of a single isolated spot much higher in

magnesium than the 49 other spots. The safest guide to choosing

between these alternatives would be to repeat that analysis.

In fact a duplicate analysis of that spot was made and gave the

value 0.072. The duplicates differ by 0.010.

We may get some more help in this situation by examining

the other 49 differences between the duplicates. The analyst ran all 50 spots once and then made a set of repeat determinations. When the results of the second set are subtracted from

the first results as shown in Table 7, an interesting state of affairs

is revealed. Plus differences predominate. There are 40 plus

differences and only ten negative differences. As a rule, the

entire second set seems to be lower than the first set. One might assume that under normal conditions there would be no reason

to expect the second measurement on a spot to be smaller than

the first one. It would be more reasonable to expect that it would

be a toss up as to which would be the larger, the first result or the second. Again it is like tossing a coin 50 times and observing

the number of heads obtained. Theory predicts that we should

expect to get close to 25 heads.

A surplus or a deficit of seven heads would be rather rare. 43
Figure 4. Histogram of 100 analyses for magnesium

15 -

10 -

5 - D

Scale values are in thousandths of one per cent

EL JZL

62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 A surplus or deficit of ten heads would be most unusual. A

surplus of 15 heads would make most of us very suspicious about

the fairness of the coin or the improper skill of the one tossing it. Thus we cannot ascribe this large preponderance of plus differ- ences to mere chance. But how can we account for it? First of

all, we note that when the second measurement is subtracted

from the first measurement the average difference, taking ac- count of the sign, is +0.0022. This together with the fact that the

first entry for each pair was run as a group supplies the clue to

the mystery.

Evidently something happened to the apparatus between

doing the first group of 50 and the second 50 determinations.

Apparently between the first and second runs there was some

small shift in the apparatus or environment and this introduced

a bias. As a result the apparatus began to give lower results.

The surprising fact is that the suspect high value of 0.082 was

made in the second run and is larger, not smaller, than its com44
Table 7. Difference between duplicate determinations of magnesium.

First duplicate minus the second duplicate. Entries show the number

of plus and minus differences. The four zero differences have been

divided between plus and minus.

difference plus minus

0.000 2 2

0.001 5 —

0.002 8 3

0.003 7 2

0.004 5 1 0.005 8

0.006 2 1 0.007

0.008

0.009 2

0.010 1 1*

Total 40 10

*The asterisk identifies the difference associated with the suspect measurement

0.082. panion first determination of 0.072. This large difference is ten

units off in the wrong direction from the +2.2 average differ- ence. (We multiplied the average difference by 1000 to drop

the zeros. ) The 50 differences listed in Table 7 are exhibited as the histogram at the top of Figure 5. The large difference of ten

units of the wrong sign is crosshatched. It stands apart and

furthest removed from the average difference of +2.2. It should

be evident by this time that not only is the 0.082 an out-of-line

high value but also that it is responsible for the largest difference

of any of the 50 pairs — and of the wrong sign. Surely these facts

justify the deletion of this result. The single first determination

is left to represent the seventh test point on bar No. 50.

Before you conclude that we are being fussy about a small

error, remember we are showing that mistakes do occur. Sometimes mistakes tip the scales of judgment one way or the other.

We are all prone to lapses in technique. Therefore we need to be prepared to search for questionable values.

45
Figure 5. Upper histogram shows distribution of the 50 differences

between the duplicates shown in Table 7. Lower histogram shows distribution of the 50 differences found between

randomly selected the pairs drawn from Table 6. See also Table 8. n DJh

E

-10 -8 -6 -4 -2 0 2 4 6 8 10

Units in last place

8 — n i .10 -8 -6 -4 -2 0 2 4 6 8 10

Histograms and the Error of Measurement

The analysts doing this work very wisely planned to do the

determinations in two complete groups. Suppose that the first group of 50 determinations had consisted of duplicate analyses

of the first five test points on each bar. The consequence would

have been to make test points six through ten appear to be lower

in magnesium than test points one through five. Thus there

would have been real doubt about the homogeneity of the

alloy sample. However, the plan used by the analysts kept the

comparison between test points fair. Each of the 50 test spots is represented in the first 50 determinations and again in the

second 50.

46
We still have not answered the question in the mind of the

experimenters who made these determinations. We want to

know if the stock of material, as represented by these 50 spots

actually tested, is acceptably uniform to serve as a standard for

testing other samples of magnesium alloy. To answer this question we must know whether the differences between spots can be ascribed to the measurement error or whether they represent

real differences in magnesium content. If the values of differeiices found between determinations on different spots are

similar to the values of the differences found between duplicates

on the same spot, we would be satisfied with the uniformity.

There may be minute differences in concentration of magnesium at the various test spots. Clearly they are not important if the variation in concentration is considerably less than the error of measurement, i.e., the error inherent in the technique.

The question is, how do we determine which of these alternatives is the case? A direct test may be made in the following

way. Write the numbers 1 to 50 on 50 cards and shuffle them

well. Cut the deck of cards in two places and read the two

numbers. These correspond with a pair of test spots. Copy one

determination for each number, but make sure that both are

either first or second determinations. Can you explain why?

Repeat this process 50 times. How many possible pairs are there

to choose from?

Suppose you cut the cards and turn up numbers 19 and 33.

Look back to Table 6 and read the values opposite these numbers, making sure you take readings from the same set of 50.

If you select the first run of 50 (the five left-hand columns),

the values would be 0.070 (for number 19) and 0.069 (for

number 33 ) . Subtract the second value from the first to obtain

a difference of +0.001. Replace the cards, shuffle, and cut again.

This time you cut numbers 30 and 46. Still using the first set of 50 determinations, you find the values 0.064 and 0.070. This

time the difference is —0.006. Continue until you have 50 differences in any pattern. If duplicates turn up, cut again.

47
Table 8. Differences between fifty pairs of determinations picked at random from fifty spots subject to the condition that both members of

the pair are from the same group of fifty. Difference is equal to the first member of the pair minus the second member of the pair.

difference plus minus

0.000 1 1 0.001 9 7

0.002 4 4

0.003 2 4

0.004 3 2

0.005 5 3

0.006 2

0.007 2

0.008

0.009

0.010 1 Total 26 24

When this game was actually tried, it produced the results

shown in Table 8. These results are represented in the histogram

at the bottom of Figure 5. If we compare the two histograms,

we see that they are spread over nearly the same width and are remarkably similar in form. The histogram for the duplicates is displaced to the right 2.2 units as a consequence of the shift in

readings between the first and second groups of measurements.

This shift does not change the form or width of the diagram.

The displacement in the top histogram exists because the

average for the first set of 50 results is 2.2 units higher than the

average for the second set of 50 results. The duplicate spots

selected for the bottom histogram were alwarys chosen from the

same 50 results (either the first 50 or the second 50) so if we

take the sign into account, the average difference should be zero. You may verify this statement by subtracting 2.2 units from

each of the 50 differences listed in Table 7 and making a third

histogram which shows the differences corrected for the shift. So, regardless of the shift, the width of the top histogram truly

48
represents the error of the measurement. Now, the plan of work

has eliminated the shift error from the comparison of test points

with the result that the shift error can also be excluded.

Duplicate determinations were made on the same spot so there could have been no actual change in magnesium concentration, only changes as a result of measurement error. The differences arising from comparing different parts of the bars are shown in the lower histogram of Figure 5. This histogram is about the same width as the top histogram for duplicates. If, on

the other hand, there had been a marked variation in magnesium concentration in the rod, the differences between determinations from different locations would have exceeded the duplicate differences. Hence we see that since the determinations

made on different parts of the bar agree about as well as repeat

determinations made on one spot, we conclude, therefore, that

the bar is homogeneous enough for our purpose.

The experiment just discussed is comparatively simple as experiments go. Nevertheless the interpretation of the data

required a good deal of care. What have we learned that will

guide us in future measurements? First, we see that no matter

what we measure or how we measure it, we will always require a knowledge of the measurement error itself. We have learned

that shifts in the apparatus may occur and that we can protect

the experiment from such shifts by a proper plan of work.

Visible, too, is the general similarity in shape between the histogram for the magnesium analyses and for the measurements on

paper thickness. We also devised quite a useful method of

testing whether the concentration of magnesium varied more

from test point to test point than could be accounted for by the

error in the measurement itself. This procedure can be adapted

to a large variety of measurements.

The similarity noticed in the general shape of the histograms

for paper measurements and for spectrographs chemical analyses was not accidental. Whether the measurements are approximate or very precise the shape persists. Nor does it matter

49
whether the measurements are in the fields of physics, chemistry, biology, or engineering

.

The histogram in Figure 6 was made from the results obtained

3200- 3400- 3600- 3800- 4000- 4200- 4400- 4600- 4800- 5000- 3399 3599 3799 3999 4199 4399 4599 4799 4999 5199 Figure 6. Histogram for cement tests reported by 102 laboratories.

-15 -12 -9 -6 -3 0 3 6 9 12 15 18 Figure 7. Histogram showing cubic inches of error in 923 gasoline

pumps tested by inspectors.
when 102 laboratories tested samples of the same batch of

cement. This was done to track down the source of disagreement between tests made by different laboratories. From the

histogram made from the data it is clear that a few laboratories

were chiefly responsible for the extremely high and low results.

All states and many cities provide for the regular inspection

of gasoline pumps to ensure that the amount of gasoline delivered stays within the legal tolerance for five gallons. From these

tests a large amount of data becomes available. Remember that

the manufacturer first adjusts the pump so that it will pass

inspection. Naturally the station owner does not want to deliver

more gasoline than he is paid for. A small loss on each transaction over a year could be disastrous. However, the pump itself

cannot be set without error; nor can the inspector who checks

the pump make measurements without error. The scatter of the results exhibited by the histogram in

Figure 7 reflects the combined uncertainty in setting and checking the pump. In this group of 923 pumps only 40 had an error greater than one per cent of the number of cubic inches ( 1155)

in five gallons. This was the amount pumped out for these tests. Some extremely fine measurements are displayed as histograms in Figure 8. These have to do with a redetermination of

the gravitational constant g

( the acceleration due to gravity

)

in Canada. The procedure involves timing the fall of a steel bar

and measuring the speed with which it falls. To time the bar's

fall, a light beam and photoelectric cells are used. As the bar

drops, the light beam is reflected from precisely placed mirrors

on the bar and signals the photoelectric cells. These precisely

timed signals measure with great accuracy how fast the steel

bar is falling. Thus the acceleration due to gravity is calculated.

The scientists, in the course of exploring possible sources of error in their work, made 32 measurements with each of two

different bars. Inspection of the histograms shows a good deal of

overlap of the results with the two bars. The average for bar

number 1 is 980.6139 cm./sec. 2

, and for bar number 2 is 980.6124

51
Figure 8. Two groups of 32 measurements of the acceleration due

to gravity at Ottawa, Canada.

10

10

Measurements made with

bar No. 2, December 1959 n05 10 1 15 20 25 30 35 40 45 50 55 60

Measurements made with

bar No. 1, August 1958

1 1 1 05 10 15 20 25 30 35 40 45 50 55 60

Mid-values of intervals are 3rd and 4th place decimals to be added to 980.6100 cm/sec. cm./sec. 2

. The difference between these averages is about 15

parts in almost ten million. The bars agree to almost one part

in a million. Small though this difference is, the evidence leads

to the conclusion that there is a real difference between the

results with the two bars. The typical shape of the histogram

appears again in these superb measurements.

Your author, in his younger days, was a research chemist in a

biological research institute. The experiments carried out there

often required substantial collections of plants. But even when

plants are started at the same time, in the same soil, and grown

on the same bench in the same greenhouse, they vary a great

deal in their growth. The biologists always asked the greenhouse

staff to start more plants than would be needed. When an

experiment was started, the biologist would pick out a supply of

52
plants with very closely the

same growth. The extremely

small and large plants would be

discarded. Usually some plants

were used for controls and

others given various experimental treatments. It was an

advantage to begin each experiment with a stock of fairly uniform plants, so that control and

treated plants would start off even.

The Normal Law of Error In experiments of this type

great care must be taken to

avoid personal bias in selecting

experimental and control plants.

If extra-good plants are unconsciously assigned to a particular

treatment the experiment cer- tainly will be biased.

Suppose there are 50 plants to

be divided into five groups of

ten plants. One group will be

used as controls; the other four

groups are to be assigned to

treatments 1, 2, 3, and 4. How

shall the groups be formed? Prepare a deck of 50 cards. Mark C

on ten of the cards, mark 1 on

ten more, and so on. Shuffle the

deck thoroughly. Arrange the 50

plants in a row and deal out
—3a —2a —ia o la 2j 3a

the deck of cards as you move down the row. Chance, not

unconscious personal bias, determines the outcome. There are many variations of this technique of assigning the plants to the control and treatment groups. Unless precautions are taken

at this stage, the most searching study of the data may still lead to incorrect conclusions. We will never know for sure whether the performance of the best group is really the result

of the experimental treatment, or because it was favored in

the assignment of the plants.

One day, just before the stock of plants was about to be culled

of the very small and very large ones, a simple experiment was

tried. The smallest plant in the lot was singled out and placed at the left end of the bench. Then the largest plant was placed at the opposite right end. There was room for 15 plants between.

A selection of 15 plants was made that provided a regular in- crease in size from the smallest to the largest plant. This took a

little while, but finally a satisfactory "scale" was established

along the edge of the bench. A summer high school assistant

was assigned the task of "matching up" the remaining plants

54
with the 17 plants used to establish a scale of size. The pots

were placed so that lines of plants of the same size extended

back toward the center of the bench. When the sorting was over, what do you think the arrangement of the plants looked like?

