PB 151 37*^

^ecknical ^lote 16

SOME APPLICATIONS OF

STATISTICAL SAMPLING METHODS

TO OUTGOING

LETTER MAIL CHARACTERISTICS

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

Functions ami Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

'5, 1901., as amended bv Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods lor making measurements consistent with these standards: the determination of physical constants and

properties of materials; the development of methods and instruments for testing materials, devices.

and structures; advisory services to government agencies on scientific and technical problems: in- vention and development of devices to serve special needs of the Government; and the development

of standard practices, codes, and specifications. The work includes basic and applied research,

development, engineering, instrumentation, testing, evaluation, calibration services, and various

consultation and information services. Research projects are also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the bureau's unique competence is required. The scope of activities is suggested by the listing of

divisions and sections on the inside of the back cover.

Publications

The results of the Bureau's work take the form of either actual equipment and devices or published papers. These papers appear either in the Bureau's own series of publications or in the journals

of professional and scientific societies. The Bureau itself publishes three periodicals available from

the Government Printing Office: The Journal of Research, published in four separate sections,

presents complete scientific and technical papers: the Technical News Bulletin presents summary

and preliminary reports on work in progress; and Basic Radio Propagation Predictions provides

data for determining the best frequencies to use for radio communications throughout the world.

There are also five series of nonperiodical publications: Mo?iographs, Applied Mathematics Series,

Handbooks, Miscellaneous Publications, and Technical Notes. Information on the Bureau's publications can be found in NBS Circular 460, Publications of the

National Bureau of Standards ($1.25) and its Supplement ($1.50), available from the Superintendent

of Documents, Government Printing Office, Washington 25, D.C.
NATIONAL BUREAU OF STANDARDS

technical <^lote

16

JULY 1959

SOME APPLICATIONS OF STATISTICAL SAMPLING METHODS

TO OUTGOING LETTER MAIL CHARACTERISTICS

Norman C. Severo

Arthur E. Newman

Shirley M. Young

Marvin Zelen

The work described in this Report was sponsored

by the Office of Research and Engineering, Post

Office Department.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D. C.

Price $2.75
PREFACE

The National Bureau of Standards is developing

equipments and systems for improving letter sorting by

automation. Therefore it is necessary to determine the

nature and characteristics of mail in post offices.

Since the volume of mail is much too large for

complete piece counts to be feasible, statistical

sampling methods of known and adequate accuracy must be

used. The present paper is the first step in the effort

to develop such methods as applied to letter size mail

characteristics

.

ISRAEL ROTKIN , Coordinator, Post Office Project

(ii)
ACKNOWLEDGMENT

The authors wish to express appreciation to Inspector

John Falconer, Post Office Department, and Mr. Martin Stark,

Section 12.5, NBS, for their assistance in supervising the

collection of data involved in the sampling procedures; to

Mr. Stanton Russell, Section 12.5, NBS, for mechanized

recording of top and bottom clearance space and perfecting

the device for measuring the size characteristics of letters;

to Inspectors Frank Wright, Washington, D.C., R.T. Rock,

Los Angeles, and Ronald Nielson, San Francisco, for their

assistance in recording the data; to Dr. Churchill Eisenhart,

Chief of the Statistical Engineering Laboratory (Section 11.3),

NBS, for many helpful suggestions concerning the analyses of

these studies and for a critical reading of the report; to

other members of the Statistical Engineering Laboratory for

consultation and assistance in preparing this report, especially

to Mrs. Lola S. Deming for a detailed statistical review of

the original manuscript; and to the members of the NBS Fact

Finding Group for the Post Office Project for their contribution

in the tabulation of the data. Finally, gratitude is expressed

to the Postmasters and staffs of the Washington, D. C, Los

Angeles, and San Francisco Post Offices for their cordial

cooperation.

(iii)
1.1.1 Cancel

1.1.2 Cull

1.1.3 Bulk

1.1.4 Color

1.1.5 Recommendations

TABLE OF CONTENTS

Page

1. Conclusions 1

1.1 Letter size and color characteristics 1

3

3

4

4

4

1.2 Ratio of hand canceled to machine canceled mail

at Washington, D. C. 5

1.3 Top and bottom Clearance Space of an addressed

envelope 5

1.3.1 Washington, D. C. memorandum 6

1.3.2 San Francisco and Los Angeles memorandum 6

1.4 Proportions of Long and Short letters 6

2. General introduction 7

2.1 Specific studies 8

2.1.1 Letter size and color characteristics 8

2.1.2 Ratio of hand canceled mail to machine

canceled mail at Washington, D. C. 8

2.1.3 Top and bottom Clearance Space of an

addressed envelope 8

2.1.4 Proportions of Long and Short letters 8

2.2 General remarks on statistical sampling 9

2.2.1 Purpose of a sampling study 9

2.2.2 Nature of a statistical sampling study 10

2.3 Definition of terms 13

3. Letter size and color characteristics 16

3.1 Introduction 16

3.2 Sampling methods and procedures 17

3.2.1 Cancel mail 19

22

23

25

3.3 Analysis 26

3.3.1 Length, height, and thickness characteristics 26

a. Presentation of data 26

b. Comparison of cities 52

c. Tolerance limits 65

(iv)

3

3

3

,2

.2

.2

.2

.3

.4

Metered mai

Cull mail

Bulk mail

1

Ana!Lys is
Page

3.3.2 Color characteristic 94

4. Ratio of hand canceled to machine canceled mail at

Washington, D. C. 97

4.1 Introduction 97

4.2 Sampling plan and results 101

5. Top and bottom Clearance Space of an addressed envelope 102

5.1 Introduction 102

5.2 Sampling procedures 104

5.3 Y/ashington, D. C. memorandum 106

5.4 San Francisco - Los Angeles memorandum 110

6. Proportions of Long and Short letters 120

6.1 Introduction 120

6.2 Sample method and procedure 120

6.3 Analysis 121

References 124

Appendix 125

(v)
LIST OF FIGURES

Page

1. Measuring device for letter size characteristics 21

2. Control charts for averages of Height and Length 47

for Cancel Short mail data from S.F. and L.A.

3. Control charts for averages of Height and Length 48

for Cancel Long mail data from S.F. and L.A.

4. Control charts for averages of Height for Cull 50

mail data from S.F. and L.A.

5. Control charts for averages of Length for Cull 51

mail data from S.F. and L.A.

6. Control charts for averages of Height and Length 53

for Bulk mail by Bundles data from S.F. and L.A.

7. Cumulative percentage curves for Height and Length 55

of Cancel Short Regular mail data of S.F., L.A.,

and D.C.

8. Cumulative percentage curves for Height and Length 56

of Cancel Short Air mail data for S.F., L.A., and

D.C.

9. Cumulative percentage curves for Height and Length 57

of Cancel Long Regular mail data for S.F., L.A.,

and D. C.

10. Cumulative percentage curves for Height and Length 58

of Cancel Long Air mail data for S.F., L.A., and

D.C.

11. Cumulative percentage curves for Thickness of 59

Cancel mail data for S.F., L.A., and D.C.

12. Cumulative percentage curves for Height and Length 60

of Cull Regular mail data for S.F. and L.A.

13. Cumulative percentage curves for Height and Length 61

of Cull Air mail data for S.F. and L.A.

14. Cumulative percentage curves for Height and Length 62

of Bulk mail by Bundles data for S.F. and L.A.

(vi)
List of Figures (Continued) Page

15. Cumulative percentage curves for Height and Length 63

of Bulk mail by Letters data for S.F. and L.A.

16. Cumulative percentage curves for Thickness of Bulk 64

mail and Cull mail data for S.F. and L.A.

17. Relative frequency histograms for Length, Height, 88

and Thickness of Cancel Short mail for combined

data of S.F., L.A., and D.C.

18. Relative frequency histograms for Length, Height, 89

and Thickness of Cancel Long mail for combined

data of S.F., L.A., and D.C.

19. Relative frequency histograms for Length, Height, 90

and Thickness of Cull Regular mail for combined

data of S.F. and L.A.

20. Relative frequency histograms for Length, Height, 91

and Thickness of Cull Air mail for combined data

of S.F. and L.A.

21. Relative frequency histograms for Length, Height, 92

and Thickness of Cull Regular plus Air mail for

combined data of S.F. and L.A.

22. Relative frequency histograms for Length, Height, 93

and Thickness of Cancel mail for combined data

of S.F. and L.A.

23. Relative frequency histograms by designated 100

Colors for the combined data of S.F. and L.A.

24. A letter illustrating the distance from the top 105

of the first line of intelligence of the address

to the top edge of the envelope, and from the

bottom of the last line of intelligence of the

address to the bottom edge of the envelope.

25. Histograms showing top and bottom Clearance 109

Space for data from Washington, D. C.

26a. Histograms showing top Clearance Space for 116

data from S.F. and L.A.

(vii)
List of Figures (Continued) Page

26b. Histograms showing top Clearance Space for 117

data from S.F. and L.A.

27a. Histograms showing bottom Clearance Space for 118

data from S.F. and L.A.

27b. Histograms showing bottom Clearance Space for 119

data from S.F. and L.A.

(viii)
LIST OF TABLES

Page

1. A summary for Cancel, Metered, Cull, and Bulk 2

mail of (i) sample size, (ii) average height,

length, and thickness, (iii) predicted lower

and upper 99% tolerance limits with confidence

coefficient .95.

2. Frequency (f) and cumulative percentage (%) by 27

designated Height of Cancel Short Regular and

Cancel Short Air mail at S.F. and L.A.

3. Frequency (f) and cumulative percentage (%) by 28

designated Length of Cancel Short Regular and

Cancel Short Air mail at S. F. and L.A,

4. Frequency (f) and cumulative percentage (%) by 29

designated Height of Cancel Long Regular and

Cancel Long Air mail at S.F, and L.A.

5. Frequency (f) and cumulative percentage (%) by 30

designated Length of Cancel Long Regular and

Cancel Long Air mail at S.F. and L.A,

6. Frequency <f) and cumulative percentage (%) by 31

designated Height of Cancel Short and Cancel

Long mail at D.C.