A picture of the arrangement is shown as Figure 9. The assistant, who had never heard of histograms, had arranged the plants

in the familiar shape we have been finding for our histograms of

paper measurements, magnesium determinations, and measure- ments of g. If we drew a curved line over the histograms that we have

examined so far, we would find that the curves would be symmetrical about their mid-points and have long tails on either

side. A curve of this type is shown in Figure 10. The equation

which determines the height of this curve and its extent on

either side of the mid-point is known as the normal law of error. Only two constants are needed to determine the height and

width of this curve. These two quantities, which make it possible to construct a curve representing a histogram, can be calculated from any collection of data. We seldom go to the trouble

of actually constructing the curve. Generally, we make direct use of the two constants which, together with statistical tables, are sufficient to interpret most collections of data. The equation

for the constants is one of the great discoveries of science. Its importance in bringing meaning to collections of observations can hardly be overestimated.

Your author has a small printing press for a hobby. He set in

type his opinion of the importance of the normal law of error.

THE NORMAL

LAW OF ERROR

STANDS OUT IN THE EXPERIENCE OF MANKIND

AS ONE OF THE BROADEST

GENERALIZATIONS OF NATURAL

PHILOSOPHY IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES

IN THE PHYSICAL AND SOCIAL SCIENCES AND

IN MEDICINE AGRICULTURE AND ENGINEERING

IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE

INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION AND EXPERIMENT
5. Mathematics

of measurement

By

this time we should be familiar with the construction and

appearance of histograms. We now follow up the idea that a particular type of equation can be written for the curve that can be made to fit these histograms. This equation, called the

normal law of error, is an exponential one of the form

i C*-/*) 2
Do not let the formidable appearance of this equation

alarm you; y and x are simply the y axis and x axis coordinates.

You are already familiar with the constant, ^, the ratio of the

circumference of a circle to its diameter. The constant e with

a value of 2.7183 is the base of Napierian, or natural logarithms.

The other constants, (mu) and (sigma) depend on the

experiment. For a given experiment ^ and a are fixed numbers,

but their values are generally not known. In fact, the main

purpose of an experiment is often to find out what these values

really are. In that case the data from the experiment is used

to provide estimates of their value.

Estimates of these two quantities are given the symbols m

and $. Together with the mathematical tables for the values

of x and y based on the equation above, these estimates are used universally both for routine measurements and the interpretation of research data.

One of the histograms used to represent the data obtained

in the study of the acceleration due to gravity is shown in

Figure 11. Superimposed on the histogram is the graph of the

normal law of error constructed or "fitted" to the histogram

by using the estimates of and a calculated from the data.

I hope you will agree that the curve is a good approximation

to the outline of the histogram. The successful fitting of the

normal error curve to the data using just two numbers justifies

finding what these two numbers are and how to calculate them.

We know how to calculate one of them. This is the estimate, m,

which is our old friend the arithmetic average of the data under

a different name. This estimate is about as close as we can come to finding the true value of the constant n in the question.

Recall the measurements you made on the thickness of paper.

It seems reasonable, does it not, to assume that there is a real

but unknown value for the thickness of the paper? The value

of the constant is a quantity that is made up of the unknown

true thickness of the paper plus any biases arising from using an imperfect scale in an individual manner. The distinction may

57
Figure 11. Normal curve

of error fitted to histogram of measurements

made in a study of the

gravitation constant.

seem unnecessary but it is important to understand that n is not the "true" thickness of the paper. Rather, n is the value

that the average, m, tends to approximate more and more

closely as the number of measurements is increased. Of course,

we hope that /x is very close to the true value. Nevertheless ft is utterly dependent on the error of the scale and on any bias

in using the scale. We found that two individuals — each using

his own scale — obtained averages that disagree substantially no matter how many measurements are made. Each individual

apparently has his own ti. The true value for the thickness is indeed an elusive quantity.

The Estimate of Sigma

The second quantity calculated from the data is an estimate

of <r, the standard deviation. This estimate of a is given the

symbol s; it determines the width of the normal error curve. Turn back to Figure 10 and examine the curve sketched there.

You will observe that if we drew a vertical line through the

highest point on the curve, the curve would be symmetrical

about it. The vertical line or the high point of the curve represents the value of the average for the data. It is customary to use this vertical line as a reference point for marking off multiples of the standard deviation to the right and to the left. You will also see that when we have proceeded as much as 58
three standard deviations on either side of the center line, the curve has dropped to about one per cent of its height at the

center. Table 9 gives the ordinates ( y values ) for the curve at k multiples of the standard deviation on either side of the mean. The table also gives the fraction of the area inclosed by any

chosen pair of ordinates. Suppose that we take the location of

the average to be the zero point, and we erect an ordinate at —1 standard deviation and another at +1 standard deviation.

By referring to Table 9 we see that 68.27 per cent of the total area under the curve is included between these two ordinates.

Ordinates erected at plus two standard deviations and minus

two standard deviations include 95.45 per cent of the area. Similarly, ordinates at 2.57 standard deviations inclose 99 per

cent of the area. We have observed how the curve approximates the outline of a histogram. Histograms are based on the

number of measurements found in the chosen intervals. So we

may use the above percentages (and others taken from Table

9) to obtain an indication of the expected per cent of the measTable 9. Ordinates and areas for the normal curve of error

x values per cent of area given in y values included between

multiples, k, given in multiples ordinates

of the of at - h

standard deviation 1/standard deviation and + (tj ± 0.00 0.3989 0.0000

0.25 0.3867 0.1974

0.50 0.3521 0.3829

0.75 0.3011 0.5467

1.00 0.2420 0.6827

1.25 0.1826 0.7887

1.50 0.1295 0.8664

1.75 0.0863 0.9199

2.00 0.0540 0.9545

2.25 0.0317 0.9756

2.50 0.0175 0.9876

2.75 0.0091 0.9940

3.00 0.0044 0.9973
urements that will fall within any given interval on the horizontal scale expressed in terms of the standard deviation.

Calculating s In order to express an interval on the horizontal scale in

terms of $, the estimate of we must first calculate the value

of s in terms of the units actually employed for the measurements. The formula for s is:V n- 1 Capital sigma, 5, indicates a sum — in this case — of d

2

, the

squares of differences obtained by subtracting the average m

for each measurement. Suppose we represent n measurements

by die symbols %u *2> *3, «> .

. , xn . Then the average is found by

adding the measurements and dividing the sum by n. xx +x2

- + —+xn Average = m = — =

n n

Finding s is simply a matter of substitution, as shown in Table 10.

But, in spite of the simple arithmetic, we are leading up to a truly remarkable generalization. We claim that in most sets of measurements, and without regard to what it is that is being

measured, about two out of three measurements will differ

from the average by less than one standard deviation. Similarly,

about 19 measurements out of 20 will be within two standard

deviations of the average. Only one measurement in one hundred can be expected to depart from the average by more than

2.57 standard deviations. Furthermore, these statements apply

to the most varied sorts of measurements, whether they are precise or approximate. This property of the normal error curve

is of great value for * v

e interpretation of data.

60
Table 10. Calculating s, the estimate of the standard deviation measurement average difference square of difference *i m x 1 — m = d«| x2 m X2 — m — c/2 m x3 — m = d3

• • • • Xn m Xn m — dn d„2

Sum of squared differences = 2d2

We will illustrate the calculation of the standard deviation

by using five made-up measurements.

measurement 27

26

23

19

15

measurement

minus average 5

4

1 -3

-7

square of difference from average 25

16

19

49

110 0 100

Average 22

Now we can substitute the values in the equation and solve

for s. The sum of the squares of the deviations from the average

is 100. Divide this sum by one less than the number of meas- urements (5 — 1=4) and obtain 25 as the quotient. The square

root of 25 gives 5 as s, the estimate of the standard deviation.

The differences between each of the measurements and their

average can now be expressed in multiples of s. All we do is divide by 5, i.e., by the estimate of the standard deviation.

These differences become 1.0s, 0.8s, 0.2s, —0.6s, and —1.4s.

61
Three of these five differences are less than one standard deviation and that is as good a check on the theoretical 68.27 per

cent as can be obtained with so few measurements.

We cannot say with confidence that the actual value of a

is close to the estimate five when so few measurements have

been used to calculate s. In order to get a really satisfactory

estimate of s we like to have at least 30 measurements, but there are many experiments in which it is not possible to get this

many. In dealing with problems when the measurements are few in number, however, a special allowance has to be made

for the uncertainty of our estimate of s. It is not actually necessary to fit the cur ve to the data. Recall

the 95 measurements made on the thi ckness of paper used in a book. If m and s are determined for this collection, the ex- pected number of measurements in each interval can be calFigure 12. Solid line outlines actual histogram, dotted

line outlines calculated histogram for measurements on

the thickness of paper.

25 —
.culated directly. These calculated values are shown in Figure

12 as a dotted-outline histogram superimposed over the actual

histogram. The predicted values obtained by calculation con- form fairly closely to the counts made on the measurements.

Thus we see that either the curve or the histogram may be

computed by using only two estimates calculated from the

original data or from the data grouped into intervals. On the

other hand, having s and m we can get a very good picture of

the measurements without doing either.

Consider carefully what we have achieved in the way of

condensing many measurements to a more useful form. First,

the 95 measurements were sorted into eleven intervals. The

mid-values of these intervals, together with the number of

measurements in each interval, replaced the 95 measurements.

This procedure in itself often makes for a very substantial

reduction in the space required to report the data. The data

may be reported either graphically as a histogram, or in brief

tabular form of two columns; one column listing the mid-values

of the intervals, the other showing the number of measurements

for each mid-value. As a second step, the histogram or con- densed table can be replaced by just two numbers, m and s, which permit us to reconstruct the original histogram, although

the reconstruction is not perfect. The advantage is that the

entire collection of 95 measurements has been reduced to just

two numbers which convey the information spread over die

95 measurements.

We may summarize the above remarks by saying that the

standard deviation is a direct measure of the variation exhibited

by the measurements. Earlier we used the range as an indication of the spread from the largest to the smallest result. Why

abandon such a simple quantity as the range in favor of the

standard deviation which required more work to compute?

As we pointed out at the time, to a considerable extent the

range also depends on the number of measurements in the

collection. Study the following two statements about two oper63
ators, each making 50 measurements of the same kind. Operator

A, who makes his measurements in sets of ten, has an average

range of 18 units, while Operator B, making sets of four measurements, has an average range of 14 units. Which one has

the measurements with the wider histogram? Let's consider

the problem for a moment. We see that the sets of ten are more likely to include extreme values than the sets of four.

Thus if Operator B had selected sets of ten measurements, instead of four, his average range would have been increased by

50 per cent. It would have been 21, not 14, units and he would

have the wider histogram. If the range is used — and it sometimes is for quick work — comparisons will be misleading unless

the size of the collection is kept constant.

Using the Standard Deviation

The standard deviation does not suffer from the limitation

just mentioned for the range, the number of measurements in

the collection being automatically allowed for in the formula.

Furthermore, the standard deviation uses all the measurements,

while the range uses only the two extreme results.

The standard deviation — our estimate of sigma — is a very

useful number indeed. First of all, it must never be forgotten

that it describes the scatter of individual measurements around

the average. Suppose our collection consists of 96 measurements.

We might divide the 96 values by lot into 24 sets, each with four

measurements. Take the average for each set of four measurements. We now have 24 averages. If these averages were given

to someone without his being told that they were averages, he

might calculate a standard deviation for them.

Can we predict fairly closely the result of this calculation,

knowing the value for s that we obtained from the 96 individual

readings? The answer is that we can, and by a simple operation.

We just divide our estimate, s, for the individual measurements

by the square root of four, the number of measurements used in

64
Table 11. Form for calculating the standard deviation. Data taken from

Table 2

1m1 iri-va 1 1 IP 1 IU VO 1 uc mid-value number product: square product:

of interval of interval in difference of the (diffja (from Table 2) minus the the by differ- by

times 10* average interval number ence number

d f f X a

_I2 o t X a

659.5 -112 1 -112 12544 12544

679.5 - 92 2 -184 8464 16928

699.5 - 72 0 0 5184 0

719.5 - 52 10 -520 2704 27040

739.5 - 32 12 -384 1024 12288

759.5- - 12 18 -216 144 2592

779.5 8 24 192 64 1536

/yy.o CO 1 A HQ A loy/o

819.5 48 8 384 2304 18432

839.5 68 4 272 4624 18496

859.5 88 2 176 7744 15488

Totals 95 0 136320

= ^/ 1450 = 38*

*Carry square root extraction to two figures.

Note:The sum of the fourth column should be zero. This provides us with a check

on the values of d. each average. Since the square root of four is two, we see that

the 24 averages will be spread over about one half the range of

values found for the individual measurements. More generally,

the average of n measurements will be assigned a standard

deviation equal to s divided by the \/~n. Often the standard

deviation of an average is given the label standard error. The examples given in the last two chapters showed that no

matter how carefully a measurement is repeated, the results

obtained in a series of measurements are spread over a range of

values. The actual width of the distribution of the measurements

65
varies with different types of measurement, with the care used,

and with the quality of the equipment. Intuitively, we conclude

that the width of this distribution gives us some clue as to how

closely we may have approached the correct value for

Clearly if the measurements stay within a narrow band of

values we feel more confidence in our technique of measure- ment than when the measurements are distributed over a wide

range of values. Suppose that a class of seventh grade students

made some measurements on the thickness of paper and calculated the standard deviation for their collection of measurements. Also, suppose a senior class in high school made measure- ments on the thickness of the same paper and calculated thenstandard deviation. Which class do you think might have the

smaller standard deviation? It seems plausible that additional

maturity and experience would enable the senior class to make

more precise measurements. The histogram for the senior class

might be narrower than the one made by the junior class. The

magnitude of the standard deviation for each class provides us with mathematical measurement for comparing the two histograms. In fact, if we use the standard deviation, we need not

construct histograms to compare the two sets of measurements.

Finding an s for Our Measurements

Now let us undertake to calculate the estimate <jf the standard

deviation of the collection of paper measurements given in

Chapter 3. By sorting the actual measurements into eleven

classes in Table 3, we have already greatly simplified the number work of finding the average. Sorting into classes makes an

even greater saving in arithmetic when calculating s. The arithmetic is shown in Table 11. This short cut gives a numerical

result for s that is slightly different from the one obtained by

using ungrouped data. The slight difference, however, is of no

consequence. In Table 11 the mid-values and the average taken

from Table 2 are temporarily multiplied by 10,000. This serves 66
to get rid of the decimals and a lot of zeros right after the

decimal point. When we get our answer we simply divide it by

10,000 to put the decimal point back where it belongs. Thus our

answer, 38, is really .0038. This temporary multiplying by 10,000

greatly reduces the chance of numerical error. The arithmetic in Table 11 is a bit time consuming but not

difficult. The average, 771.5, is subtracted in turn from the

eleven mid-values. The difference is squared and multiplied by

/, the number of measurements in the interval. The sum of the

squares of these differences is 136320. After substituting in the

formula for the standard deviation, this leads to a value for 5 of

38 or 0.0038 after putting back the decimal point. Remember

that s is not <r, the standard deviation, but only the best estimate

we can obtain from these data. However, from here to the end

of the book we will use the terms s and a interchangeably.

You are now in a position to test an earlier claim made for

the standard deviation — that about two out of three measure- ments will differ from the average by less than one standard

deviation. We found that the 95 measurements of paper thick- ness had an average value of 0.07715. Now add to and subtract

from the average the value 0.0038 which we found for the

standard deviation. The result is a lower limit of 0.07335 and

an upper limit of 0.08095. These two limits are just one standard deviation away from the average. If you now turn back to

Table 2 in Chapter 3, you may count up all the measurements

that fall between these two limits. The number of individual

values between these limits is 66. Thus 69.5 per cent of the

95 measurements fall between these limits, and this checks in a very satisfactory manner the theoretical percentage of 68.3

(Table 9). Two standard deviations amount to 0.0076. When

this quantity is added to and subtracted from the average, we

obtain the upper limit, 0.08475 and the lower limit, 0.06955.

The per cent of the measurements expected to fall within these

limits is 95.4. Therefore we should expect 4.6 per cent of the 95

measurements to be outside these limits. This gives 4.4 measure-67
ments. Examination shows that just five measurements in the

collection are beyond these 2<j limits.

Although this collection of measurements did not quite fulfill the requirements of complete independence, we find that the

standard deviation can be used to predict the number of meas- urements included in any chosen interval centered on the

average.

We take data in the hope that we can get the answer to a

scientific question. We want to know whether or not our data

provide a satisfactory answer to the scientific question we had

in mind. Suppose that someone were doing a project that involved the effect of plant hormones on plant growth. One of

the things he might want to know is, how large are the leaves

of the treated plants? If it seems worthwhile answering the

question, measurements will be made in an attempt to get a

satisfactory answer. Of course, one can simply take the average

of the measurements and report this. But in general, such an answer is not adequate in scientific investigations.

Student's t At the very least, it would seem, we should think about what

might happen if we repeated the set of measurements. Suppose

this were your project and you did repeat the set and got a second average. By this time you are prepared, I trust, to find

that there would be some difference between the two averages.

What should we do in such a case? You may reply that you would

report a grand average based on the two averages. But you

should not conceal the fact that there was a difference between

the two averages. The simple fact is that if only the first set of

measurements had been taken, a repeat of the work will give a

different answer. The essential question is "How different?"

It appears that it is not quite enough just to report averages.

Something is missing. We would like to make a statement, if possible, that would give some idea of how close our estimate m

68
has come to the value It would be nice if we could say that our average m does not differ from /x by more than some small

amount that we shall call A, the Greek letter delta. Now we

can not say this and be absolutely sure of being right. We can

pick a A large enough so that we may have a fairly high degree

of confidence that our statement is correct. However, we would

like to keep A small and still keep our confidence high.

The statement we make has to be based upon the data we

have obtained. The indispensable element in making any state- ment of confidence in our average is s, our estimate of the standard deviation. We have already seen that tables based on the

normal law of error make it possible to make statements about

the per cent of measurements that fall in an interval centered on the average. Our problem now is somewhat different but

closely related. The estimate of v tells us how much the individual measurements deviate from the average. Our present problem is to make a statement about the average and its closeness

to The more measurements in the collection, the better the

chance that the average will lie close to m. There is one way to

obtain /x with absolute confidence. That would be to make an

infinite number of measurements. Since we will always have

some limited number of measurements, the chances that aver- ages of small collections coincide with ^ are extremely remote.

So let us now study this problem of making a statement about

our average that will somehow relate the average to

For a long time many investigators did not attempt to make

any statement about the average, particularly if the average was

based on very few measurements. The mathematical solution to

this problem was first discovered by an Irish chemist who wrote

under the pen name of "Student." Student worked for a company that was unwilling to reveal its connection with him lest

its competitors discover that Student's work would also be advantageous to them. It now seems extraordinary that the author

of this classic paper on measurements was not known for more

than twenty years. Eventually it was learned that his real name

69
was William Sealy Gosset (1876-1937).

Suppose we had a collection of n measurements where n may

be as small as two. We know how to calculate the average,

m, and s, the estimate of We now calculate the quantity, A.

s When A is added to and subtracted from the average, this

gives an interval which may or may not include within it the

unknown m for which our average, ra, is an estimate. We are already familiar with all the symbols in the formula for A except

for the multiplier t. This number is known as Student's t and is obtained from tables first published by Student.

Suppose we have only four measurements and we desire to

have a 50-50 chance that the limits determined by our A enclose

/x. Student found that in this case the proper value for t is 0.765.

If we had eight measurements, the proper value for t would be

0.711. Observe that t is smaller with more measurements as is only reasonable. More measurements give a better average and

a better value of s. Suppose we wish to increase our chance from 50 per cent to a higher probability that our limits include If we want to

increase the chance that the interval includes the unknown

we must make the interval wider. To raise the probability to 90 per cent ( nine chances out of ten )

, t must be increased to 2.353. Table 12 is a brief table of t. Inspection shows how the

value of t depends on a number of measurements and the

desired confidence that the interval includes

The first column in Table 12 is headed "Degrees of Freedom"

and not "Number of Measurements." Observe that if we have

only two measurements each measurement differs from their

average by the same amount. If we know one difference, we

know that the other difference must be the same.

If there are three measurements, the three differences from

their average must sum up to zero if we take the sign into

70
account. Consequently if we are told the values for two of the

differences, the third difference is easily found. We can see that

the number of independent differences from the average is one

less than the number of measurements. The number of independent differences is called degrees of freedom.

Table 12. A brief table of t- degrees of probability

freedom

.50 .90 .95 .99

1 1.000 6.314 12.706 63.657

2 .816 2.920 4.303 9.925

3 .765 2.353 3.182 5.841

4 .741 2.132 2.776 4.604

5 .727 2.015 2.571 4.032

6 .718 1.943 2.447 3.707

7 .711 1.895 2.365 3.499

15 .691 1.753 2.131 2.947

30 .683 1.697 2.042 2.750

99 .676 1.660 1.984 2.626

X 674 1.645 1.960 2.576

Bv calculating A we can set limits above and below the aver- age for a set of data with some confidence that n lies within this

interval. The width of the interval enclosed bv these limits

depends on the value we find for s\ on the number of measurements, n, and on the degree of confidence that we wish to have

that these limits enclose p. Naturally every experimenter would

like to have these limits as close to m as possible and still have

high confidence that n is included.

The expression

s A = t \ n shows quantitatively how the limits depend upon the standard

deviation and the number of measurements made.

There are two ways by which an experimenter can narrow

71
these limits. First of all, he may increase the number of measurements. This soon becomes unprofitable because we can see from the formula for A that the effect of increasing the number

of measurements depends on the square root of the number of

measurements. Thus increasing 100 measurements to 200 has a negligible effect. Obviously the most efficient method of nar- rowing the limits enclosed by plus and minus A is to make the

numerator, s, smaller. That is the goal of every experiment —

and the art of measurement.

With a small set of measurements, say four, we may calculate

A for the 90 per cent or even the 99 per cent limits. We have no

way of knowing for sure that these limits actually do include /* in any given case. On the average, i.e., nine times out of ten or 99 times out of 100, such limits will include We will proceed

now to try out this technique and see how well it works.

We return to the 95 measurements made on paper thickness.

Again we simplify matters by multiplying each value for thick- ness by 10,000 to get rid of all the decimal points and zeros. Write these 95 values on 95 cards. After shuffling the cards, deal

out four of them and consider this a set of four measurements

that might have been made. Calculate average m and s for the

four measurements. Then calculate for 50 per cent limits

$ s A = 0.765 ——- = 0.765

V^T 2

Subtract A from the set average and add A to the set average.

We can now say that there is a 50-50 chance that these limits

include

What is a* for this investigation? We really don't know, but

95 is a rather large number and, therefore, the average, m, will

probably be a close approximation to /*. Thus, it is fairly safe to use m for ft in this experiment. The average is 771 and we may

note whether or not the limits we compute do include 771.

Now shuffle the deck again and deal out another four cards

and consider these to be another set of measurements. (Be sure 72
to return the first four cards before you shuffle them. ) Repeat

the computations as before and observe whether or not these

new limits for this second set inclose 771, We may continue

this process to find out if about half the sets have limits which

inclose 771.

You may not want to undertake all this arithmetic, so I have

done it for you for 20 sets of four measurements each. In Table 13

I have tabulated for each set its average, its standard deviation,

and A for 50 per cent limits of confidence, and the limits deterTable 13. Calculation of 50 per cent and 90 per cent limits for twenty

sets of four measurements drawn by lot from the 95 values in Table 2. All values in Table 2 have been multiplied by ten thousand.

set set set A

= 50% A

= 90%

no. ave. S.D. 0.765 s

. limits 2.353 s limits m s vT VT

1 767 7 3 764-770* 8 759-775

2 772 33 13 759-785 39 733-811

3 793 13 5 788-798* 15 778-808*

4 764 14 5 759-769* 16 748-780

5 776 25 10 766-786 29 747-805

6 782 25 10 772-792* 29 753-811

7 780 22 8 772-788* 26 754-806

8 785 26 10 775-795* 31 754-816

9 803 46 18 785-821* 54 749-857

10 774 21 8 766-782 25 749-799

11 784 38 15 769-799 45 739-829

12 753 33 13 740-766* 39 714-792

13 759 20 8 751-767* 24 735-783

14 769 19 7 762-776 22 747-791

15 789 33 13 776-802* 39 750-828

16 794 33 13 781-807* 39 755-833

17 749 57 22 727-771 67 682-816

18 799 23 9 790-808* 27 772-826*

19 774 19 7 767-781 22 752-796

20 762 52 20 742-782 61 701-823

* Values marked with an asterisk did not bracket the grand average of 771. Note: 0.765 and 2.353 are values of t taken from Table 12.

73
mined by A. An asterisk marks those limits which do not include

771. There are twelve sets with asterisks instead pf the expected

10. That is not 50-50, is it? Tossing a coin should give heads or

tails on a 50-50 basis. Still we would not be too surprised to get

as many as 12 heads (or tails) out of 20 tosses. These 20 sets provide a reasonable verification of our claim.

The last two columns of Table 13 show the 90 per cent limits

calculated by using t = 2.353. There are only two asterisks in

this column marking those sets whose limits did not include the

grand average 771. This time we have hit it exactly. Nine times

out of ten the limits inclosed the average that was based on a

large number of measurements.

We have gone a good way beyond merely reporting an average. We can now attach to an average a pair of limits corre- sponding to some chosen confidence that the limits will inclose

/a. These limits reveal the quality of the measurements and guard

us against making undue claims for our average.

Suppose we think of the 95 measurements as constituting one

large set out of many large sets of 95 measurements that might

have been made. It is easy to set up limits around the average

of the 95 measurements. The number of measurements, n, is now 95. Let us select 90 per cent limits. The appropriate value

for t may be taken from the next to the last line of Table 12.

This is the line for 100 measurements, not 95. But there is very

little change in t as n becomes larger. Consequently we may

use the value 1.66, to calculate A as follows:

38

A = 1.66 = 6.5 VW

The value 6.5 is subtracted from and added to 771 to get the

limits 764.5 and 777.5. We may convert these back to millimeter

units by dividing by 10,000 and obtain the values 0.07645 and

0.07775. The interval between these two limits is a bit more

than one thousandth of a millimeter. A statement that the thick74
ness is probably between 0.076 mm. and 0.078 mm. is much

more useful than reporting an average of 0.077 mm. without

any limits to indicate how uncertain the average may be.

The above computations of a were made on the assumption

that the 95 measurements were independent. We found previously that some of the girls appeared to have individual biases.

A more conservative view of these data would be to consider

that we had just 24 measurements, one from each girl. Naturally

this would be the average for each girl. You may find it interesting to calculate limits using the 24 averages and taking n = 24. The limits will be somewhat wider.

There is still another important way the standard deviation

is put to work. Consider two samples of paper that appear to be

of much the same thickness. Are thev? How could vou detect a

difference between them and substantiate your claim? Or, how

could you say that there is no difference in thickness? Using the

technique of measuring a stack of sheets, you could obtain four

measurements for each paper. A set of such measurements is shown in Table 14 together with the numerical operations.