7. Frequency (f) and cumulative percentage (%) by 32

designated Length of Cancel Short and Cancel

Long mail at D.C.

8. Frequency (f) and cumulative percentage (%) by 33

designated Thickness of Cancel Short Regular

and Cancel Short Air mail at S.F, and L.A,

9. Frequency (f) and cumulative percentage (%) by 33

designated Thickness of Cancel Long Regular

and Cancel Long Air mail at S.F. and L.A.

10. Frequency (f) and cumulative percentage (%) by 34

designated Thickness of Cancel Short and Cancel

Long mail at D.C.

11. Frequency (f)^ relative frequency (r.f.), and 35

cumulative percentage (%) by designated Height

of Metered mail at S.F.

(ix)
List of Tables (Continued) Page

12. Frequency (f), relative frequency (r.f.), and 35

cumulative percentage (%) by designated

Thickness of Metered mail at S.F.

13. Frequency (f), relative frequency (r.f.), and 36

cumulative percentage (%) by designated Length

of Metered mail at S.F.

14. Frequency (f) and cumulative percentage (%) by 37

designated Height of Cull Regular mail and Cull

Air mail from S.F. and L.A.

15. Frequency (f) and cumulative percentage (%) by 38

designated Length of Cull Regular and Cull Air

mail at S.F. and L.A.

16. Frequency (f) and cumulative percentage (%) by 39

designated Height of Bulk mail by Bundles at

S.F. and L.A.

17. Frequency (f) and cumulative percentage (%) by 40

designated Length of Bulk mail by Bundles at

S.F. and L.A.

18. Frequency (f) and cumulative percentage (%) by 41

designated Height of Bulk mail by Letters at

S.F. and L.A.

19. Frequency (f) and cumulative percentage (%) by 42

designated Length of Bulk mail by Letters at

S.F. and L.A.

20. Frequency (f) and cumulative percentage (%) by 43

designated Thickness of Cull Regular and Cull

Air mail at S.F. and L.A.

21. Frequency (f) and cumulative percentage (%) by 43

designated Thickness of Bulk mail by Bundles

and Bulk mail by Letters at S.F. and L.A.

22. Number and percentage of Post Cards at S.F. , 45

L.A. , and D.C.

23. Predicted lower and upper limits for 90%, 95%, 68

and 99% of Cancel Short Regular mail based on

data from S.F., L.A., and D.C.

(x)
List of Tables (Continued) Page

24. Predicted lower and upper limits for 90%, 95%, 69

and 99% of Cancel Short Air mail based on data

from S.F., L.A., and D.C.

25. Predicted lower and upper limits for 90%, 95%, 70

and 99% of Cancel Long Regular mail based on

data from S.F., L.A., and D.C.

26. Predicted lower and upper tolerance limits for 71

90%, 95%, and 99% of Cancel Long Air mail based

on data from S.F., L.A„, and D.C.

27. Predicted lower and upper limits for 90%, 95%, 72

and 99% of Metered mail based on data from S.F.

28. Predicted lower and upper limits for 90%, 95%, 74

and 99% of Cull Regular mail based on data from

S.F. and L.A.

29. Predicted lower and upper limits for 90%, 95%, 75

and 99% of Cull Air mail based on data from

S.F. and L.A.

30. Predicted lower and upper limits for 90%, 95%, 76

and 99% of Bulk mail by Bundles based on data

from S.F. and L.A.

31. Predicted lower and upper limits for 90%, 95%, 77

and 99% of Bulk mail by Letters based on data

from S.F. and L.A.

32„. Percentages with specified dimensions and the 78

99% confidence limits of Cull Regular and Cull

Air mail.

33. Percentages with specified dimensions and the 79

99% confidence limits of Bulk mail by Bundles

and Bulk mail by Letters.

34. Frequency (f) and relative frequency (r.f.) by 81

designated Length of Cancel Short and Cancel

Long mail at S.F., L.A., and D.C. combined.

35. Frequency (f) and relative frequency (r.f.) by 82

designated Height of Cancel Short and Cancel

Long mail at S.F., L.A., and D.C. combined.

(xi)
List of Tables (Continued) Page

36. Frequency (f) and relative frequency (r.f .) by 82

designated Thickness of Cancel Short and Cancel

Long mail at S.F., L.A., and D.C. combined.

37. Frequency (f) and relative frequency (r.f .) by 83

designated Length of Cull mail at S.F. and

L.A. combined.

38. Frequency (f) and relative frequency (r.f.) by 84

designated Height of Cull mail at S.F. and

L.A. combined.

39. Frequency (f) and relative frequency (r.f .) by 84

designated Thickness of Cull mail at S.F. and

L.A. combined.

40. Frequency (f) and relative frequency (r.f.) by 85

designated Thickness, Height, and Length of

Cancel Regular mail (includes both Long and

Short letters) at S.F. and L.A. combined.

41. Predicted lower and upper limits for 95% of 86

Cancel mail based on data from S.F., L.A., and

D.C. combined.

42. Predicted lower and upper limits for 95% of Cull 87

Regular, Cull Air, and Cull Regular and Air mail

based on data from S.F. and L.A. combined.

43. Predicted lower and upper limits for 95% of 87

Cancel mail (Long and Short letters combined)

based on data from S.F. and L.A. combined.

44. Frequency and relative frequency by designated 95

Color of Cancel Regular and Cancel Air mail at

S.F. and L.A.

45. Frequency and relative frequency by designated 96

Color of Cull Regular and Cull Air mail at

S.F. and L.A.

46. Frequency and relative frequency by designated 96

Color of Metered mail at S.F.

47. Percentages and their 95% confidence limits by 98

designated Color of Cancel Regular and Cancel

Air mail at S.F. and L.A. combined.

(xii)
List of Tables (Continued) Page

48. Percentages and their 95% confidence limits by 98

designated Color of Cull Regular and Cull Air

mail at S„F„ and L.A. combined.

49. Percentages and their 95% confidence limits by 99

designated Color of Metered mail at S.F.

50. Summary table from First Class Hopper. 103

51. Frequency tabulation of distance from the top 107

edge of the envelope to the top of the first

line of intelligence.

52. Frequency tabulation of distance from bottom 108

edge of the envelope to the bottom of the last

line of intelligence at Washington, D.C.

53. Top clearance space tolerance limits. 112

54. Bottom clearance space tolerance limits and 113

proportion of distribution less than (<) .2

inches.

55. Frequency distribution of clearance data at 114

Los Angeles.

56. Frequency distribution of clearance data at 115

San Francisco.

57. Sample data showing proportion of Long to Long 122

plus Short letters by A.M. and P.M. periods at

S.F. and L.A.

(xiii)

Some Applications of Statistical Sampling Methods

To Outgoing Letter Mai' Characteristics

Norman C. Severo

Arthur £. Newman

Shirley M. Young

Marvin Ze ! en

This paper presents applications of statistical

sampling procedures especially devised to procure

information about the characteristics of outgoing

letter mail. The results of four separate studies

carried out in the Washington, D. C, San Francisco,

and Los Angeles post offices are herein summarized.

The techniques used in the various studies were

developed so that the required information would be

of predetermined reliability and could be gathered

without the use of a large staff and without interrupting the flow of mail. The four studies presented

concern: 1) letter size and color characteristics,

2) ratio of hand canceled mail to machine canceled

mail, 3) top and bottom clearance space of an addressed

envelope, 4) proportions of long and short letters.

-1-
1 . Conclusions

This section presents what appear to be the major

conclusions of this report. These conclusions are elaborated

upon in detail in the following sections.

1.1 Letter size and Color characteristics . The study of

the characteristics of letter size mail from the cities of San

Francisco, Los Angeles, and Washington, D.C. led to the

conclusions that follow.

Table 1 is a summary showing all categories of mail

sampled, the sample size, size characteristic averages, and

tolerance limits. The table gives, for each type of mail

studied, the average height, length, and thickness of the

letters collected in the study. In addition, statistical

tolerance limits are given for both the length and height

characteristics. These limits are 99% limits implied by the

data at hand with confidence coefficient .95. Thus, for

example, for Cancel Long Regular mail from San Francisco,

where a total of 291 letters were sampled, the tolerance

limits of 3 9/16 and 4 5/16 inches are recommended for height.

This means that we expect 99% of all Cancel Long Regular mail

to have heights between 3 9/16 and 4 5/16 inches - and we

shall be correct in this expectation unless the sample from

which we deduced this result was anomalous to an extent

that would arise no more than 1 time in 20.

* Care should be exercised in attempting to draw precise

conclusions to other post offices.

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1.1.1 Cancel . There appear to be no differences

among the three cities with respect to the letter size

characteristics (height, length, and thickness) studied for

Cancel mail. Furthermore, the control charts for height

and length show a remarkable amount of agreement from sample

to sample within a specific type of mail for a given city.

The fact that the sample averages fall within their respective control limits is strong evidence that the differences

observed from sample to sample are due to chance rather than

some assignable cause which would tend to alter or change

the letter size characteristics (see Figures 2 through 6 )

It is recommended that wherever information is desired

regarding the characteristics of Cancel mail that the sampling

plan used in this report for that type mail be adopted. This

sampling plan has been designed to obtain information about

certain types of mail while at the same time utilizing

efficiently current postal methods of canceling letter mail.

1.1.2 Cull . In general there are no differences

between San Francisco and Los Angeles with respect to the

letter size characteristics studied for Cull mail.

The study of Cull mail was started in Los Angeles after

sampling had been initiated to obtain letter size characteristics data on Cancel mail. It was believed that the Cull

study would provide information about a much larger class of

- 3 -
mail than studied at the facing table (i.e., Cancel mail).

The results, while informative and unbiased in that the

methods provide good estimates of the averages, are subject

to more variability than is evidenced in the Cancel mail

study. If additional information of this type is desired,

then it is recommended that further investigation be made

of the sampling procedures with the aim of devising, if

possible, a method subject to less variability. It may

well be that the method used here is adequate for the

purposes intended.

1.1.3 Bulk . There are apparently significant

differences between San Francisco and Los Angeles with

respect to the characteristics studied for Bulk mail.