Our purpose is to determine whether these data provide substantial evidence of a difference in thickness between the two

papers. We must remember that even if we find that the differTable 14. Thickness measurements on two papers; the measurements

have been multiplied by 10,000.

paper cliff. square paper Cliff. square A from ave. of cliff. B from ave of diff. 772 -7 49 765 17 289

759 -20 400 750 2 4

795 16 256 724 -24 576

790 11 121 753 5 25

Total 3116 0 826 2992 0 894

Ave. 779 748

Average for A minus average for B = 31 Combined sum of squares of the differences = 826 - 894 = 1720

75
ence between the two papers apparently does not exceed the

error of measurement, we still have not proved the papers are equal in thickness. We could make more measurements on each

paper. This would give more stable averages and thus might

make the evidence for a difference more convincing. On the

other hand, there is no point in doing a lot of work if it appears

that the difference is too small to be of any practical importance.

Inspection of the two sets of measurements in Table 14 shows

that the largest determination of thickness in set B is larger

than the smallest in set A, and that the measurements overlap.

This suggests that the two papers may, indeed, be of equal

thickness. The difference between their averages is 31.

Decisions and Confidence

Our problem now is to determine whether 31 represents a

real difference between the averages, or whether it arises simply

through the errors in measurement.

If the two samples of paper are of equal thickness, the differ- ence between them would be zero. One way to solve our problem would be to calculate a limit, A, with some chosen degree

of confidence and see whether zero is included in the range

between 31 plus A and 31 minus A. The two sets of measurements have been made by the same

observer using the same equipment, and therefore should have

the same or. Each set provides an estimate, s, of <j based on four

measurements. We will combine the two individual estimates

of <r into one estimate in the following manner. Add together the two sums of squares of the differences to

get 1720. Divide this sum by 6. And where does the 6 come

from? If we were determining s for each set, our divisor would

be 3, (n — 1). Since we are looking for s of the combined

measurements we use the sum of the divisors which is 6. So we

have V 1720 / 6 = 17. This gives a combined estimate of the

standard deviation for this method of measurement. The prob76
lem is to find the correct s for the difference between two aver- ages of four measurements. Differences increase the s by the

factor VIZ", and averages reduce s by l/v^n where n is the

number of measurements in each average. The s to use for A is VT y/1

We will use this number (already divided by V"n) to set up

limits around the observed difference of 31.

What value of t shall we use? We need values of t for six

degrees of freedom, the divisor in our estimate of s. That is, there are three degrees of freedom from each set of four measurements, making a total of six of freedom altogether. So we

turn back to Table 12 and select values for t with six degrees

of freedom for the 90, 95, and 99 per cent probabilities.

prob. 90% 95% 99%

t 1.943 2.447 3.707

A = s x t 23.3 29.4 44.5

Upper limit 54.3 60.4 75.5

Lower limit 7.7 1.6 -13.5

The limits are obtained by adding s X t to 31 and subtracting

s X * from 31.

If we pick a probability of 95 per cent we will use a value

for * that will, 95 times out of 100, give limits that include the

true value of the difference. In this case we use t = 2.447 and

get the calculated limits around the average of 60.4 and 1.6. We notice that the range 1.6 to 60.4 does not include zero, i.e., no difference between the papers. Thus we can conclude, with

95 per cent confidence, that there is a real difference in thickness

between the papers. Strictly speaking, what we have shown is that there is only a small probability (five per cent) that we

would obtain a difference as large or larger than 31 if the papers

were, in fact, of equal thickness. Consequently we would give

up the assumption of equal thickness.

Another investigator might be more cautious about claiming

77
to have shown a difference between the papers. He elects to

work at the more conservative level of 99 per cent probability,

In this event his report indicates limits from —13.5 to 75.5.

Zero is a possible value between —13.5 and 75.5, so he is unwilling to report that there is a difference at the 99 per cent

level of confidence. The choice lies with the investigator, and

the importance of the decision greatly influences his choice of a level of confidence. Additional measurements pay off in reducing the standard deviation of the difference between the averages.

There is an extremely important comment to make about this

comparison of the thickness of the papers. The difference in

thickness is not influenced by any bias that may happen to

afflict all the measurements. Suppose there was a bias +B in

the values obtained. This bias will appear in both averages,

making each average too large by +B. Consequently the differ- ence between the averages is just what would have been found

if the measurements had no bias at all. Comparative measure- ments have this enormous advantage over absolute measure- ments such as the determination of the gravitation constant, g. Indeed, it is possible to measure and compare the differences

between the gravitation constants at two latitudes much more

accurately than the constant can be determined at any one

latitude.

Measurements and the Work of Scientists

The work of the scientist is tremendously aided by having

available very careful measurements on certain standard substances. For example, a chemist who has prepared a new liquid

organic compound may be interested in the viscosity of this

new substance. Using very simple apparatus, the viscosity can be determined by comparing it with the viscosity of water.

The viscosity of water has been very carefully measured. One

way to determine the absolute viscosity is to measure the time

78
required for a given volume of water to flow through a capillary

tube of known diameter and length under a given pressure. It

is quite an undertaking to establish all these quantities. But if equal volumes of water and the test liquid are compared in

identical capillaries under identical conditions of temperature,

it is only necessary to measure the time of flow of each liquid

and the specific gravity of each liquid. Thus the viscosity of the

new compound can be obtained relative to the viscosity of

water. When carefully performed, such a comparison is practically free from bias. Of course any bias in the viscosity value

assigned to water will be carried over in the value assigned to

the new liquid. For this reason the very greatest care is taken

in establishing the values of the physical constants for certain

reference materials.

What we have been examining is a small part of the theory

of measurements. The role of the computations we have just

made is to give the investigator an objective basis for making

statements about his experimental results.

You may find all this mathematics pretty tiresome and not

nearly as much fun as assembling your apparatus and getting it to work properly. And you may ask, "Is it really necessary to go

into all these complications?" There are three alternatives, all of which have been widely used in the past and are still used

to some extent today. These alternatives are:

1. Arbitrarily make the limits ridiculously wide

2. Guess at the limits

3. Ignore the whole matter of giving your fellow scientists a measure of the quality of your work. That is, just report

averages without any limits.

You will agree, I hope, that these alternatives come in a poor

second to a piece of work well done and supported by a standard

deviation and the narrowest possible limits at the highest possible level of confidence.

79
6. Instruments for

making measurements

You may now appreciate why scientists would like to reduce

experimental errors in their measurements to the point

where they could be ignored. One way to reduce error is to use

better instruments. In the paper thickness study we used only

a simple metric scale. This may be replaced by a vernier caliper

similar to the one shown in Figure 13. The stack of paper can be caught between the jaws of this instrument so there is no
need to hold one end of the scale on one edge of the stack. The

other advantage — a big one — comes from the auxiliary vernier

scale, named after the French mathematician Pierre Vernier

(1580-1637).

The auxiliary vernier is a short scale which divides nine

divisions on the main scale into ten equal parts. Each whole

division of the auxiliary scale equals nine tenths of a main-scale

division. The auxiliary scale is used to estimate the tenths of the

millimeter. Move the auxiliary-scale zero up to the position on

the main scale that is to be estimated. In Figure 14 this position

is between 11 and 12 mm. To find the tenths directly, run your

eye along the scale until you find a mark on the main scale in

line with a mark on the auxiliary scale and read the auxiliary

scale. The reading is 11.2 mm. Can you prove that this scheme

is sound?

As an example of the usefulness of the vernier caliper, consider the set of measurements made on paper thickness shown

in Table 15. The agreement for thickness per sheet is so good

in the fourth place that estimates can be given to the fifth

decimal place. In this set of measurements the s for the thick- ness per sheet is 0.00015, one twentieth of the 0.0038 obtained

Table 15. Measurements on paper thickness with a vernier caliper

number of thickness thickness sheets mm. per sheet 215 20.0 0.09302

184 17.1 0.09293

146 13.6 0.09315

120 11.2 0.09333

103 9.6 0.09320

Average = 0.09313

81
by students using a millimeter scale. The vernier caliper makes

it much easier to compare the thickness of two papers. Notice,

too, how the standard deviation provides a measure of the

improvement.

Still further improvement could be obtained with the micrometer, shown in Figure 15. This is a more elaborate vernier and a standard piece of equipment for precision machine work. The

shop version reads directly to thousandths of an inch, and tenths

of a thousandth may be estimated.

Before we leave the five measurements on paper thickness

made with the vernier caliper, let's try another way of looking

at them. If a graph is constructed and the actual thickness of

each stack is plotted against the number of sheets per stack, the

five points should — in theory — lie in a straight line through the

origin of the graph. Actually, the points will not lie exactly on a

line because of small errors in the measurements. We may ask,

Figures 13 and 14. Objects to be measured

with the vernier caliper are held between the

jaws. Approximate

reading is made on

main scale, and readings to nearest tenth

are made in auxiliary

scale shown in detail at right.

C\J

O o

OJ — <T> — CD — K m

— <6 — m — * — too O

to£

E

82
1

Figure 15. With micrometer,

measurements can be made

directly to thousandths of an

inch and estimated to ten

thousandths.

why should we expect them to do so?

The equation for a straight line through the origin is y — bx

where b is the slope of the line. (Slope is the constant that

defines the rise of y values in terms of the increase of x. ) The

increase in y for one additional sheet on the stack gives the

slope. Therefore the slope is an estimate of the thickness of the

paper.

The Method of Least Squares

One part of the mathematics of measurement deals with getting the best fit of a line ( or curve ) to a set of points. What is meant by "best fit"? We can explain fitting a line in terms of

the arithmetic average which we accept as the best single number to represent a collection of data. Consider three possible

measurements; 13, 8, and 9, represented by the average, 10.

The differences between the three measurements and the aver- age are 3, —2, and —1. The sum of the squares of these differ- ences is 14. If any number other than ten is used to get the

differences, the sum of their squares becomes larger than 14.

The arithmetical average is the number that makes the sum of

the squares of the differences a minimum.

Suppose that rather than using the average to represent this

set, we use 9, the middle or median number. The differences

then are 4, —1, and 0. The sum of the squares of these differ- ences is 17. The graph in Figure 16 shows that the sum of the

83
squares of the differences has a minimum value when the aver- age is chosen to represent the entire collection. You should

confirm the graph shown in Figure 16. Perhaps you can give a general proof.

Now let us see how we can use this generalization to interpret a collection of measurements. For our example we will try to

find the equation of a line that best fits the plot of the thickness

of a stack of paper against the number of sheets per stack. Our

data will be the vernier readings taken from Table 15. If we let y

equal the thickness of the stack in millimeters and x equal the

number of sheets per stack, our problem becomes one of considering the various values of b in the equation y = bx.

Suppose that we arbitrarily take b as equal to 0.09. Substituting this value in the equation makes it possible to calculate the

thickness, */, for any given number of sheets.

X calculated y observed y difference

(no. sheets) (0.09x) (thickness in mm.) (Obs. y-calc. y) 103 9.27 9.6 +.33

120 10.80 11.2 +.40

146 13.14 13.6 +.46

184 16.56 17.1 +.54

215 19.35 20.0 +.65

All the calculated values for y are below the observed values;

without doubt the coefficient b has been given too small a value.

The sum of the squares of the differences between observed y

and calculated y is 1.1946. By analogy with the arithmetic aver- age, it seems that if we found a Value for b that would make the

sum of the squares of the differences a minimum, we would

have a line that best represents the collection. This procedure

is frequently used and it is called the method of least squares.

We could cut land try various values of b, but this is trouble- some. There is a formula that gives the desired value of b

directly for lines that pass through the origin.

2i2 84
Figure 16. Graph shows 2d2 when an arbitrary number is subtracted

from 13, 9, and 8. Note 2,d2 is a minimum for x=\Q, the average.

Y

50 r—

40

30

20

10 —

Value chosen to represent data _L_1 I I I

I L

10 12 Multiply each x by its corresponding observed y and sum up

the five products (11764.8). Square each x and sum the five

squares ( 126406). The ratio of these two sums gives b for a line

through the origin. The value found for b is 11764.8/126406 =

0.09307. The equation can now be written y = 0.09307x. The

graph for this equation is shown in Figure 17. The new set of

calculated y s and their differences from observed y's show a much improved fit.

103

120

146

184

215

calculated y (0.09307*)

9.586

11.168

13.588

17.125

20.010

observed y 9.6

11.2

13.6

17.1

20.0

difference

(obs. -calculated) + .014

+.032

+ .012

-.025

-.010

The sum of the squares of these differences has gone down to 0.002089. This sum of squares also leads to an estimate of the

85
standard deviation of the measurement of the width of the stack.

The constant b plays the role of the arithmetic average in our

earlier calculations of s. The estimate, s, of the standard deviation is / 0.002089

s — w — = nn0Q 0.023 mm.

Since our measurements were recorded only to the nearest

tenth of a millimeter, we probably were rather lucky to get

this small s. If you are sharp eyed you may have noticed that the value

0.09307 for the slope b (the thickness per sheet) is a little less than the average given in Table 15. This is due to the fact that

the average in the table gives equal weight to each measurement, while the thicker stacks get more weight in this calculation for the slope.

36
Which is the better estimate — the average or the slope? The

slope is, if we assume that the error of measurement is not

influenced by the thickness of the stack. If this assumption is true, the calculations of thickness per sheet should be more

reliable for the thicker stacks. The arithmetic average ignores

this advantage of the thicker stacks. Usually, however, the discrepancy in the two estimates is unimportant unless the stacks

vary greatly in thickness.

As an exercise, try adding one millimeter to all the y values

actually observed. That is, imagine that all these measurements

are biased by +1 mm. The graph and fitted line (the dotted

line) for the adjusted data are shown in Figure 18. What is the

slope of the line? The line, in an effort to compromise since it must go through the origin, runs below the points near the

origin and above the points farther out. Even the eye can see

that a line not forced to go through the origin would be a better

fit to the points. This line has an intercept on the y axis at about

87
one millimeter. An incorrect zero setting for the vernier caliper

would make all results high ( or low )

, and thereby introduce a constant error. Perhaps the above discussion will suggest to you

one way to detect a constant error or bias.

One major concern in all refined measurements is the calibration of the instrument. Little good would come from carefully reading fractions of a scale division if the instrument itself

is in error. The sad news is that errors in the instruments are not

revealed by repeated measurements. If a thermometer is in error by half a degree when read at 25° centigrade, this error

will not be eliminated by taking the average of many readings

all estimated to tenths of a degree. All the readings will have

this hidden constant error. There are two ways around this

dilemma. One way is to have a thermometer checked by a competent testing laboratory. The laboratory will supply a certifi- cate that gives the corrections to be applied at periodic points

along the thermometer scale.

Another way to reveal constant errors is to have one or more

similar instruments. One thermometer is used and then replaced

by another thermometer. If readings are divided among two or more thermometers, inconsistencies among the thermometers

will ultimately be revealed. If two vernier calipers are available,

each should be used for half the readings. We may find, just as we did for the magnesium analyses and for the measurements

on g discussed in Chapter 4, that there is a diflFerence between

the two sets of readings.

Very often this sort of check on instruments can be introduced into experiments without adding appreciably to the labor.

If the instruments are in agreement there is, of course, the possibility that both instruments are in error to the same amount

and of the same sign. This coincidence is generally regarded as unlikely; so agreement between sets of measurements made by

using two or more instruments gives us confidence that constant

errors of appreciable magnitude in the instruments are not being

overlooked.

88
An Instrument of Your Own

One way to learn about instruments at first hand is to make

one. We have chosen a simple one, so that you can easily make

two or three and compare them. Back in 1887, Captain Andreas

Prytz, a Dane, invented a very simple instrument for measuring

the area of an irregular plane figure. We all know the formula

for the area of a square, rectangle, triangle, and circle. But suppose we need to get the area of a leaf or an irregularly shaped

plot of land. The outline could be traced or drawn to scale on

graph paper and the number of squares counted.

The measurement of irregular areas is very important in engineering. There are instruments that trace the rise and fall of

pressure in an automobile cylinder throughout the piston stroke.

The area under the pressure curve must be measured. This

information is needed to evaluate the performance of the engine. Civil engineers laying out modern highways use their

Figure 19. Irregular areas can be measured with polar planimeter.

89
transits to determine the profile of a hill. From this profile the

engineer will be able to compute the quantity of earth that must

be removed. But first, he will need to measure the area enclosed

by the profile of the hill. In technical and engineering work

irregular areas are measured by a beautiful and expensive instrument called the polar planimeter. See Figure 19.

You will have to make your own planimeter. The hatchet

planimeter invented by Captain Prytz can be made very quickly

from a piece of coat hanger or other stiff wire. From the lower

straight portion of a coat hanger cut a piece about 30 centimeters long. File one end flat and straight across, similar to the

base on a right cylinder. File the other end to a tapered but

very slightly rounded point so that it will not scratch paper.

The blunt end is then hammered out to a hatchet shape. The

rough wedge shape should be filed until a sharp arc-shaped edge

is formed on the blade as shown in Figure 20.

Now comes the only part requiring care. Bend down about

five centimeters of the hatchet end at right angles. Be most

particular to have the edge of the hatchet in the same plane as the long piece. Then bend down five centimeters of the pointed

end so that it is parallel to the bent-down hatchet end. The

hatchet planimeter looks like a very low, very wide letter U.

One preliminary test of this planimeter may be made by

drawing it along a straight line. Place hatchet and point on the

line. Hold the planimeter vertical by the pointed end and draw

the point along the line. The hatchet should follow and stay on

the line or very nearly so. If the instrument performs satisfactorily in the first test, test

it on simple figures of known areas. A circle of four-centimeter

radius or an equilateral triangle seven centimeters on a side

will serve nicely. Graph paper is especially useful in this experiment. Tape two sheets end to end. Draw the figure to be measured on one piece and rest the hatchet end of the planimeter

on the other piece.

The measurement is made by picking a point close to the

90
center of the area to be measured. Draw a straight line from

this point to the perimeter.

Place the pointed end of the

planimeter on the center point

of the figure and the hatchet

end on the other sheet of graph

paper. Generally it is convenient to have the hatchet, the

pointed end, and the line to the

perimeter all in line and coincident with one of the rulings on

the graph paper. Now press the

hatchet end down so that it makes a faint but definite mark

on the paper. Hold the planimeter upright by the pointed

end and move the point along

the line from the center out to

the perimeter. The hatchet

should track along the graph

paper ruling. Now trace the outline of the figure and return to

the center point. Let the hatchet

end follow as it will; do not

force it. When the pointed end

is back at the center, press down

the hatchet to make another

indentation in the graph paper.

The area of the figure is computed by multiplying the distance between the two indentations by the distance between

the two arms of the planimeter.

The hatchet end will have been
displaced sideways, i.e., at right angles to its original position.

The displacement can be read directly from the graph paper.

If the distance between the two arms is also measured by the

graph paper, the area can be given in whatever units the graph

paper is ruled. The area of a circle or rectangle is also easy to

get in square graph paper units.

Repeat the measurement by going around the figure in the

opposite direction. You will find the hatchet is displaced in the

direction opposite to that in the first trial. The displacements

may disagree. Probably this is because hatchet blade and point

are not in perfect alignment. Use the average of the two tries as your answer. Repeat such complementary pairs until you

have a series of estimates of the area. Be sure to prepare a good

form on which to record your data. Determine the error for

this method of measuring the area. Refer back to Chapter 5

for the formulas.

One rather satisfying thing about this experiment is that the

areas of regular figures drawn on the graph paper are known

fairly exactly — much more so than the homemade planimeter measures them. So for all practical purposes, we know the

true area. Consequently we can discover if our planimeter has a "constant" error. Does it always miss by about the same

amount and of the same sign?

We can in effect "calibrate" this simple instrument. Certainly it would be appropriate to investigate two or more sizes

of area and two or more shapes of area. The results of such a

calibration study are indispensable in any serious effort to

determine the area of some very irregular-shaped area. You can

see that mere agreement between repeat tracing on the irregular area does not protect you from some constant error. By

testing the instrument and your technique of using it on at least

two different, regular figures of known area, you can detect

hidden constant errors. Sometimes it is not easy to have a reference item that is accurately known so you can check yourself. If you make a 92
second planimeter, you will probably find some difference

between the sets of measurements made with the two planimeters. The discrepancy between the two sets is a direct warning that there are other errors in addition to those revealed by

the scatter of the measurements, all made with the same planimeter. Every investigator who is trying to get very accurate

data is confronted with exactly the problems you face in

calibrating a homemade planimeter.

We are now at the point where we see that the error in a measurement may be a complex matter. First we studied the

variation among repeat measurements. We learned to compute

the standard deviation as a measure of the variation among

the measurements.

Precision, Accuracy, and Truth

The deviations of the individual measurements from their

average determine the precision of the measurements. These

deviations do not reveal either bias or constant error that may

be present in every one of the measurements. Scientists try to

arrange their experiments so that the precision, or standard

deviation, is the only source of error they need to worry about.

This is often achieved by comparing one or more test items

with some reference material of known values.

Thus if someone, with much labor, has measured very

accurately the thickness of a stack of paper, a sample of this

paper may be compared with a paper of unknown thickness.

The difference in thickness found between the reference paper

and the unknown paper is added to ( or substracted from ) the

value assigned to the reference paper. The only error to consider here is the precision error, since any bias in the measure- ments does not affect the difference between the two papers.

The scientist who undertakes to establish the correct thick- ness for the reference paper faces a more difficult problem.

Possible biases now become a matter of real concern, and a 93
great deal of effort is required to detect such biases and eliminate them as far as possible. The object is to arrive at an

accurate value — one that is close to the true value.

The term accuracy involves the error as measured from the

true value, not as scatter measured around the average of the

data. Even if we knew the true value, it is most undesirable

to take the differences between the individual measurements

and the true value. The differences should always be taken from

the average of the data. The t tables only apply when the

standard deviation is calculated by using the average. The constant error is revealed by the difference between the average

of the measurements and the true value, if one is lucky enough

to know the true value. Notice that good precision is required

to detect small constant errors.

Table 16. Different values reported for the Astronomical Unit (values

1-12, from SCIENTIFIC AMERICAN, April 1961)

source of A.U. in experimenter's

number measurement millions estimate of and date of miles spread

1 Newcomb, 1895 93.28 93.20 - 93.35

2 Hinks, 1901 92.83 92.79 -92.87

3 Noteboom, 1921 92.91 92.90 - 92.92

4 Spencer Jones, 1928 92.87 92.82 -92.91

5 Spencer Jones, 1931 93.00 92.99 -93.01

6 Witt, 1933 92.91 92.90 - 92.92

7 Adams, 1941 92.84 92.77 - 92.92

8 Brower, 1950 92.977 92.945 - 93.008

9 Rabe, 1950 92.9148 92.9107 -92.9190

10 Millstone Hill, 1958 92.874 92.873 - 92.875

11 Jodrell Bank, 1959 92.876 92.871 - 92.882

12 S. T. L, 1960 92.9251 92.9166 - 92.9335

13 Jodrell Bank, 1961 92.960 92.958 - 92.962

14 Cal. Tech., 1961 92.956 92.955 - 92.957

15 Soviets, 1961 92.813 92.810 -92.816

94
The standard deviation and the constant error should be

reported separately. Quite different remedies are required to

improve the precision and to reduce constant errors. Both tasks

challenge the skill of the experimenter. The experimenter finds

it very useful to have some means of demonstrating that improvements in the measurement technique have been achieved.

Our discussion is only an introduction to the mathematical

treatment of the errors in measurements.

A particularly revealing compilation of measurements made

since 1895 of the value of the Astronomical Unit (average distance of the earth from the sun) is shown in Table 16. The

table reveals a spread of values reported by the astronomers.

This spread refers to the precision of the work and is not a measure of accuracy. The "best" value reported by a later

worker is often far outside the limits assigned by an earlier

worker.

Make a graph by taking the number of the measurement as x and the reported value as y. The scale on the t/-axis should

extend from 92.70 to 93.20. You will see that the later values

show much better agreement with each other than those in the

early part of the century. We have here an impressive demonstration of the increasing refinements of the measurement

process. Man is never satisfied. Men will always strive to

achieve one more decimal point as they seek to penetrate deeply

into the nature of the universe.

95
7. Experiment with

weighing machines

One day your author wished to demonstrate the advantage

of a carefully planned experiment. There had been some

discussion about the accuracy of the weighing machines found

in drugstores and other public places. He and three colleagues

weighed themselves on four different machines. Each man was,

weighed on each machine. Each man read his own weight just once. The other three times, his weight was read by his com-
panions. A schedule was set up so that each mans weight was

read by all four participants. Each man was weighed on every

machine and each man made one reading on each machine.

A number of questions could be answered from the 16 weights

obtained in this investigation.

First, the weights of the men could be compared. Second,

the machines could be compared with one another. Third, the

men could be compared with respect to their method of reading a scale. In particular we could discover whether an individual had a tendency to get consistently high or low readings,

that is, whether he had a bias.

A schedule was made by dividing a square into 16 smaller

squares. Each column of small squares was assigned to one of

the men. Each row of squares was assigned to a different

machine. The sketch shows the plan at this point.

Man getting weighed

J.C. J.Y. CD. M.D. Machine I Machine II Machine III Machine IV

Why did we complicate this investigation by having each

j man read his own weight only once and then have it read the

other three times by his companions? This device prevented

the second and following readings on a man's weight from

being influenced by preceding readings. If a man read his own

weight each time, he might have been tempted to make the

I readings agree a little better with each other than was actually

the case. So, as part of the experiment, the readings recorded

97
by any reader were not revealed until the experiment was over. This made sure that we had four readings for any one man's

weight by four different readers, none of whom knew what

the others had put down.

This foresight, which added nothing to the work, had one

other important consequence : if a man had a reading bias and

he read his own weight each time, this bias would enter into

his own average and into no other average. The difference

between the average weights for two men would include their

reading biases. An arrangement for each man to read the

weights of all four men eliminates this source of error from the

comparison of the weights.

The problem was how to enter in the proper box the initials

of each individual who was to make the reading. Each man was

to read once every man's weight, including his own, and to make

one reading on each machine. Clearly each man's initials must

appear in all four rows and in all four columns. There are 576

ways in which this can be done. An arrangement of this kind is called a Latin square. The particular Latin square used in this

experiment is shown in the next sketch.

J.C. Man getting weighed

J.Y. CD. M.D. Machine I J.C. CD. M.D. J.Y. Machine II J.Y. M.D. CD. J.C. Machine III CD. J.C J.Y. M.D.

Machine IV M.D. J.Y. J.C. CD.

The initials in each square designate the man who made the

98

j
reading for the particular machine and man associated with that

column. We see that J.C. read his own weight on machine I, and read the weight of J.Y. on machine III. The plan was not yet complete. In what order should the

men get on the scales? You may ask, "what difference will that

make?" It should not make any difference, is the reply. Whenever possible we try to introduce into our experiments elements

that are extremely unlikely to alter the results. This is done as a check. If it turns out that the results are altered by this operation, there may be something wrong with the plan of the experiment, or the data.

How were we to introduce the order of being weighed so as

to be perfectly fair to all involved? The obvious way to be fair

was to let each man be first once, second once, third once, and

last on the scales just once. Furthermore — and this takes a bit

of thinking — if a man in his role as reader should read the

weight of a first man on the scales, then a second on, a third on, and a last one on, this would further even things out. Therefore,

we assigned in each box a number 1, 2, 3, or 4 which told each

man listed at the top of each column when it was his turn to get on a particular scale.