However, more theoretical consideration needs to be given

to the problem of analyzing the Bulk data.

1.1.4 Color . In general there appears to be no

difference between the cities studied, San Francisco and

Los Angeles, with regard to the color of envelopes. On the

average, 80% of the envelopes are white.

1.1.5 Recommendations . If it is desirable to make

each of the cities already studied for letter size characteristics more complete and comparable, then additional studies

should be made for:

a. Air mail in San Francisco

b. Metered mail in each of the three cities

- 4 -
c. Color in Washington, D. C.

d. Cull mail in Washington, D. C.

The measuring device used in all three cities was not

large enough to enable measurements to be made of oversized

mail. Thus any letter that was larger than either scale on

the template (approximately 7 1/2" x 12") was lumped in a

catch-all class. Therefore, much information concerning

this type mail was lost. Additional study should be made of

the oversize category if more detailed information concerning

its distribution is desired. This, of course, would involve

using a larger measuring device.

1.2 Ratio of hand canceled to machine canceled mail for D.C.

The results of the sampling do not strictly apply to letter

size mail, but instead include mail that may be considered

as "slightly larger." The results of the statistical

analyses were that the ratio, expressed as a percentage, of

hand canceled mail to machine canceled mail was 2.11% +0.21

for A.M. , 3.94% + 0.88 for P.M., and 3.34% + 0.61 for All Day.

We strongly recommend that a sampling procedure such

as used here be employed whenever similar type ratios are

desired. This sampling method would provide ratios that are

otherwise obtained by complete enumeration,

1.3 Top and bottom Clearance Space of an addressed

envelope . Essentially two studies were conducted to determine top and bottom Clearance Space, i.e., the distances

- 5 -
from the top edge of the first line of intelligence of the

address to the top edge of the envelope and from the bottom

edge of the last line of intelligence of the address to the

bottom edge of the envelope. The San Francisco - Los Angeles

study represents a refinement of the one initiated in

Washington, D. C.

1.3.1 Washington, D. C. memorandum

1„ A distance of 3/4 of an inch be used

for the top Clearance Space of an

addressed envelope.

2. No tolerance limit is recommended

for the bottom Clearance Space.

1.3.2 San Francisco and Los Angeles memorandum

1, A distance of 0.9 of an inch be used

for the top Clearance Space of an

addressed envelope,

2. No tolerance limit is recommended for

the bottom Clearance Space of an

addressed envelope for stamped mail.

A tolerance limit of 0.4 of an inch

is recommended for the bottom Clearance

Space of an addressed envelope for

Metered mail.

1.4 Proportions of Long and Short letters. This studywas concerned with determining the ratio of Long letters to

- 6 -
Long plus Short letters for machine cancel mail. The ratio

is not the same in the morning as it is in the evening.

Approximately 30% of A.M. letters (from 10 A.M. to 4 P.M.)

are Long and approximately 50% of the P.M. letters (from

4 P.M. to 11 P.M.) are Long. The overall daily percentages

are 46% for San Francisco and 45% for Los Angeles and the

confidence limits for each ratio are approximately 41% and

50%.

2. General Introduction , This report is the second in a series which presents

applications of statistical sampling procedures especially

devised to procure information about the characteristics of

outgoing letter mail. The results of four separate studies

carried out in the Washington, D, C, San Francisco, and

Los Angeles post offices are herein summarized. The techniques used in the various studies were developed so that

the required information could be gathered without the use

of a large staff and without interrupting the flow of mail.

It is hoped that these statistical sampling plans will be used

again whenever the same type of information is desired. Four

studies are described in this report. Unless otherwise indicated these studies were conducted during the peak A.M. and

peak P.M. periods of mail flow

- 7 -
2. 1 Specific studies . 2.1.1 Letter size and color characteristics . This

study was initiated to determine the length, height, thickness,

and color characteristics of outgoing letter mail. The

details are presented in Section 3 The samples taken at

both the San Francisco and Los Angeles post offices were

studied for all four of these characteristics. The samples

taken in Washington, D t C. were studied only for length,

height, and thickness.

2.1.2 Ratio of hand canceled mail to machine

canceled mail e This study was conducted at the Washington,

D.C. post office to determine the proportion of hand canceled

mail to machine canceled mail. The study is presented in

Section 4 . 2.1.3 Top and bottom Clearance Space of an

addressed envelope . This study was conducted to determine

the top and bottom Clearance Space of an addressed envelope.

Two separate studies were made, one conducted in Washington,

D.C, and the other in San Francisco and Los Angeles. The

details are given in Section 5.

2.1.4 Proportions of Long and Short letters. This

study is confined to samples taken at the San Francisco and

Los Angeles post offices. Ratio figures for Long and Short

letters are given for both morning and evening periods. The

details are given in Section 6.

- 8 -
Each of these studies utilizes statistical sampling

techniques for acquiring the appropriate information. No

elaborate counts of mail are made in any case. Using

statistical methods relatively few letters need to be studied

as compared to a complete enumeration. By applying the theory

of probability and statistics, the desired information can be

provided with a predetermined reliability. Furthermore we

are able to predict how well our results will agree with

results obtained by a future study — providing, of course,

that no substantial changes occur in the characteristics

studied.

2.2 General remarks on statistical sampling

2.2.1 The purpose of a sampling study . The purpose

of any sampling study is to give information about the population of interest, without having to make a complete

enumeration of the population involved. The sample is not

taken for the information it provides in itself. It is of

the utmost importance that the population to be studied (the

target population) be carefully defined. The definition of

the target population dictates the types of mail to be

sampled.

In one of the applications, . we are interested in characterizing envelope dimensions by collecting data on a few

- 9 -
hundred randomly selected letters. A complete enumeration

would have required examination of several million envelopes.

We have been particularly careful to define the target

populations from which samples are taken and about which

conclusions may be drawn.

2.2.2 The nature of a statistical sampling study . A statistical sampling procedure is designed to obtain a

reliable estimate of what would have been found from a complete (and completely accurate) enumeration of the population.

It describes a procedure for collecting data to estimate with

known reliability the characteristics of the population,

without having to examine the entire population.

A distinctive feature of statistical sampling plans

(aside from the savings in cost) is that the property of

representativeness is inherent in the sampling plan itself,

not in the particular sample at hand. This enables one to

state how well the results of a particular sample are likely

to agree with results of other samples for which the same

sampling plan is used.

In a statistical sampling procedure, the samples are

selected at random . It is this random selection that gives

assurance that the results of the sample can be related to

the population with a known degree of reliability. More

details about methods of selecting a random sample are given

below

- 10 -
A random sample is selected in accordance v/ith fixed

rules and must not be confused with a haphazard selection of

samples. If we were interested only in the characteristics

of the sample itself, we might simply walk around the post

office and select any batch of mail that meets our fancy.

The information thus obtained would be limited strictly to

the sample itself and could not be applied to any other

batch of mail in the post office. Using a random procedure

for selecting the samples from some well defined population

enables us to make general statements about whole

classes of mail.

Each of the studies of this report utilizes sampling

procedures designed specifically to determine the information

required. The sampling must, in all cases, be done in

accordance with the well designed rules of randomization.

Thus, whenever at all possible —- and this was almost always -

we have employed some mechanical device such as dice, a

lottery system, or a table of random numbers to insure the

randomness of the selected items of the samples. The

importance of objective randomization cannot be overs tressed,

and is well illustrated in a rather lengthy quotation from

an article by Cochran, Mosteller and Tukey [2] — .

1/ Figures in brackets refer to the list of references

given on page 124 .

- 11 -
•TVhether by biologists, sociologists, engineers,

or chemists, sampling is all too often taken far too

lightly. In the early years of the present century

it was not uncommon to measure the claws and carapaces

of 1000 crabs, or to count the number of veins in each

of 1000 leaves, and then to attach to the results the

"probable error" which would have been appropriate had

the 1000 crabs or the 1000 leaves been drawn at random

from the population of interest. Such actions were

unwarranted shotgun marriages between the quantitatively

unsophisticated idea of sample as "what you get by

grabbing a handful" and the mathematical precise notion

of a "simple random sample . " In the years between we

have learned caution by bitter experience . We insist

on some semblance of mechanical (dice, coins, random

number tables, etc.) randomization before we treat a

sample from an existent population as if it were

random. We realize that if someone just "grabs a

handful," the individuals in the handful almost always

resemble one another (on the average) more than do the

members of a simple random sample. Even if the "grabs"

are randomly spread around so that every individual has

an equal chance of entering the sample, there are

difficulties. Since the individuals of grab samples

resemble one another more than do individuals of random

samples, it follows (by a simple mathematical argument)

that the means of grab samples resemble one another less

than the means of random samples of the same size. From

a grab sample, therefore, we tend to underestimate the

variability in the population, although we should have

to over-estimate it in order to obtain valid estimates

of variability of grab sample means by substituting

such an estimate into the formula for the variability

of means of simple random samples. Thus using simple

random sample formulas for grab sample means introduces

a double bias, both parts of which lead to an unwarranted

appearance of higher stability.

Returning to the crabs, we may suppose that the

crabs in which we are interested are all the individuals

of a wide-ranging species, spread along a few hundred

miles of coast. It is obviously impractical to seek to

take a simple random sample from the species - no one

knows how to give each crab in the species an equal

chance of being drawn into the sample (to say nothing

of trying to make these chances independent) . But this

does not bar us from honestly assessing the likely range

- 12 -
of fluctuation of the result. Much effort has been

applied in recent years, particularly in sampling

human populations, to the development of sampling

plans which simultaneously,

(i) are economically feasible

(ii) give reasonably precise results, and

(iii) show within themselves an honest measure

of fluctuation of their results.

Any excuse for the dangerous practice of treating nonrandom samples as random ones is now entirely tenuous.

Wider knowledge of the principles involved is needed

if scientific investigations involving samples (and

what such investigation does not?) are to be solidly

based. Additional knowledge of techniques is not so

vitally important, though it can lead to substantial

economic gains.