The next sketch shows this number entered in the box, and

the weight of the man at the top of the column as it was read

by the man whose initials are in the box. For example, the numbers tell us that when we came to machine II, CD. got on the

scale first and read his own weight, next J.Y. got on the scale

and M.D. read his weight, and so on. Some of the advantages of this planned experiment appear

immediately. The average weight found for each man was a consensus based on all four machines. Since each man was

weighed on every machine, the variation among the machines

did not enter into the comparison of the weights of the men.

That is, if a machine read two pounds too high, the machine

introduced this increment for every man. Each man's average

would therefore have been increased by half a pound without

99
Man getting weighed

1 c 1 Y r n lYI .D, Machine

I 1J.C. 100. /D

3

CD.

i co or 4

M.D.

lOo.U

2

J.Y.

1 CO c Machine

average

i cn a lbU.4

II 4

J.Y.

i cc n 2

M.D.

i CD.

3

J.C.

1 "7 1 f\ 1/1.1 C 1 1

III 2

CD.

100.

4

J.C.

1 CI K lOl.O

3

J.Y. 131.0

1 M.D.

1 CI oc lb/.^3 1 CO o

IV

3

M.D.

155.0

1

J.Y. 162.75

2

J.C. 152.25

4

CD.

169.0 159.8

Man Ave. 154.9 162.9 152.5 169.2

changing the difference between the average weights for the

men. This state of affairs is strictly true only if all the men are approximately the same weight. If one of the men had been

replaced by an 80-pound boy, difficulties might have arisen.

A machine might have given readings two pounds too high at 160 pounds and also have given readings 1.5 pounds low at

80 pounds. In that case all the men would have gained an extra

two pounds and the boy would have lost 1.5 pounds. So the

differences between the boy and the men would be altered.

The machines, too, could be fairly compared only in the

neighborhood of the average weight of the men. There was a

difference of almost three pounds between the averages for

machines II and III. The highest weight for each man was given

by machine II and the lowest by machine III. Notice that for

each machine we form the total weight of all four men. That

the men differ in weight does not matter, provided only that

100
they are not too far removed from the over-all average of 160

pounds.

What about the performance of the men as readers of the

scales? The scheme was arranged so that each reader took a reading on every machine and for every man. Therefore the

same total was put before each reader but in different combinations of machine and man being weighed. The weights read by

J.Y. are 169.5, 156.0, 151.5, and 162.75. The total is 639.75

pounds and the average of his four readings is 159.9. Similarly

the order on the scales may be examined. The four weights

associated with the number 1 include a weighing of each man,

a reading by each man, and a weighing from each machine.

The weights are 155.75, 153.25, 167.25, and 162.75 and the

average is 159.8.

We compute the average for each reader and for the numbers

1, 2, 3, and 4, using the data displayed in the last sketch. All

the averages have been assembled together for comparison in

Table 17. The men were expected to differ in their weights.

The extreme difference between machines was 2.8 pounds. As

readers, the men showed excellent agreement; the maximum

difference between readers was 0.5 pounds. This difference was

just about the same as the 0.4 maximum difference found between the average weight of the men when they were third on

and their weight when they were first on the machine. As might

be expected, there were small errors in estimating to a quarter

of a pound, otherwise the averages for the readers would.have

1 agreed exactly.

We cannot explain the differences between machines as arising solely from reading errors. The reading error is clearly quite

small by comparison with machine differences. It is safe to

conclude that at least some of the machines are slightly in error.

* The maximum error in a machine, as estimated by the difference

between it and the average of all four machines, is 1.5 pounds.

This is not enough to worry most users of the machines.

Can we conclude that no machine gives a weight that is off

101
Table 17. Averages for men, weighing machines, readers, and order

of being weighed

men machines readers order

J.C. 154.9 I 160.4 J.C. 160.1 1 159.8 a 160.1

J.Y. 162.9 II 161.1 J.Y. 159.9 2 159.8 b 159.7

CD. 152.5 III 158.3 CD. 159.6 3 160.2 c 159.9

M.D. 169.2 IV 159.8 M.D. 159.9 4 159.9 d 159.8

Maximum

difference 2.8 0.5 0.4 0.4

by more than 1.5 pounds from the truth? Certainly not. Perhaps

the machines were all set by the same mechanic and have a common error. Suppose the machines were purposely all set to read low in order to please those wr ho are concerned about being

overweight. And, of course, we have data only in the neighborhood of 160 pounds and no information for much smaller or much larger weights. The easy way to check the machines

would be to get hold of some "standard" weights. These are weights that have been checked against the official standards

of weight.

The Importance of Experimental Design

In Table 17 there is a fifth column of averages rather mysteriously labeled a, by c, and d. Recall that we were able to enter the numerals 1, 2, 3, and 4 in the boxes so that there was

a 1 assigned once to each machine, man, and reader. The same

holds good for 2, 3, and 4. In a similar way the letters a, b, c, and

d can be put in the boxes so that the letter a is assigned once to

each machine, man, reader, and numeral.

The averages for the letter a, by c, and d ought to agree

within the error of reading because no physical action is asso- ciated with these letters. The maximum difference for these

averages is 0.4 pound. This difference is about the same as the

maximum difference found among readers, or for order of get102
ting on the scales. Such a device provides good evidence that

the averages for readers agree within the error of reading, and

also that the order of getting on the scales did not matter.

The placing of these letters is not shown but is left as an

exercise for the reader. Hint: place the a's in the diagonal

starting at the top left corner. When additional symbols are entered in the Latin square,

it is then called a Graeco-latin square. Both are widely used in

experimental design. A 3 x 3 square permits only four sets of

averages; a 5 x 5 square permits six sets of averages. Curiously

a 6 x 6 square can be constructed with only one set of symbols

in the boxes.

Even without formal analysis, the sixteen measurements in

Table 17 have revealed a good deal of information. The primary

object of the study may be considered to have been a comparison of the men s weights. The same data permitted a check on

the weighing machines in regard to possible disagreement

among the machines. The data also made it possible to check on possible biases the men might have had as readers. There was

no convincing evidence of such biases. The differences among

the men as readers were about the same as the small differences

associated with the order the men got on the scales, and order

should not have had an effect. Most important, the comparison

of the men's weights was not impaired by disagreement among

the machines. Neither would reading biases have altered the

differences found between the weights of the men. Try adding

a small bias to any man's readings to verify this statement.

103
8. Selection of items

for measurement

The problem of getting good measurements and finding ways

to describe them concisely has been our chief concern in

the preceding chapters. Only indirectly has there been any

suggestion that there is sometimes a problem of picking representative items to be subjected to measurement. In Chapter 4, to determine the homogeneity of the magnesium alloy, we

tested 50 spots on five bars chosen from 100 bars. Was this a
fair sample? We omitted any reference to the question of

whether the 923 gasoline pumps could be considered an adequate sample of the total collection of pumps in the country.

Taking a large number of items does not guarantee getting a representative selection of the whole supply, or population.

Many investigators have found this out the hard way.

Before tackling the general problem of how to get a good

selection of items for measurement, let us investigate a special

case. Your author went to the bank and got two rolls of newly

minted 1961 cents. For the moment, let us accept without argument that these 100 coins give an adequate picture of the coins

minted during the work period in which they were made. Undoubtedly the coins accumulated in a big tray and got mixed

up in the process. There is a specified weight for cents, together

with legal tolerances for minimum and maximum weight of a

coin. The nominal weight is 3.110 grams, with a permitted

tolerance of 0.130 grams above and below the nominal weight.

The problem was to determine whether or not this sample

of 100 pennies fell within the permitted tolerance of 3.110 grams

± 0.130 grams. To do this experiment your author had access

to a very fine balance ( see Figure 21 ) that could be read to the

millionth part of a gram. Such precision was quite unnecessary.

The weights were recorded to tenths of a milligram; that is, to

the fourth place of decimals. Even that was really unnecessary.

Weights to the nearest milligram would have been quite good

enough. Why?

The actual weights of the coins vary over a range of about

250 milligrams. Weighing to the nearest milligram would surely

be good enough since the weighing error could, at most, extend

the range of actual weights by only a milligram or two in 250

milligrams. The weights are correct to four places of decimals.

How do we know that? If we reweighed the 100 coins using any

other fine balance capable of weighing accurately to six decimal places, we would get exactly the same weights over again

out to four decimal places. The possibility of a constant error 105
Figure 21. This sensitive balance is capable of accurate

weighings to one millionth of a gram.

common to all weights was eliminated by checking the balance

with a standard weight.

All the above detail is directed to establish that the observed

scatter of the weights is not a result of errors in weighing. The

measurement error is nil for this inquiry. An effort to determine

whether the weights of the coins do or do not conform to specification must depend on examining a number of coins. For this

106
purpose we need a balance good enough so that a coin will not

be called outside the tolerance limit because our weighing introduces additional scatter into the results. There is no measure- ment error in the weights as recorded. The variation among the

weights found is a property of the coins, and in no wise reflects

measurement error. Table 18. Weights of new U.S. cents weighed on an accurate balance

fiacc viass interval mud veil number of coins in grams 1st 50 2nd 50 100 coins 2.9800-2.9999 0 1 1 3.0000-3.0199 1 3 4

3.0200-3.0399 3 1 4

3.0400 - 3.0599 2 2 4

3.0600 - 3.0799 5 2 7

3.0800 - 3.0999 9 8 17

3.1000-3.1199 13 11 24

3.1200-3.1399 8 9 17

3.1400-3.1599 6 7 13

3.1600-3.1799 3 3 6

3.1800-3.1999 0 2 2

3.2000-3.2199 0 1 1 Total 50 50 100

The individual weights are given in class intervals of 20 milligrams for each roll of coins. These are displayed in Table 18.

A glance at the totals in the last column suggests that the actual

weights of the coins are distributed among the class intervals

in very much the same manner as are the measurement errors on one object such as the paper thickness measurements.

When the coins themselves were arranged in columns cor- responding to the class intervals, they formed a histogram that

is indistinguishable from those exhibited in Chapter 2. We have

here, not 100 crude measurements on one object, but one very

careful measurement on each of 100 objects. Nevertheless the

100 results are distributed in the same form as the normal law

of error. 107
The same calculations that were made on repeated measure- ments on one object are appropriate for this collection of single

measurements on each of 100 objects. A standard deviation may

be calculated and the same probability statements made that

were explained in Chapter 5. So there is no difficulty in arriving

at a concise description of this collection of weighings. The

<N cs coco coco coco coco coco COCO COCO CO CO CO CO CO CO CO CO 108
reader may calculate the average and standard deviation. Compare the average with the nominal weight, 3.110.

Some may wonder what would be the result if these coins

were weighed on a rather crude balance. If the balance is such

that repeated weighing of the same object gave a standard deviation of more than 20 milligrams, the spread of the distribution

would be substantially increased. If sc is the standard deviation

of the coins, and Sb the standard deviation for weighings made

with the crude balance, then the standard deviation of the

observed weights for coins weighed with the crude balance is easily calculated. It is Vs

2

c + s

2

&. If duplicate weighings are made on each of, say, 50 coins, it is an easy matter to solve for

both Sb and sc. We can calculate what the standard deviation

for coins would be if they were weighed on an errorless balance.

We delayed answering the question as to whether these 100

coins are completely representative of the total population of

newly minted cents. The chances are very good, of course, that

the coins are representative of some rather short interval in the

total annual production of cents. If we desire to report on the

cents minted in any year, we should plan to put aside a few

coins from each day's work. These coins should be thoroughly

mixed and a sufficient number drawn from the collection. The

coins are enough alike so that there is virtually no danger of bias

or intentional selection, either for those going into the collection or for those drawn for weighing.

Coins are easy to set aside and mix thoroughly. However,

there are many cases in which it is physically impossible to do

this. For example, bales of raw wool weigh several hundred

pounds, and there may be hundreds in a shipment. Custom

duties are assessed on the basis of the per cent of clean wool in

the shipment. Since the bales vary considerably, it becomes

necessary to sample the shipment by taking two or more cores from each of a number of bales. Economy of effort requires that

no more bales be sampled than is necessary to obtain a satisfactory estimate of the per cent of clean wool in the entire ship109
ment. We have already learned one way to sample fairly. The

bales may be marked 1 to n. Then n cards are numbered 1 to n. If k bales must be sampled, k cards are drawn from the carefully

shuffled pack.

Random Digits

There is a more convenient way to attain a random selection

without using a pack of numbered cards. You can use random

number tables, just page after page filled with the digits 0 to 9

put down in random order. Do not think this random order is easy to get. Oddly enough, very special efforts have to be made

to avoid excessive inequalities in the frequencies of the digits

and to avoid recurring patterns. A very small selection of 1600

random digits is given in Table 19.

A table of random numbers can be used in various ways.

Suppose we wish to draw 12 bales from a shipment of 87 bales.

The bales are first numbered 1 to 87. Now go down a column of

paired random numbers. Start at the top left of Table 19 and

write down pairs of digits. Omit pairs 88, 89, ... , 90, and 00,

because there are no bales corresponding to these numbers. We

find the pairs 44, 84, 82, 50, 83, 40, 33, (50), 55, 59, 48, 66, and

68. The second 50 is omitted because that bale is already in the

sample. These are the bales to be sampled. This method avoids

any deliberate attempt to put good, or poor, bales in the sample.

One may start at any part of the table so there is no possibility

of anyone's influencing the selection. In fact every effort must

be made to start in different places.

If there were 302 bales, we would take the digits in triplets.

Can you see the triplet, 441, in the two numbers 44 and 17?