2.3 Definition of Terms.

1. Clearance Space - the distances from the top of the

first line of intelligence of the address to the top

edge of the envelope, and from the bottom of the

last line of intelligence of the address to the

bottom edge of the envelope.

2. Long letters - any letter equal to or greater than

7 9/16 inches in length.

3. Short letters - any letter less than 7 9/16 inches in

length,

4. Regular mail - all first class letter mail that

passes over the facing table to be canceled.

5. Air mail - all letter mail given air transportation

at the legal rate of six cents per ounce.

13 -
6. Metered mail - all first class letter mail bearing

a metered imprint. This mail usually arrives at

the post office already faced and bundled.

7. Cancel mail - all letters and cards that receive

a machine imprint bearing the time and date along

with the identity of the originating post office.

8. First Class Hopper - a place in the Washington, D.C.

post office where first class mail, which is

culled before facing because it cannot be canceled

on the regular canceling machines adjacent to the

facing tables, is transported by conveyor belt.

Here, mail is separated according to its size in

preparation for hand cancellation. A very small

amount is canceled on a special canceling machine.

9. Cull mail - the remaining collection mail after

packages and second class matter have been removed

at the culling tables during the initial stage of

handling. Hand Cancel and Oversize mail are included

in Cull mail.

10. Bulk mail - third class mail which has separately

addressed identical pieces bound in bundles as in

accordance with Section 134.22 of the Postal Manual.

11. Bundles vs. Letters Each shipment of identical

pieces from a given distributor is called a Bundle.

Each piece within a Bundle is called a Letter.

- 14 -
12. "Greater than'* class - refers to the letters whose

height and/or length exceeded the measuring device

(7 9/16" x 12 1/16").

13. Oversize class - all Cull mail greater than 5 9/16

inches in height and/or 11 1/16 inches in length.

14. Tolerance limits - predicted limits, determined by

the samples, between which at least a certain

proportion of a population is predicted to lie.

These limits are calculated with a predetermined

confidence coefficient.

15. Confidence limits - statistically calculated limits

which give an interval estimate within which some

population characteristic will lie„ These limits

are calculated with a predetermined confidence

coefficient.

16. Control limits - limits determined from the samples

within which successive sample values drawn from

the same distribution should fall, using a predetermined probability level.

17. Target population - a class or category of mail

for which certain characteristics are to be studied

and to which it is intended that the conclusions are

to apply. In this report nine classes (i.e., target

populations) are defined in order to study outgoing

letter size mail.

- 15 -
3. Letter size and color characteristics . 3.1 Introduction . From June 4 to 13, 1956, a statistical

sampling study was conducted at the Washington, D. C. post

office. A year later, similar studies were conducted at the

Los Angeles post office (June 12 to 18, 1957) and the San

Francisco post office (June 21 to 28, 1957). Each sampling

study was conducted over at least a five day period. The

purpose was to determine the size and color characteristics

of envelopes for letter size mail originating in each of the

cities.

In this report we shall present in several ways the data

that were collected during the studies. Each way of presentation is used to bring out one or more particularly important features of the sample data. Included are tables which

list the data. Control charts slhow the uniformity in the

sampling methods and make possible visual comparisons between

cities and between types of mail. Cumulative percentage graphs

further enable comparison of cities to be made. Frequency

histograms graphically portray the data and tolerance limits

show the predicted sizes between which we can expect at least

a certain proportion of a particular type of mail to fall.

Section 3.3 on Analysis goes into detail

about each topic that is briefly mentioned here.

- 16 -
3.2 Sampling methods and procedures . The procedures and

techniques of the statistical sampling plans, which are

designed to acquire the appropriate data for determining

letter size characteristics, are herein discussed.

To begin with, we must

1. Specify exactly the nature of the populations

that are to be studied.

2. Select appropriate sampling points and methods

of sampling in which the property of representativeness is inherent.

The nature of post office operations reveals that it

would be almost an impossible feat to attempt to select one

sampling point or to devise one sampling method to describe

all of the letters which pass through a post office. The

operations are so diversified and the types of mail handled

vary throughout the day so that one composite picture could

hardly hope to predict the results for any given situation.

For example, Cancel mail, which appears to be the most

homogeneous and most easily predictable type, cannot be described in terms of one picture for the whole day because even

for this mail the characteristics change throughout the day.

Thus, we choose to define not one but several different

populations that when taken together should describe a fairly

sizable amount of mail handled in the post office. This

- 17 -
report confines itself only to that mail specifically given

in the following: (terms are defined on page 13)

1. Cancel Long Regular mail

2. Cancel Long Air mail

3. Cancel Short Regular mail

4. Cancel Short Air mail

5. Metered mail

6. Cull Regular mail

7. Cull Air mail

8. Bulk mail by Bundles

9. Bulk mail by Letters

Below are summarized the number of letters collected in

each of the three cities studied for each of the nine populations listed above.

Total Sample Sizes (in pieces)

San Francisco Los Angeles Washington, D.C. Total

1. 291 581 1253 2125

2. 51 289 222 562

3. 463 800 1545 2808

4. 66 330 248 644

5. 616 616

6. 1133 1924 3057

7. 197 322 519

8. 290* 414* 704*

9. 985,704 1,620,823 2,606,527

* Bundles

- 18 -
3.2.1 Cancel mail . The samples of Cancel mail were

taken at the stackers of the cancellation machines at such a

time when two or more machines were operating. The mail

accumulating in these stackers is fed from a moving conveyor

belt that passes seven or eight persons each of whom faces

and places on the belt letters selected from those within

his reach. Thus the letters undergo a fairly thorough

mixing as they are being stacked so that any "bunch" or

"bite" of mail sampled at this point would tend to have the

property of randomness which is necessary in sampling

studies. At a predetermined time one two-inch bite of Long

and one two-inch bite of Short letters were drawn from each

of the two selected machines making a total of four two-inch

bites of mail. Both Long and Short letters were canceled on

the same machine but stacked separately in San Francisco and

Los Angeles. In Washington, D. C, Long and Short letters

were canceled on separate machines. In order to further

eliminate the possibility of personal bias, conscious or

unconscious, or personal responsibility for actual allocations,

a lottery was employed to make a random selection of the two

operating machines to be sampled and the samples were always

taken at a preselected spot on the stackers.

The only departure from these procedures was made in the

drawing of Air mail samples for San Francisco. In San Francisco

- 19 -
Air mail was not separated from the Regular mail until it

had been canceled and distributed through the Primary cases.

No special effort was made to draw a separate sample of Air

mail letters. Instead, it was treated as a part of the

Regular mail sample at the canceling machines.

Samples were taken during the period from 10 A.M. to

7 P.M. Although samples were collected throughout this time

interval there were two periods of concentration, about 12

noon and near 6 in the evening which corresponded in part

with the peak periods in handling outgoing mail.

For San Francisco and Los Angeles the size characteristics

were recorded from a metal template used as a measuring device.

See Figure 1 . The device has a vertical and horizontal

scale marked in quarter inch intervals. A letter was placed

on the flat surface and fitted into the ninety degree left

hand corner. The scale was visible at the top edge of the

left side of the envelope and at the bottom edge of the right

side of the envelope. The left hand vertical scale showed

the height (to within a quarter of an inch), and the right

hand scale showed the length (to within a quarter of an inch)

[One should not mistakenly assume that the letter sizes were

recorded in units of 1/16 of an inch. Due to some

difficulty in reading the measuring device, the inch scale had

- 20 -
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envelope, was recorded.

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described above was used with the exception that the height

and length dimensions were recorded to within 1/2 inch

intervals, as compared to 1/4 inch intervals for San

Francisco and Los Angeles. The color characteristic was not

recorded for Washington, D. C.

3.2.2 Metered mail . The samples of Metered mail were

taken from trays in the metered section at such a time when

two or more areas were operating. Metered mail is usually

already faced and therefore does not undergo the mixing

process characteristic of the Cancel mail at the facing

table. Furthermore Metered mail tends to "run", that is,

entire clusters of mail would have many of the same destination and size characteristics. Therefore a "bite" or

"bunch" of Metered mail would not have the required property

of randomness. What was done was to select, from the surface

tier of mail stacked in trays on a cart, successive letters

- 22 -
every two inches apart until approximately two inches of

letters were obtained. Again a lottery was employed to make

a random selection of the two areas to be sampled and the

sampling was always started at a preselected spot on the

tier.

3.2.3 Cull mail . Before letter mail reaches the

facing tables, a preliminary and rough separation is made.

This initial separation process is called culling. In the

three post offices studied in this report, this activity is

carried out on the mezzanine above the facing tables. Sacks

of collection mail are emptied on culling tables and

various types of mail, such as second and third class,

parcel post, meter, and all other mail given special

handling (undersize and oversize pieces are in this group)

are removed. The remainder is processed on the facing

tables below.

A knowledge of the distribution of mail (whatever the

classification) at this point in the processing stage would

provide information about a much larger class of mail than

studied at the facing table. Reported here is a sampling

study carried out directly at the culling table. The results,

while informative and unbiased, in that the methods provide

good estimates of the averages, are subject to more variability than is evidenced in the Cancel mail study.

- 23 -
Cull mail samples were selected in the following manner.

At a predetermined time, three bags of outgoing mail were

chosen at random and emptied on a clean table. The packages

and second class mail were removed and the remainder

thoroughly mixed by hand. A plan for drawing samples from

the remainder is representative of a general picture of

first class mail, air mail, and flats during the first

stages in the handling of outgoing mail. After the pile

was thoroughly mixed, the sampler reached into it and pulled

out about fifty pieces of mail (about one to three handf uls)

Each letter was measured for its height, length, thickness,

and the color was recorded as was described in the previous

discussion for Cancel mail.

But, one might ask, "Is this a good method of selecting

a subsample from the three bags of mail?*' A ready answer

was obtained as follows:

Three bags of outgoing mail were selected at random and

a complete enumeration was made of the combined contents

for the five categories of mail: Long Regular, Short Regular,

Long Air, Short Air, and Flats. The respective proportions

were calculated.

Then the combined contents of the three bags were

thoroughly mixed and a sampler pulled out about fifty letters

(one to three handfuls) . These results and their respective

- 24 -
proportions were calculated. Comparing the proportions

obtained by the two procedures showed that they were much

more in agreement than was first suspected. This comparison

was made on the spot.

3.2.4 Bulk mail . All Bulk mail that accumulated at the

post office within the approximate time period, 10 A.M. and

7 P.M., was considered in the sample. As each Bundle of Bulk

mail arrived, one Letter was removed and the total number of

Letters within the Bundle was estimated according to the

standard procedure by weighing the corresponding Bundle. The

Letter that had been removed was set aside with the information

about the total number of Letters within the Bundle. Later,

the Letters which had been removed from Bundles and allowed

to accumulate, were measured for height, length, and thickness

according to the procedure discussed under Cancel mail.

[Actually, this type of mail was not sampled statistically

because no random selection of letters was made. A partial

complete enumeration, which included all Bulk mail within a

certain time interval, was made. This is one instance where

a complete enumeration was simpler than resorting to a

statistical sampling plan.]

-25-
3.3 Analysis

3.3.1. Length, height, and thickness character istics

a) Presentation of data . Length, height, and thickness

are three variables of interest in this study. The data for

each variable for Los Angeles, San Francisco, and Washington,

D. C. are presented in Tables 2 to 21 . Each table gives

the frequency corresponding to the indicated class interval.

Some of the tables also present the relative frequency of

each class interval, i.e., the ratio of the number of pieces

in the class interval to the total sample. Other tables

give the cumulative percentage for each class interval, i.e.,

the sum of all percentages for all class intervals up to

and including the one in question.

We shall have occasion to use the term sample distribution, by which we mean the classification according to

one of the letter size characteristics of all the sample

data in one city and for one type of mail. For example, all

the data from Cancel Short Air mail for Los Angeles when

classified according to Length, is considered a sample distribution. Table 3 has the data for the example that has just

been cited, Cancel Short Air mail for Los Angeles by Length.

The column headed frequency shows the number of letters whose

lengths fall in the corresponding interval of measurement. The

column headed cumulative percentage shows the sum of all

- 26
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TABLE 6

Frequency (f) and cumulative percentage (%) by designated

Height of Cancel Short and Cancel Long mail at D.C.

Short Letters

Interval in Inches Regular Air Mail

f % f %

2 1/16 to 2 9/16

2 9/16 to 3 1/16 25 1.6 3 1.2

3 1/16 to 3 9/16 565 38.2 53 22.6

3 9/16 to 4 1/16 766 87.8 151 83.5

4 1/16 to 4 9/16 120 95.5 29 95.2

4 9/16 to 5 1/16 59 99.4 11 99.6

5 1/16 to 5 9/16 10 100.0% 1 100.0%

Total Sample Size 1545 248

Interval in Inches Long Letters

Regular Air Mail

f % f %

1 0.5

8 0.6 4 2.3

290 23.8 34 17.6

947 99.4 182 99.5

6 99.8 99.5

2 100.0% 1 100.0%

2 1/16 to 2 9/16

2 9/16 to 3 1/16

3 1/16 to 3 9/16

3 9/16 to 4 1/16

4 1/16 to 4 9/16

4 9/16 to 5 1/16

5 1/16 to 5 9/16

Total Sample Size 1253 222

- 31 -
TABLE 7

Frequency (f) and cumulative percentage (%) by designated

Length of Cancel Short and Cancel Long mail at O.C.

Short Letters Long Letters

Regular Air Mail Regular Air Hail

Interval in Inches f % f % f % f %

3 9/16 to 4 1/16 2 0.1

4 1/16 to 4 9/16 28 1.9

4 9/16 to 5 1/16 36 4.3 8 3.2

5 1/16 to 5 9/16 451 33.5 57 26.2

5 9/16 to 6 1/16 251 49.7 33 39.5

6 1/16 to 6 9/16 503 82.3 112 84.7

6 9/16 to 7 1/16 169 93.2 11 89.1

7 1/16 to 7 9/16 105 100.0% 27 100.0%

Total Sample Size 1545 248

7 9/16 to 8 1/16

8 1/16 to 8 9/16

8 9/16 to 9 1/16

9 1/16 to 9 9/16

9 9/16 to 10 1/16

10 1/16 to 10 9/16

10 9/16 to 11 1/16

11 1/16 to 11 9/16

11 9/16 to 12 1/16

6 0.5 4 1.8

22 2.2 2 2.7

257 22.7 34 18.0

947 98.3 177 97.7

3 98.6 3 99.1

13 99.6 2 100.0%

4 99.9

99.9

1 100.0%

Total Sample Size 1253 222

- 32 -
TABLE 8

Frequency (f) and cumulative percentage (%) by designated

Thickness of Cancel Short Regular and Cancel Short Air mail

at S.F. and L.A.

San Fra ncisco Los Ange les

Interval in

Inches Regular Air Mail Regular Air Mail

f % f % f % f %

to 1/8 436 94.2 53 80.3 762 95.3 297 90.0

1/8 to 2/8 23 99.1 13 100.0% 36 99.8 32 99.7

2/8 to 3/8 3 99.8 2 100.0% 1 100.0%

3/8 to 4/8 1 100.0%

4/8 to 5/8

Total Sample

Size 463 66 800 330

TABLE 9

Frequency (f) and cumulative percentage (%) by designated

Thickness of Cancel Long Regular and Cancel Long Air mail

at S.F. and L.A.

Interval in

Inches

San Francisco Los Angeles

Regular Air Mail Regular Air Mail

f % f % f % f %

to 1/8 252 86.6 34 66.7 528 90.9 252 87.2

1/8 to 2/8 26 95.5 11 88.2 51 99.7 32 98.3

2/8 to 3/8 11 99.3 6 100.0% 2 100.0% 5 100.0%

3/8 to 4/8 2 100.0%

4/8 to 5/8

Total Sample

Size 291 51 581 289

- 33 -
TABLE 10

Frequency (f) and cumulative percentage (%) by designated

Thickness of Cancel Short and Cancel Long mail at D. C.

Washington, D. C.

Short Letters Long Letters

Interval Regular Air Mail Regular Air Mail

f % f % f % f %

to 1/8

1/8 to 2/8

2/8 to 3/8

3/8 to 4/8

Total Sampl<

Size

1405

127

12

1

1545

90.9

99.2

99.9

100.0%

224

22

2

248

90.3

99.2

100 . 0%

1060

158

31

4

1253

84.6

97.2

99.7

100.0%

184

32

6

222

82.9

97.3

100.0%

- 34 -
TABLE 11

Frequency (f), relative frequency (r.f.), and cumulative

percentage (%) by designated Height of Metered mail at S.F,

Interval in Inches San Francisco

r.f. %.

< 1 13/16

1 13/16 to 2 1/16

2 1/16 to 2 5/16

2 5/16 to 2 9/16

2 9/16 to 2 13/16

2 13/16 to 3 1/16

3 1/16 to 3 5/16

3 5/16 to 3 9/16

3 9/16 to 3 13/16

3 13/16 to 4 1/16

4 1/16 to 4 5/16

4 5/16 to 4 9/16

4 9/16 to 4 13/16

4 13/16 to 5 1/16

5 1/16 to 5 5/16

5 5/16 to 5 9/16

5 9/16 to 5 13/16

5 13/16 to 6 1/16

6 1/16 to 6 5/16

6 5/16 to 6 9/16

6 9/16 to 6 13/16

6 13/16 to 7 1/16

7 1/16 to 7 5/16

7 5/16 to 7 9/16

> 7 9/16

2 .00 0.3

15 .02 2.8

105 .17 19.8

182 .30 49.4

121 .20 69.0

160 .26 95.0

23 .04 98.7

4 .01 99.4

2 .00 99.7

1 .00 99.8

00 100.0%

Total Sample Size 616 1.00

Table 12

Frequency (f), relative frequency (r.f,), and cumulative

percentage (%) by designated Thickness of Metered mail at S.F

Interval in Inches San Francisco

r.f. %

to 1/8

1/8 to 2/8

2/8 to 3/8

3/8 to 4/8

4/8 to 5/8

> 5/8

501 .81 81.3

47 .08 89.0

51 .08 97.2

11 .02 99.0

2 .00 99.4

4 .01 100.0

Total Sample Size 616 1.00

- 35 -
TABLE 13

Frequency (f), relative frequency (r.f.), and cumulative

percentage (%) by designated Length of Metered mail at S.F.

Interval in Inches San Francisco

r.f. %

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4 1/16 to 4 5/16

4 5/16 to 4 9/16

4 9/16 to 4 13/16

4 13/16 to 5 1/16

5 1/16 to 5 5/16

5 5/16 to 5 9/16

5 9/16 to 5 13/16

5 13/16 to 6 1/16

6 1/16 to 6 5/16

6 5/16 to 6 9/16

6 9/16 to 6 13/16

6 13/16 to 7 1/16

7 1/16 to 7 5/16

7 5/16 to 7 9/16

7 9/16 to 7 13/16

7 13/16 to 8 1/16

8 1/16 to 8 5/16

8 5/16 to 8 9/16

8 9/16 to 8 13/16

8 13/16 to 9 1/16

9 1/16 to 9 5/16

9 5/16 to 9 9/16

9 9/16 to 9 13/16

9 13/16 to 10 1/16

10 1/16 to 10 5/16

10 5/16 to 10 9/16

10 9/16 to 10 13/16

10 13/16 to 11 1/16

11 1/16 to 11 5/16

11 5/16 to 11 9/16

11 9/16 to 11 13/16

11 13/16 to 12 1/16

> : L2 i/ie

2

16

5

72

151

1

2

4

145

6

1

1

4

22

1

170

8

2

1

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.28

.00

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.00

.00

0.3

2.9

3.7

15.4

39.9

40.1

40.4

41.1

64.6

65.6

65.7

65.9

66.6

70.1

70.3

97.9

98.2

99.5

99.8

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Total Sample Size 616 1.01

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percentages for all class intervals up to and including the

one in question. There was one letter whose length fell in

the interval 3 13/16 to 4 1/16 inches and this represented

,3% of the total sample, 330 letters. There were four

letters whose length measured between 4 13/16 and 5 1/16

inches and this, along with the two previous letters,

comprised 1.8% of the total sample.