Any number from 001 up to and including 302 gets in the

sample. There will be many numbers greater than 302. If a number is greater than 302 and less than 605, subtract 302 from

it to get your number. If the number is 605 up to 901, subtract

604 to get your number. Ignore numbers above 907. In this way

110
Table 19. Random Numbers

44 17 16 58 09

84 16 07 44 99

82 97 77 77 81

50 92 26 11 97

83 39 50 08 30

79 83 86 19 62

83 11 46 32 24

07 45 32 14 08

00 56 76 31 38

42 34 07 96 88

06 76 50 03 10

20 14 85 88 45

32 98 94 07 72

80 22 02 53 53

54 42 06 87 98

55 23 64 05 05

10 93 72 88 71

93 85 79 10 75

86 60 42 04 53

35 85 29 48 39

40 33 20 38 26

96 83 50 87 75

88 42 95 45 72

33 27 14 34 09

50 27 89 87 19

13 89 51 03 74

97 12 25 93 47

16 64 36 16 00

45 59 34 68 49

20 15 37 00 49

17 76 37 13 04

70 33 24 03 54

04 43 18 66 79

12 72 07 34 45

52 85 66 60 44

07 74 21 19 30

97 77 46 44 80

94 77 24 21 90

99 27 72 95 14

38 68 88 11 80

55 74 30 77 40

59 29 97 68 60

48 55 90 65 72

66 37 32 20 30

68 49 69 10 82

44 22 78 84 26

71 91 38 67 54

96 57 69 36 10

77 84 57 03 29

53 75 91 93 30

04 33 46 09 52

13 58 18 24 76

96 46 92 42 45

10 45 65 04 26

34 25 20 57 27

68 07 97 06 57

15 54 55 95 52

97 60 49 04 91

11 04 96 67 24

40 48 73 51 92

83 62 64 11 12

06 09 19 74 66

33 32 51 26 38

42 38 97 01 50

96 44 33 49 13

67 19 00 71 74

02 94 37 34 02

79 78 45 04 91

87 75 66 81 41

34 86 82 53 91

60 47 21 29 68

76 70 90 30 86

16 92 53 56 16

40 01 74 91 62

00 52 43 48 85

02 02 37 03 31

38 45 94 30 38

02 75 50 95 98

48 51 84 08 32

27 55 26 89 62

64 05 71 95 86

75 73 88 05 90

33 96 02 75 19

97 51 40 14 02

15 06 15 93 20

11 05 65 09 68

52 27 41 14 86

07 60 62 93 55

04 02 33 31 08

01 90 10 75 06

76 83 20 37 90

22 98 12 22 08

59 33 82 43 90

39 54 16 49 36

40 78 78 89 62

57 16 00 11 66

07 52 74 95 80

49 37 38 44 59

47 95 93 13 30

02 67 74 17 33

22 35 85 15 13

09 98 42 99 64

54 87 66 47 54

58 37 78 80 70

87 59 36 22 41

71 41 61 50 72

23 52 23 33 12

31 04 49 69 96

31 99 73 68 68

94 58 28 41 36

92 03 51 59 77

61 71 62 99 15

73 32 08 11 12

42 10 50 67 42

26 78 63 06 55

12 41 94 96 26

96 93 02 18 39

10 47 48 45 88

35 81 33 03 76

45 37 59 03 09

59 56 78 06 83

06 51 29 16 93

44 95 92 63 16

32 17 55 85 74

13 08 27 01 50

44 95 27 36 99

07 02 18 36 07

13 41 43 89 20

24 30 12 48 60

90 35 57 29 12

52 91 05 70 74

58 05 77 09 51

29 56 24 29 48

94 44 67 16 94

15 29 39 39 43

02 96 74 30 83

25 99 32 70 23

97 17 14 49 17

18 99 10 72 34

82 62 54 65 60

98 80 33 00 91

73 81 53 94 79

73 82 97 22 21

22 95 75 42 49

39 00 03 06 90

09 77 93 19 82

33 62 46 86 28

05 03 27 24 83

39 32 82 22 49

55 85 78 38 36

74 94 80 04 04

08 31 54 46 31

72 89 44 05 60

02 48 07 70 37

94 37 30 69 32

45 07 31 66 49

53 94 13 38 47

35 80 39 94 88

16 04 61 67 87

90 89 00 76 33
every number from 1 to 302 gets an equal chance to be drawn.

Earlier in the book you were told to use catds. I must now

confess that, instead of cards, I used a table of random numbers

because it was less work. Besides, cards stick together.

Most of you are aware that the census taker puts a lot of

questions to some people, while others are asked for only a little information. Modern measurement theory shows that properly

drawn small samples are quite satisfactory measures of the

whole population. Often these small samples are even better

than a complete count because they can be made by a few welltrained census takers who will make fewer mistakes. In all such

samples, the use of random selection is absolutely indispensable

to avoid various sources of bias in the results. This subject of

sampling is so vast that many large volumes have been written

on it in recent years.

112
9. Measurement of

thread strength

This chapter describes a method of measuring the strength

of sewing thread. The results will not add to our store of

factual knowledge, but we will become further acquainted with

the problem of making a measurement. We are going to start pretty nearly from scratch and make an apparatus out of easily

obtained parts. I am sure you will discover just how exasperating a piece of apparatus can be. This elementary problem drives
home the fact that making measurements is not child's play.

We are going to load a piece of thread until it breaks. One

way to do this would be to tie one end of the thread to a support

and attach a light container to the other end. Pour in sand very

slowly and stop as soon as the thread breaks. Now weigh the

container and contents and you have your answer. You can do

the experiment this way if you like. Of course you will repeat

the experiment many times in order to reveal the scatter of the

results. What do you think causes the scatter? Is it the difficulty

in making the measurement, or do different specimens actually

have different strengths?

Making a Testing Apparatus

The method we are going to use is more complicated. The

reason for making it more complicated is that the apparatus

involves features found in actual test apparatus. The scheme is one that multiplies the weight applied to the thread by means

of a lever. This permits testing relatively strong material with

moderate loads. Our test device will double the load applied.

The illustration (Figure 23) shows a stick, 52 cm. long with a 13 cm. crossarm mounted on a wooden base. The pieces are joined by small right-angle brackets with the upright braced

with a guy wire. A hook is screwed in the underside of the crossarm about nine cm. out from the upright. The specimen to

be tested hangs from this hook. A small eyelet is screwed in near the base of the upright. Finally ten small brads are placed

close together in d, row along the edge of the baseboard and one

more brad about 20 cm. away from the middle brad in the

group of ten. Our lever is made from a short piece of coat hanger wire.

A piece 16 cm. long will do. File a shallow notch half a cm. from each end. Take care to keep the notches in line. Now turn

the wire over and file another shallow notch halfway between

the end notches. This notch should be 180° from the other two.

114
1I
A small cardboard box 5x7 cm. x 3 cm. deep can be put

together with Scotch tape. This allows layers of six nickels to

be placed in the box. Eight layers for a total of 48 nickels were

used in the experiment. Strong linen thread loops are attached

to the box. On top of the nickels rests a false cardboard bottom.

BB shot are then added one by one until the load breaks the

specimen.

We are now ready to assemble the apparatus. All thread,

other than the test specimen, is strong linen thread. The test specimen, 30 cm. in length, is taken from a spool of size A

mercerized cotton thread. A loop is tied in one end of the test thread and the loop placed over the middle brad in the row of

brads. The other end is doubled around the lone brad in the corner of the board. Hold the thread together, slip the thread

off the brad, and tie the loop in that end of the specimen. This

standardizes the length of the test specimen at about 20 cm. Take a piece of linen thread and tie loops in each end so that

the length between the tips of the loops is about 40 cm. A

similar piece about 14 cm. between loop tips will also be needed.

Hang the test specimen from the hook. Slide the notched wire

lever through the free loop and settle the loop in the center

notch. In the notch in the end of the wire near the upright, hang

a loop of the long linen thread. Lead the free end through the

eyelet and place the loop over one of the brads so that the lever

is a little high on the free end. When the load is applied to the

other end, the test specimen stretches and the free end of the

lever will get lower. In the notch on the free end of the lever,

hang a loop of the short linen thread. In the other end attach a paper clip with an end bent out to make a hook. Hang the box

of nickels on this hook.

We are now ready to test the apparatus to see if it works

properly. Add BB shot one by one until the specimen breaks.

It should break between the loops. A break at a loop may result

if the notch has a rough edge. The box should clear the base by

about one cm. Too big a drop causes the BB ' s to jump out

116
and perhaps be lost. Place a spool against the side of die box

to prevent rotation.

BB Units

The end of the lever near the upright may be regarded as a fulcrum. The load is applied twice as far from the fulcrum as the point of support by the specimen. Hence the load on the

specimen is doubled. The load includes the box and box contents and hook. The specimen supports the wire, but the weight

of the wire is not doubled. Why? After you have adjusted the

apparatus and acquired some skill in using it, you can proceed

to measure a number of specimens.

Nickels are used as the "standard" of weight because the

nominal weight of a nickel is five grams. The lever, supported

by a linen thread in the center, was used as a balance to find

the weight of the box, of the lever, and the conversion factor

for nickels to BB's.

Several nickels were attached to a thread by cellulose tape and

hung on one end of the lever. The box and accessories (but

without nickels ) were hung on the other end. BB's were added

until box and contents balanced the nickels. Some more nickels

were added and the number of BB's were again increased to get

balance. From these data it was easy to get the weight of the

box in BB's ( 20 ) and the conversion factor of 14 BB's to one

nickel.

How could we get the weight of the lever? Easy. We cut

another piece of wire the same length, notched it, and weighed

it against box and BB's. The weight was found to be 18 BB's,

All measurements were reported in BB's.

We have rather casually taken for granted that we managed

to get the middle notch exactly midway between the two end

notches. If we did not, then the factor two will not give the load

on the specimen. We will more than double the load if the

shorter end is near the upright, and do less than double the load

117
if the longer end is near the upright. The two ends of the lever

were marked I and II so that a record could be made of whichever end was near the upright. A series of tests could be run

with the lever in one position and a second series with the lever

reversed. Unequal division of the lever would ultimately be

revealed by the averages for the two series.

In thinking about this problem a bit, it appears that if two

trials are run — reversing the lever on the second trial — the

effect of unequal division virtually cancels out when the average

of the two trials is taken. The trials were run in pairs and the

results for seven pairs of tests are given in Table 20. Examination of the 14 results shows considerable variation among the

specimens. The seven results with end I near the support may

be compared with the seven made with end II near the support.

The technique described in Chapter 5 is appropriate.

The average breaking strength of the thread in grams is obtained by dividing the number of BB's by 14 to get them con- verted to nickels. Then multiply by five to get grams. Why

bother with BB's at all? Why not just use nickels? A nickel is a

pretty big weight to drop in the box, and the shock effect would

break the thread prematurely. The BB's are also a convenient

way of estimating fractions of a nickel.

If the lever had been divided into quite unequal parts we

would expect one member of a pair to give consistently higher

results than the other member. End I is the higher four times,

and II has the higher result three times. The high variability of

the thread obscures the slight inequality of division. If the

result with end I near the support is divided by the total for the

pair, we get the seven ratios: .492, .543, .529, .542, .473, .499,

and .557. The average of these is .5193. A t test could be run to see if the limits for this average include the value 0.5000. If so, the evidence would be insufficient to show an unequal division

of the lever. You should also satisfy yourself that the way to

calculate the position of the middle notch is to take the result

with end I near the support and divide by the total for the pair.

118
Table 20. Measurement of thread strength. All weights in BB's

Weight: Lever = 18; Box = 20; 48 nickels = 672

expt. end at BB's box plus load twice lever total no. support 3dded total losd u/t WL.

1 I 150 r no 692 O A O 842 1 C O A 1684 18 1 TOO 1702

II 111 692 869 1738 18 1756

2 I 257 692 949 1898 18 1916

II 105 692 797 1594 18 1612

3

T

I O A "7 24/ 692 939 1878 18 1896

II 144 692 836 1672 18 1690

4

T

I 222 692 914 1000 1828 18 1 O A f- 1846

TT11 d9z "7"70 //2 1 C A A 1544 1 o 18 1 c c o 1562

5 I 104 692 796 1592 18 1610

II 196 692 888 1776 18 1794

6 I 183 692 875 1750 18 1768

II 185 692 877 1754 18 1772

7 I 299 692 991 1982 18 2000

II 93 692 785 1570 18 1588

1729

1764

1793

1704

1702

1770

1794

This will give the distance from end I to the middle notch as a

fraction of the distance between the two end notches.

Aside from the fun of assembling the apparatus, we have seen another wav in which a "constant" error can enter into measurements. If the middle point is not exactly halfway between the

end notches, a bias is introduced. We have seen how a proper

program of work (reversing the lever) puts us in a position

where we can correct for the bias easily and automatically. The

data also make it possible to "test" the lever for bias at the

same time you are collecting data to determine thread strength.

We have also obtained some idea of how the thread strength

varies.

The experiment could be easily extended in scope. We used

a standard length for the test specimen. Suppose we used a 119
specimen twice or half as long, do you think this would influence

the results? Think carefully. You could run the tests by permitting the box to rotate and compare with tests in which the box

is not allowed to rotate. You could try the effect on the strength

of boiling the thread, or exposing it to direct sunlight for two

weeks. You could compare different brands of thread or different colors.

The apparatus and the technique of using it are there to serve whatever line of inquiry interests you. An interesting line of

research soon diverts attention from the measurement problem

itself. Nevertheless the problem of measurement is still there.

Although it hardly needs to be said, men have found that good

research depends on good measurement.

120
EPILOGUE

I wish that I could assure you that the vast scope and variety

of measurement problems have been revealed in this book.

Instead, I must warn you that there is much more to the subject.