From this manner of presenting the data, we are able to

see how much of the sample was greater or less than a certain

interval

All of the data presented in Tables 2 to 21 include

post cards and all the analyses have been made with these

post cards included. Table 22 shows the number and percentage

of post cards in the samples.

Control charts for averages x have been included in

this report to show the consistency of averages of the samples

with regard to height and length. An average height and an

average length have been calculated for each sample that was

collected during the study. By combining all the averages of

samples collected for a particular type of mail, an overall

average x was calculated pertaining to each type. For

example, during the Los Angeles study, eleven samples were

collected of Cancel Short Air mail. The average sample size

n was 30. Altogether there were 330 letters in the total

- 44 -
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sample. The average heights x" in inches, from each of the

eleven samples were 3.86, 3.94, 4.00, 3.81, 3.72, 3.92, 3.83,

4.12, 3.98, 3.70, and 3.97. The overall average IT is 3.90.

Figure 2 shows this overall average x as a solid line.

The eleven sample averages are plotted as points which cluster

about the overall average. The broken lines above and beneath

the plotted points are the upper and lower control limits

(UCL, LCL) . Statistically speaking, these are three-sigma

limits but we shall not elaborate on this point - See

Appendix 1.1 . These control limits are not to be confused

with tolerance limits which are discussed in Part c of this

section. The fact that all of the points fall within the

control limits is strong evidence that the sampling methods

were carried out accurately and that the averages for the

two letter size characteristics, height and length, did not

change throughout the sampling period.

The control limits vary from one city and type of mail

to another. The width of the band depends upon the size of

the sample from which each sample average was calculated as

well as the sample values themselves. The smaller the sample

size, the wider the limits and vice versa. The average sample

size n is listed for each part. The agreement between cities

(Los Angeles and San Francisco) and types of letters (Regular

and Air mail) is remarkably good. See Figures 2 and 3

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48
We also see that the average height of the sample of Short

letters is slightly less than the average height of the

sample of Long letters.

The control charts for Long letters are exceedingly

uniform, and, in fact, a further statistical test showed

that there is no significant difference in the average

heights or the average lengths of Cancel mail. (A weighted

analysis of variance was used throughout Part (a) to test

differences between averages. See Appendix 1.2.)

The control charts for Short letters also exhibit

fairly good uniformity; however, statistical tests show a

significant difference in the average heights but no difference in the average lengths. It might be noted that circled

points for Cancel Short Air mail at San Francisco by length

are cases where the sample consisted of only one letter.

This is the reason why these points are so far from their

grand average and not because these points are averages of

letters with abnormally large or small lengths. Since the

average sample size was so small in comparison with the other

average sample sizes, no control limits are shown.

Referring to the control charts for averages IF for

Cull mail in figures 4 and 5 , the average lengths of

letters are not significantly different for the two cities,

Los Angeles and San Francisco. Furthermore, there is no

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significant difference between the average lengths of Regular

mail and Air mail. The variances of the heights for the four

groups are not constant; therefore no test for differences

between the average heights was made. At this point we would

like to mention that the variability in Cull mail is much

greater than that for Cancel mail. This is evidenced by the

wide control limits on Cull mail as compared to narrower

limits on Cancel mail. This further illustrates that the

control limits are a function of the variability of the

data as well as the sample size.

For Bulk mail, Figure 6 shows control charts for

Bundles. There is no significant difference in the average

lengths of Bundles between Los Angeles and San Francisco.

But the two cities do differ significantly with respect to

the average heights of Bundles.

b) Comparison of cities . It is interesting to note

that the size characteristics of Cancel mail do not appear to

differ from one region to another . We were able to show this

by comparing San Francisco, Los Angeles, and Washington, D.C.

two at a time. Statistical tests (Kolmogorov-Smirnov test

was used throughout Part b, See Appendix 1.3.) verify that

the cities do not differ significantly with respect to Cancel

mail. However, there is one exception in Cancel Short Regular

mail by length where it appears that San Francisco is much

- 52 -
Figure 6 Control charts for averages of Height and Length

for Bulk mail by Bundles data from S.F. and L.A.

S.F.

S.Zr

LA.

3S

~ UCL

10 ->- y

• *^ , • • >** * « \

k.^f • $**

Ul*

X

LCL

7S

7.5-L--

UCL

• •

• •

LCL

~n =Al/££AG-£ SAMPLE SlZ£

- 53 -
different from the others at one part of the distribution

curve (see Figure 7 ) . We believe this departure in the

curves is explainable because a closer look at the original

data indicated that an atypical sample, as compared to all

other samples for Los Angeles, San Francisco, and Washington,

D. C.j was collected wherein over 40% of the letters fell in

one size interval.

The comparison of cities just referred to is shown in

Figures 7 through 11 which are cumulative percentage

graphs of the sample distributions.

The data for all three cities have been plotted on the

same graph in order to enable the reader to visually compare

the sample distributions for the letter size characteristics.

On each graph, the vertical scale gives percentages ranging

from to 100. Horizontally the scale gives the size

characteristic measured in inches. The total number of

letters in each sample (i.e., the size of the entire sample

distribution) is recorded near each graph as N Sample Size.

Figures 12, 13, 16 show the cumulative percentage

graphs for Cull mail. For Cull mail the plotted sample

distributions show good agreement between Los Angeles and

San Francisco. In fact, statistical tests show that the two

cities do not differ significantly with respect to length and

height of Air mail, nor the height of Regular mail; however,

the two cities do differ significantly with respect to the

length of Regular mail.

- 54 -
HEJ&Ur (SHORT LETTERS)

5H0RTS

5F M-44,3

LA N~- 800

DC N- /S+5

i*//6 3% *% &k

LENGTH {SHORT LETTERS

)

1

?0

UU

T^'

^

80 f

70

Jl

\u Vb J \ &

60 1» ?5sb 1i 3* ^rj ^ )*40 / y

<> Ki.30 — f"

• 7

^20

i »/

/o

o -^3^^~-~ 1 1 1

i i *'#'/£ 4% <%'/&

WASH'NCrTOM , £.C

LOS ANGr£LB5

SAN FRANC/SCO

At- SAMPt-£ S/z£

Figure 7

Cumulative percentage curves for Height and Length of Cancel

Short Regular mail data of S.F., L.A., and D.C.

- 55 -
HEIGHT (5H0RT LETTERS)

too?o

,'Ji

80

70

.1

*v

60 —

50

40

30

20 ;

• // //

/O

» o 1 L. • ^ f i i 1 1 3% «'?/6 5% l'3

//6 ///CH£S

SHORTS

SF /•= 66

la #= yjo

DC //= £#S

*% 3% +'3/<6 ?'*//< t'fo

/NCHES

LENGTH (SHORT LE.TTBRS)

SAA/ *KAA/C/SCO

A/ ' SAMPLE S/Z.C

Figure 8

Cumulative percentage curves for Height and Length of

Cancel Short Air mail data for S.F., L.A., and D.C.

- 56 -
/oo

?o 80

^\io -

20

/O

HEIGrHT (LONG- L£Tr£PLS)

/.OA/G-S

DC. N * /Z£Z

2f%f

/<o JMCHB S

5% 6%

LFNG-rH (Lo NO- LETTERS')

_L 8% ?7/6

/NCH£S

/0//6 /ft //'*/,'/€>

WASHINGTON yO.C

LOS A/VG-^ZS

SAA/ f/ZAMC/SCO

A/- SAMPLE StZ£

Figure 9

Cumulative percentage curves for Height and Length of Cancel

Long Regular mail data for S.F., L.A., and D.C.

- 57 -
HEIGHT /"LONG LETTERS)

LONG-S

SF N*5I

LA N-2QJ

DC A/'- 222.

7% afjk 9%

i NCi-ieS

LO 5 AA/G-EI-&S

SAN ?&ANC/Sco

/V-SAMPJ-Z S/Z-£

Figure 10

Cumulative percentage curves for Height and Length of

Cancel Long Air mail data for S.F., L.A., and D.C.

- 58 -
LONG- (R.BG-ULARL) LONG- fA/£\ MAIL)

/uu

/^y^

?0 V

80

., 70

. ^

^ vb ,„

,N k ^ * 56 ^ ^ ^ <J -fO — ^ ^

^ ^ 3b

<o t^ -^20

/O - O i i i i i '% % % % %

/MCH£S

JOO

?o

80

10

\\*

s * § 5; »

/Oo 1

fc * X fc %

///exes

SHOR T (REG-ular)

100

7oBO

70

\ \

*°

V <b

Ik X- 5(3 t £

£,*> *

^

tv 30

^20

/0

-_•_* _? -£ *-

/8 % % % %

INCHES

SHORT (A/# MA/L)

/oo

*7

?o

/

80 /

7o

b ^ ^ _

" ?

^ ^i 4o -_~

20

/*o l ' 1 1 1 '£ Va Ye ?e 7s

WASHINGTON ,D.C

LOS AflJGr££-ES

^5AN FRANC/SCO

Figure 11

Cumulative percentage curves for Thickness of Cancel mail data

for S.F., L.A., and D.C.

- 59
HEIGHT

A/- SAMPLE 5/z.e

SAN FRANCISCO /V= //J3

LOS AM&FLES

Z% 3'fo +'fo 5%'/G 7%'/6

/oo

9o

So

10

V) hi ^ <!) GO

10o

LENGTH

i—^ »-»• i

sS/)// FR.ANC/SCO

_L *% 4% ^76 (>'3fc 7% 8% 9'fc /o'V* //'A*

JNCHtrs

Figure 12

Cumulative percentage curves for Height and Length of Cull

Regular mail data for S.F. and L.A.

- 60 -
HEIGHT

/oo

?o 80

v.*

7° \Vb CO -

$ % 40

20

to

>y,<,

N- SAMPLE. S/Z.&

SAN FR.ANCISCO N=/97

LOS AN6-ELES A/=S22-

I i i i i J L Z

/s//6 3% jfe £>% 6% 1%

IN C H £S

/OO

9o

80 V

K£60

zo

/oo

LENGTH

SAN FRANC/SCO

LOS AMG-£L£S

J_ 5% 4% *'%* 6'lk 7% 8'U 1'fc /oJ/6

/ N CH£ -5 //%76

Figure 13

Cumulative percentage curves for Height and Length of Cull

Air mail data for S.F. and L.A.

- 61 -
HE/G-HT

/oo

fo

80

10

3 £

*° \\ +

s £

^<?

loo I »JL«^

N- SAMPLE SIZE

SAN FRANC/SCO A/* 2?0

LOS A NGELES A/ ' 4/4

'% V% *'%

JNCHGS

s"/„'/<, <% 7%

i: *

/OOr9o80

70 -

CO

SO

40

30

20

to

o

LENGTH

SAN FG.ANC/SCO

LOS AN&EL-ES

_ i _. BBS "" 3% 4'3A S% 6'3M 7% e'V*

/A/Ct/ES

9% /o% //%

Figure 14

Cumulative percentage curves for Height and Length of Bulk

mail by Bundles data for S.F. and L.A.

- 62 -
hi EIGHT

/oo

90

80

10

GO

50

4o

30

20

10

/"

1% *% *% *%

N -SAMPLE SIZ£

SAN FRANCISCO fil- J85, 70-+ LOS ANGELES N*/,6Z0,823

I J *% '% 7%

LenGt /•

S/\N FRANCISCO

/.OS AA/<r£LES

3% 4% S% *1k 7% 8'3A ?% ,o'3/,6 //%

/7VC//£\S

Figure 15

Cumulative percentage curves for Height and Length of Bulk

mail by Letters data for S.F. and L.A,

- 63 -
BULK. MAIL (LETTERS) BULK MAIL (BUNDLES)

I-

/CO

£ ^^-~ loo .. i

~"

?a 90

80 80

10 70

6,0 5o

30

^ ^ 40

1**

20 - *, * 40 -

10 /O -

Q 1 1 1 1 1 i 1 i i i 1 1 i 1 1 1 1 1

| | % % % % % fc % ?8 % %

JNCHES INCHES

CULL MAIL (REGULAR) CULL MAIL(AIR MAIL)

4, ^J

k k ^ Hi

^3!

/oo

90

80

70 -

io - fo40

30

2.0

/Oo I I I L J I I I % % % % %

X

k

/oo

9o

80

10 ^ kO

m 4o

9C 3o

20

IOo

I I I I I I I I I

'k'3 %

JNCH£S

9 %

/A/C/S£"S

%'8 %

5AA/ FR.ANCISCO

LOS ANGELBS

Figure 16

Cumulative percentage curves for Thickness of Bulk mail

and Cull mail data for S.F. and L.A.

- 64 -
Due to the limitation of the device for measuring sizes,

there were a large number of letters that were either smaller

or larger than could be measured, (This was especially the

case for Cull mail and Bulk mail.) These letters were lumped

together in the "catch-all" classes of "less than" or "greater

than." This is particularly true of the "greater than"

category where in some cases as much as 10% of the mail

sampled was too large for the measuring device. This amounts

to loss of information when we really do not know what the

distribution is like in the "larger than" category. We,

therefore, recommend that in any future sampling studies the

device for measuring sizes be enlarged to cope with the

bigger sized envelopes.

Figures 14 through 16 show cumulative percentage graphs

for Bulk mail both by Bundles and by Letters, From the

graphs it is evident by the departure in the curves that the

length and height characteristics of Bulk mail are apparently

not the same in Los Angeles as in San Francisco. Statistical

tests verify that this departure is, in fact, statistically

significant.

c) Tolerance limits . One might next ask if we can be

sure that at least a certain percentage of the mail, for each

letter size characteristic, will lie within certain sizes.

- 85 -
The answer is yesj it is possible to predict with a prescribed

confidence coefficient that at least a fixed percentage of

the mail will lie within certain size limits. These predictions

are called tolerance limits and they show the predicted

limits within which at least a specified proportion of the

mail is contained. There are two types of tolerance limits,

two-sided limits and one-sided limits. If we are interested

in tv/o limits which contain at least a specified proportion of

the mail, we use two-sided limits. If we are interested in

only one limit, either above which or below which at least a

specified proportion of the mail will lie, then we use onesided limits. The confidence coefficient for Tables 23 to 31

and 41 to 43 was ,95, which means that we shall be correct

in expecting at least the prescribed percentage of the mail

to be within the tolerance limits unless the sample from

which we deduced this result was anomalous to an extent that

would arise no more than one time in twenty. For detailed

discussions of statistical tolerance limits see the references

referred to in Appendix 2.1

The tolerance limits for the length of Cancel mail are

one-sided. This is due to the breaking point between Short

and Long letters. A Short letter, by definition, has a

limiting length less than 7 9/16 inches and so we only are

interested in the proportion of letters that are greater than

a certain size.

- 66 -
Tables 23 through 31 show tolerance limits which are predicted sizes which contain at least 90%, 95%, and 99% of each

type of mail. We can see that for Cancel mail (Tables 23

through 26) the three cities are fairly consistent in the

size intervals that include various proportions of the mail.

The tolerance limits point out the fact that the culling or

separation procedure varies from one post office to another

with regard to the definition of "oversize . " For instance,

it is apparent from the tolerance limits on Cancel mail that

more letters are culled between the sizes 9 9/16 and 10 9/16

inches at Los Angeles than at San Francisco (see Tables 25,

26). in other words, the definitions of what constitutes

the oversized mail that is pulled out before facing are not

uniform.

Table 27 shows the 90%, 95%, and 99% tolerance limits

on Metered mail based on data from San Francisco. This is

the only Metered mail sample collected; therefore no comparisons between cities can be made.

It was not possible to place 90%, 95%, and 99% tolerance

limits on Cull mail as has been done on Cancel mail because a

sizable proportion of the sample distributions fell in the

"greater than" class. That is, there was a considerable number

of the envelopes that could not be measured because they were

larger than at least one dimension of the measuring device

and therefore these envelopes were lumped in the "greater than"

class. In order to discuss and compare Cull mail, we

- 67 -
TABLE 23

Predicted lower and upper limits for 90%, 95%, and

99% of Cancel Short Regular mail based on data from

S.F., and L.A., and D.C. Table gives 90%, 95%, and

99% tolerance limits with confidence coefficient .95.

Lower Tolerance Limit Upper Tolerance Limit

Percent of

mail covered

Height Length Height Length

(in inches) (in inches)

San Francisco

90%

95%

99%

3

3

3

2/16

1/16

1/16

5 5/16 4 1/16

5 5/16 4 5/16

4 5/16 4 13/16

Los Angeles

(a)

(a)

(a)

90%

95%

99%

3

2

2

1/16

15/16

5/16

5 5/16 4 12/16

4 5/16 5 1/16

3 13/16 6 5/16

Washington, D. C.

(a)

(a)

(a)

90%

95%

99%

3

3

3

2/16

1/16

1/16

5 2/16 4 9/16

4 15/16 4 14/16

4 4/16 5 1/16

(a)

(a)

(a)

(a) By definitionj Short letters are less than 7 9/16 inches in

length.

- 68 -
TABLE 24

Predicted lower and upper limits for 90%, 95%, and

99% of Cancel Short Air mail based on data from

S.F., L.A., and D.C. Table gives 90%, 95%, and

99% tolerance limits with confidence coefficient .95

Lower Tolerance Limit Upper Tolerance Limit

Percent of

mail covered

Height

(in

Length Height

inches) (in i

Length

nches)

San Francisco

90%

95%

99%

3

3

3

9/16

9/16

9/16

5 6/16 4 5/16

5 5/16 4 5/16

5 5/16 4 5/16

Los Angeles

(a)

(a)

(a)

90%

95%

99%

3

3

3

5/16

2/16

1/16

5 7/16 5 1/16

5 3/16 5 5/16

4 1/16 5 9/16

Washington, D. C.

(a)

(a)

(a)

90%

95%

99%

3

3

3

2/16

1/16

1/16

5 2/16 4 12/16

5 1/16 5 1/16

5 1/16 5 1/16

(a)

(a)

(a)

(a) By definition, Short letters are less than 7 9/16 inches in

length.

- 69 -
TABLE 25

Predicted lower and upper limits for 90%, 95%, and

99% of Cancel Long Regular mail based on data from

S.F., L.A., and D.C. Table gives 90%, 95%, and 99%

tolerance limits with confidence coefficient .95.

Lowei Tolerance Limit Upper Tolerance Limit

Percent of

mail covered

Height

(in

Length Height Length

inches) (in inches)

San Francisco

90%

95%

99%

3

3

3

12/16

9/16

9/16

(a) 4 5/16

(a) 4 5/16

(a) 4 5/16

Los Angeles

10 4/16

10 5/16

10 5/16

90%

95%

99%

3

3

3

13/16

10/16

9/16

(a) 4 5/16

(a) 4 5/16

(a) 4 13/16

Washington, D. C.

9 9/16

9 9/16

9 9/16

90%

95%

99%

3

3

3

10/16

9/16

9/16

(a) 4 9/16

(a) 4 9/16

(a) 5 1/16

9 8/16

9 9/16

10 7/16

(a) By definition, Long letters are equal to or greater than 7 9/16

inches in length.

- 70 -
TABLE 26

Predicted lower and upper tolerance limits for 90%, 95%,

and 99% of Cancel Long Air mail based on data from S.F.,

L.A., and D.C. Table gives 90%, 95%, and 99%

tolerance limits with confidence coefficient .95.

Lower Tolerance Limit Upper Tolerance Limit

Percent of

mail covered

Height Length Height

(in inches) (in

Length

inches)

San Francisco

90%

95%

99%

3

3

3

5/16

5/16

5/16

(a) 4 5/16

(a) 4 5/16

(a) 4 5/16

Los Angeles

10 9/16

10 9/16

10 9/16

90%

95%

99%

3

3

3

13/16

10/16

9/16

(a) 4 5/16

(a) 4 5/16

(a) 5 5/16

Washington, D. C.