Not all measurements are best described by the normal law of

error. There are many special situations and distributions that

apply to counts of radioactive particles, to seed germination,

and to opinion polls. Other distributions are used for measure- ments on the fatigue failure of metal parts, and still others apply

in the study of flood heights and in reliability studies.

There are, however, elements common to all these distributions. Consider the notion of unexplained variation. Apparently

there is something subtle in the notion of random numbers and

in using random procedures of selection. Because these concepts

of variability and randomness are common to all measurement

problems, they have been the major objects of our attention.

A large part of our attention has been given to how measure- ments are obtained. Yet, the science of measurement, like a coin,

has two sides to it. One side shows the ingenuity and skill of

experimenters in devising better methods of measurement for

their individual researches. The other side of the coin deals

with the properties common to all measurements. While this side

of the coin has dominated our discussions, the two sides are

inseparable.

As scientists explore the unknown regions beyond the present

frontiers of science, they encounter new problems and new

kinds of measurements. Often these data pose entirely new

problems to the measurement specialist. Working together,

scientist and measurement specialist push back the frontiers of

knowledge; still the frontiers grow longer. One thing appears

certain — there is no limit to the length of the frontiers of

knowledge.

121
Table of Squares

0 1 2 3 4 5 6 7 8 e 0 0 1 4 9 16 25 36 49 64 81

1 100 121 144 169 196 225 256 289 324 361 2 400 441 484 529 576 625 676 729 784 841

3 900 961 1024 1089 1156 1225 1296 1369 1444 1521 4 1600 1681 1764 1849 1936 2025 2116 2209 2304 2401

5 2500 2601 2704 2809 2916 3025 3136 3249 3364 3481

6 3600 3721 3844 3969 4096 4225 4356 4489 4624 4761

7 4900 5041 5184 5329 5476 5625 5776 5929 6084 6241

8 6400 6561 6724 6889 7056 7225 7396 7569 7744 7921

9 8100 8281 8464 8649 8836 9025 9216 9409 9604 9801

10 10000 10201 10404 10609 10816 11025 11236 11449 11664 11881

11 12100 12321 12544 12769 12996 13225 13456 13689 13924 14161

12 14400 14641 14884 15129 15376 15625 15876 16129 16384 16641

13 16900 17161 17424 17689 17956 18225 18496 18769 19044 1932]

14 19600 19881 20164 20449 20736 21025 21316 21609 21904 22201

15 22500 22801 23104 23409 23716 24025 24336 24649 24964 2528]

16 25600 25921 26244 26569 26896 27225 27556 27889 28224 2856]

17 28900 29241 29584 29929 30276 30625 30976 31329 31684 3204]

18 32400 32761 33124 33489 33856 34225 34596 34969 35344 3572]

19 36100 36481 36864 37249 37636 38025 38416 38809 39204 3960]

20 40000 40401 40804 41209 41616 42025 42436 42849 43264 4368]

21 44100 44521 44944 45369 45796 46225 46656 47089 47524 4796;

22 48400 48841 49284 49729 50176 50625 51076 51529 51984 5244]

23 52900 53361 53824 54289 54756 55225 55696 56169 56644 5712! 24 57600 58081 58564 59049 59536 60025 60516 61009 61504 6200: 25 62500 63001 63504 64009 64516 65025 65536 66049 66564 6708: 26 67600 68121 68644 69169 69696 70225 70756 71289 71824 7236: 27 72900 73441 73984 74529 75076 75625 76176 76729 77284 7784: 28 78400 78961 79524 80089 80656 81225 81796 82369 82944 8352: 29 84100 84681 85264 85849 86436 87025 87616 88209 88804 8940: 30 90000 90601 91204 91809 92416 93025 93636 94249 94864 95481 31 96100 96721 97344 97969 98596 99225 99856

122
SELECTED READINGS

A Million Random Digits, by the Rand Corporation. Glencoe, 111., The Free Press, 1955. Dip into it at random.

Design of Experiments, by R. A. Fisher. New York, Hafner Publishing Company,

1949. Read the first 100 pages. Facts from Figures, by M. J. Moroney. New York, Penguin Books, Inc., 1958. Easy to read and inexpensive.

Introduction to Statistical Analysis, by W. J. Dixon and F. J. Massey, Jr. New

York, McGraw-Hill Book Company, Inc., 1957. A widely used college textbook. Mathematical Recreations and Essays, by W. W. Rouse Ball and H. S. M. Coxeter. New York, The MacMillan Company, 1947. Read about Euler's officer's problem.

Probability and Statistics, by F. Mosteller, R. E. K. Rourke, and G. B. Thomas, Jr. Reading, Mass., Addison-Wesley Publishing Company, Inc., 1961. Textbook for Continental Classroom on Television.

Statistical Methods for Chemists, by W. J. Youden. New York, John Wiley and

Sons, Inc., 1955. A small book, not limited to chemists.

Statistics, by L. H. C. Tippett. New York, The Home University Library of Modern Knowledge. Oxford University Press, 1944. Very elementary.

Statistics: A New Approach, by W. Allen Walks and H. V. Roberts. Glencoe, 111., The Free Press, 1956. A lively introduction by professionals for amateurs.

<e Student's" Collected Papers, edited by E. S. Pearson and John Wishart. London,

The Biometrika Office, University College, 1947. Wise essays on making measurements. Technological Applications of Statistics, by L. H. C. Tippett. New York, John

Wiley and Sons, Inc., 1950. Chapter 8 gives an account ot the density measure- ments on nitrogen that led to the discovery of the rare gases.

123
GLOSSARY

Accuracy Refers to the discrepancy between the true value and the result obtained by measurement. Average Refers to arithmetic average, m, or mean. If n measurements have

been made, the average is obtained by dividing the sum of all n measurements by n. Average deviation If n measurements have the average m, the sum of the

deviations ( ignoring signs ) divided by n gives the average deviation. Bias Refers to a more or less persistent tendency for the measurements, as a

group, to be too large or too small. Deviation The difference between a measurement and some value, such as the average, calculated from the data. Class interval An arbitrarily selected interval into which measurements are grouped on the basis of their magnitude.

Error In the study of measurements "error" does not mean "mistake," but is a technical term denoting deviations from the average or some other computed

quantity. Such deviations are considered to be random errors. Bias involves the notion of a constant error. Estimate A numerical value calculated from data. The average is an estimate

of the quantity under measurement. Other parameters such as the standard

deviation, a, are often estimated from the data.

Graduation mark The marks that define the scale intervals on a measuring

instrument are known as graduation marks. Histogram A graphical representation of a collection of measurements. Equal

intervals are marked off on the x axis. A rectangle is erected on each interval,

making the heights of the rectangles proportional to the number of measurements

in each interval. Least squares A mathematical procedure for estimating a parameter from a

collection of data by making the sum of the squares of the deviations a minimum.

124
Normal law of error A mathematical equation that in many cases describes the scatter of a collection of measurements around the average for the collection. Parameter A parameter is the property or quantity that the measurements

are expected to evaluate. The word parameter is used for the correct value of the property. Precision Refers to the agreement among repeated measurements of the same

quantity.

Population Refers to a group of items belonging to a well-defined class from

which items are taken for measurement. Random A random procedure for selecting items from a population gives every member of the population equal opportunity to be chosen. Range The difference between the largest and smallest values in a collection

of measurements. Standard deviation Estimated from n measurements by calculating

2(dev)2 n — I where 2(dev) 2 means the sum of the squared deviations from the average. Standard error Sometimes used for the standard deviation of an average. It

is equal to the standard deviation divided by the square root of the number of measurements used to get the average. Symbols

2 — Capital sigma, summation sign

ix — population average, true value m — arithmetic average of the measurements, an estimate of ^

0" = population standard deviation

s = estimate of <j computed from data

e — base of natural logarithms, a mathematical constant

7r = ratio of circumference to diameter of a circle, a mathematical

constant b — conventional symbol for slope of a straight line

t — Student's t, a multiplying factor for s used to obtain probability limits about the average of a collection of data. Tolerance An agreed-upon permissible departure from specification.

Unit Every measurement is expressed as a multiple or fraction of some appropriate, well-defined unit quantity such as centimeter, volt, etc. Vernier A mechanical aid for estimating fractions of a scale interval.

125
INDEX

Age measurement 17-18

Air pressure 10

Air temperature 16

American Society for Testing

Materials 29

Archeological materials 18

Argon 21

Astronomical measurements 94-95

Astronomical unit 94, 95

Atomic weights 11

Auxiliary vernier scale 81

Average 30, 66, 68-76, 87

Balance 105

"Best fit" 83

Bias 39, 78, 79, 88, 98, 103, 119

Carat 19

Carbon 17

Carbon dating 17

Card-shuffling experiment 47-48

Cement testing „ 51

Chance, laws of 36

Chemical analysis 10-11, 19-20,

42-47, 78-79

Chlorine 10

Coin-tossing experiment -..36-38 Coin-weighing experiment 105-109

Constant error 87

Degree 24

Degrees of freedom 70-71, 77

Delta (A) 69, 70, 72

Density _ 21

Digits, random 110-112

Equations 37, 56, 60, 70,

71, 74, 84, 86

Error 6,11-12,47,49,94

constant 87

normal error curve 58-59, 61

normal law of 53-54

standard „ 65

Estimating 24-27

Experimental design .....102-103

Fever thermometers 24

Formulae 56, 60, 70, 71, 74, 84, 86

Freezing point 25

Gasoline pump measurements 51

Gold content ..19,20

Gosset, William Sealy 69

Graeco-Latin square 103

Gravitational constant 51

Hatchet planimeter 89-92

calibration 91-92

construction 90

measuring with , 91-92

testing 90

Height 17

Histogram 34, 35, 41-42,

46-51, 60, 65

Ice 25

Impurity 25

International Bureau of Weights

and Measures .. 16

Interval 32, 34, 60, 63, 71

modal 33

Isotopes 17

Kilogram 16

Latin square . 98

Laws of chance 36

Least-squares method 83-86

Limits of confidence 73

Magnesium-alloy

experiment ~ 42-47, 49

Mathematical laws 22

Measurements 6-14, 15

absolute 78

accuracy of 7, 11-12, 29, 94

comparative 78

errors in 6, 11-12, 47, 49, 94

frequencies 38

independent 35

individual 64-65,36

laws 7,22

126
need for 17

precision 93

repetition in 28

record of 26

units of 10, 15-17

uses 17, 78

Measuring instruments 10, 80, 89-92

auxiliary vernier scale 81

calibration - ... 88, 92

fever thermometer 24

hatchet planimeter 89-92

micrometer , 29, 82

millimeter scale 12, 26-33

polar planimeter 88, 89

thermometer 10, 16, 24

tire gauge 10

transit „ 89

vernier caliper 81

Median number 83

Meter 10

Method of least squares 83-86

Metric system 16

Micrometer „ 29, 82

Mid-values 32, 41

Milligram ~ 105

Millimeter „ 10, 12, 26-33

Modal interval 33

Mu Od „ 57, 58, 72

Nitrogen 21

Normal error curve 58-59, 61

Normal law of error 53-54

Number system „ 16

Obsidian tools 17

Paper thickness experiment 25-40,

49, 62-63, 65, 75-76

bias 39-40

disagreement in measurements 27-28

form for recording

measurements „ 26

frequency table „ 31, 34

thickness per sheet 31-32

Planimeter 89-92

hatchet 90

polar 89

Plant growth measurements 52-54

Polar planimeter 88, 89

Prytz, Captain Andreas 89

Rayleigh, Lord 21

Random numbers 110-112

table Ill

Range 29-30, 63

Rare gases 21

Room temperature 16

Sampling 14, 104-105

Scale 34

Scatter 30, 35

Sewing thread experiment ...113-119 Sigma (^) (a) 58-60, 76

Slope m 86, 87

Small differences 17-22

Squares, table of 122

Standard deviation 58, 61, 63

form for calculating 66

Standard error 65

Steel bar measurements 51-52

Student's t 68-74

Table of squares „ 122

Testing apparatus 114-117

Testing laboratory 88

Thermometer 10, 16, 24

Thickness 25-26

Tire gauge 10

Tolerance 20

Transit „ 89

True value 94

Values 32

Vernier, Pierre 81

Vernier caliper « 80

Vernier scale 81 Water „ 79

viscosity measurement 79

Weighing machine experiment

96-102

Weight measurement 10-11, 17

127
W. J. YOUDEN

1900 - 1971 Among the small number of recognized ex- perts in the field of statistical design of ex- periments, W. J, Youden's name is likely to come up whenever the subject is mentioned.

A native of Australia, Dr. Youden came to the United States at a very early age, did his graduate work in chemical engineering at the

University of Rochester, and holds a Doctorate in Chemistry from Columbia University. For almost a quarter of a century he

worked with the Boyce-Thompson Institute for Plant Research, where his first interest in

statistical design of experiments began. An

operations analyst in bombing accuracy for the Air Force overseas in World War II, he is now statistical consultant with the National Bureau of Standards, where his major interest

is the design and interpretation of experi- ments. Dr. Youden has over 100 publications,

many in the fields of applied statistics, is author of Statistical Methods for Chemists,

and is in constant demand as a speaker,

teacher, columnist, and consultant on statistic cal problems.
"An experiment is a considered course of action aimed at answering one or more carefully framed

questions," writes Dr. W. J. Youden, the author

of Experimentation and Measurement. Measure- ment is both an art and a science. There are "laws

of measurement" just as fascinating as the laws of

science. There is only one way to learn the laws and tech- niques of scientific measurement and that is to make measurements. When you open this book

you will begin a series of fascinating experiments.

You will make instruments, calibrate them, and

put them to work. By experimenting with the "art" of measurment, you will learn something of

the laws of measurement that apply no matter what it is that is being measured.