9 9/16

9 9/16

9 9/16

90%

95%

99%

3

3

3

9/16

5/16

1/16

(a) 5 1/16

(a) 5 1/16

(a) 5 1/16

9 9/16

9 9/16

10 1/16

(a) By definition, Long letters are equal to or greater than 7 9/16

inches in length.

- 71 -
TABLE 27

Predicted lower and upper limits for 90%, 95%, and

99% of Metered mail based on data from S.F. Table

gives 90%, 95%, and 99% tolerance limits with

confidence coefficient .95.

Lower Tolerance Limit Upper Tolerance Limit

Percent of Height Length Height Length

mail covered (in inches) (in inches)

90%

95%

99%

3 5/16 6 5/16 4 6/16 9 9/16

3 4/16 5 7/16 4 9/16 10 5/16

3 1/16 5 1/16 5 9/16 11 5/16

- 72 -
arbitrarily chose natural cut-off points which coincide with

the Cancel mail sample distributions. For example the distributions of Cull, Cancel Short, Cancel Long mail by length have

somewhat the patterns shown below (for the sake of comparison

we have distorted the fact that the area under each of the

distribution curves is equal to one)

CULL MAIL

LBNG TH

Greater, than

' class

LENG-TH

A value (11 l/16 ft ) in the upper tail of the Cancel Long sample

distribution was thus chosen as the cut-off point for the length

of Cull mail. Similarly, the distributions of Cull mail and

Cancel mail by height have somewhat the following patterns:

- 73 -
HEIGHT

"G-KEA TEJl THAN" CLA 55

CANCEL MAIL

HEIG-HT

Again a value (5 9/16") in the upper tail of the Cancel

sample distribution was chosen as the cut-off point for the

height of Cull mail. These cut-off points are shown as

dotted lines in Tables 14 and 15. Hence, letters greater

than 5 9/16 inches in height, or greater than 11 1/16 inches

in length are considered Oversize. Tables 28 and 29 show the

sizes which include at least 90%, 95%, 99%, of all the Cull

mail, but the reader should keep in mind that these are

tolerance limits on truncated distributions as described above.

- 73a -
With few exceptions, there again seems to be fairly good

agreement between Los Angeles and San Francisco with

respect to the predicted sizes that will include at least

certain proportions of the mail. Also, Table 32 shows the

proportions (expressed as a percentage) with confidence

limits, of the mail that we can expect to fall in the

Oversize category. Los Angeles appears to have a larger

proportion of Oversize mail than San Francisco.

Tables 30 and 31 give the tolerance limits that have

been placed on Bulk mail. Again, three estimates have been

given which include at least 90%, 95%, and 99% of all Bulk

mail. Furthermore, here as in Cull mail there is a "greater

than" class. Table 33 shows the "greater than" proportions

and limits associated with these proportions.

Wherever permissible, as judged by the statistical

tests cited in the above discussion, the data have been

combined. In table 41 we present 95% tolerance limits for

- 73b -
TABLE 28

Predicted lower and upper limits for 90%, 95%, and

99% of Cull*Regular mail based on data from S.F.

and L.A. Table gives 90%, 95%, and 99% tolerance

limits with confidence coefficient .95.

Lower Tolerance Limit Upper Tolerance Limit

Percent of Height Length

mail covered

Height Length

(in inches) (in inches)

90%

95%

99%

3 2/16

2 13/16

2 13/16

San Francisco

5 3/16

4 15/16

3 13/16

4 5/16

4 10/16

5 5/16

9 9/16

9 9/16

11 1/16

90%

95%

99%

3 4/16

3 2/16

2 12/16

Los Angeles

5/16

5 1/16

4 3/16

4 5/16

4 11/16

5 1/16

9 12/16

9 13/16

10 9/16

These limits have been determined by using the sample

distributions truncated as described on page 73.

- 74 -
TABLE 29

Predicted lower and upper limits for 90%, 95%, and

99% of Cull*Air mail based on data from S.F. and

L.A. Table gives 90%, 95%, and 99% tolerance

limits with confidence coefficient .95.

Lower Toleranee Limit Upper Tolerance Limit

Percent of

mail covered

Height

(in

Length He

inches)

ight

(in

Length

inches)

San Francisco

90%

95%

99%

3 8/16

3 5/16

3 5/16

5 6/16 4

4 15/16 4

4 13/16 5

Los Angeles

5/16

13/16

1/16

9 9/16

9 9/16

9 9/16

90%

95%

99%

3 8/16

3 6/16

3 5/16

5 6/16 4

5 3/16 5

5 1/16 5

4/16

1/16

5/16

9 13/16

10 5/16

10 13/16

These limits have been determined by using the sample

distributions truncated as described on page 73.

- 75 -
TABLE 30

Predicted lower and upper limits for 90%, 95%, and

99% of Bulk mail by Bundles based on data from

S.F. and L.A. Table gives 90%, 95%, and

99% tolerance limits with confidence

coefficient .95.

Lower Tolerance Limit Upper Tole'ranee Limit

Percent of

mail covered

He ight Length Height

(in inches) (in

Length

inches)

San Francisco

90%

95%

99%

3

3

3

6/16

3/16

1/16

5 6/16 (b)

5 5/16 (b)

4 5/16 (b)

Los Angeles

11 5/16

(b)

(b)

90%

95%

99%

3

3

2

4/16

1/16

13/16

5 6/16 (b)

5 5/16 (b)

4 1/16 (b)

11 14/16

11 16/16

(b)

(b) No upper limits could be calculated because of the limitations of the measuring device.

- 76 -
TABLE 31

Predicted lower and upper limits for 90%, 95%, and

99% of Bulk mail by Letters based on data from

S.F. and L.A. Table gives 90%, 95%, and 99%

tolerance limits with confidence coefficient .95

Lower Tolerance Limit Upper Toleranee Limit

Percent of

mail covered

He ight Length Height

(in inches) (in

Length

inches)

San Francisco

90%

95%

99%

3

3

3

8/16

6/16

3/16

5 7/16 (b)

5 6/16 (b)

5 5/16 (b)

Los Angeles

10 9/16

11 4/16

(b)

90%

95%

99%

3

3

2

9/16

5/3.6

13/16

5 7/16 (b)

5 5/16 (b)

4 1/16 (b)

11 9/16

11 14/16

(b)

(b) No upper limits could be calculated because of the limitations

of the measuring device.

- 77 -
Table 32 Percentages*with specified dimensions

and the 99% confidence limits of Cull Regular

and Cull Air mail

Regula r Air Mai.1

San

Francisco

Los

Angeles

San

Francisco

Los

Angeles

99%

Percent

> 5 9/16 inches

in height

Confidence

limits

3.1

2%, 4%

6.3

4%, 8%

4.1

1%, 9%

6.2

3%, 11%

99%

Percent

> 11 1/16

inches in length

Confidence

1 imits

1.9

1%, 3%

4.3

3%, 5%

3.0

1%, 7%

4.7

3%, 9%

Percent

> 5 9/16 inches

in height and

> 11 1/16 inches

in length

1.7 4.2 2.5 4.3

99% Confidence

1 imits

1%, 3% &/0f O A> 1%, 7% 3%, 9%

These are estimates of the percentages of mail cut-off in

the truncation process described on page 73.

- 78 -
TABLE 33

Percentages*with specified dimensions and the 99%

confidence limits of Bulk mail by Bundles and

Bulk mail by Letters

Bundles Letters

San Los San Los

Francisco Angeles Francisco Angeles

Percent

> 7 9/16 inches

in height

10.3 16.9 5.6 22.1

99% Confidence

limits 7%, 16% 13%, 23% 3%, 11% 17%, 27%

Percent

> 12 1/16 inches

in length

99% Confidence

limits

1.4

0%, 3%

0.7

0%, 2%

0.6

0%, 2%

0.7

0%, 2%

These are estimates of the percentages of mail whose

measurements are greater than the limitations of the

measuring device.

- 79 -
Cancel mail based on the combined San Francisco, Los

Angeles and Washington, D. C. data. Table 42 gives 95%

tolerance limits for Cull mail for the combined data of

San Francisco and Los Angeles. Note that this latter table

also presents tolerance limits based on the Regular and Air

mail recombined as they were originally collected. The

O/ersized Cull mail is not represented here. Table 43

presents 95% tolerance limits based on all the Cancel

Regular mail samples collected at San Francisco and Los

Angeles. This includes both Long and Short letters. These

can be combined because the samples of Long and Short letters

were taken approximately in proportion to their relative

volume as estimated by the method of Section 6

Figures 17 through 22 are frequency histograms of

the combined data discussed in Tables 41 through 43

The original data for these figures appear in Tables 34

through 40 . The histograms show the relative frequencies

of the data for each corresponding size interval. In addition Figures 17 and 18 also portray the 95% tolerance

limits previously tabulated. No tolerance limits are given

on thickness; however, it is interesting to note the following

interval estimates (i.e., .99 confidence limits) for the

proportion of a given type mail less than or equal to 2/8

inches in thickness

- 80 -
TABLE 34

Frequency (f) and relative frequency (r.f.) by designated

Length of Cancel Short and Cancel Long mail at S.F.,

L.A., and D.C. combined

l Inches

Short Letters Long Letters

Interval ir Regular Air Mail Reg alar Air Mail

f r.f. f r.f. f r.f. f r.f.

3 9/16 to 4 1/16 13 .01 1 .00

4 1/16 to 4 9/16 50 .02

4 9/16 to 5 1/16 63 .02 13 .02

5 1/16 to 5 9/16 674 .24 90 .14

5 9/16 to 6 1/16 394 .14 111 .17

6 1/16 to 6 9/16 1130 .40 317 .49

6 9/16 to 7 1/16 287 .10 44 .07

7 1/16 to 7 9/16 197 .07 68 .11

7 9/16 to 8 1/16 13 .01 5 .01

8 1/16 to 8 9/16 37 .02 14 .02

8 9/16 to 9 1/16 405 .19 66 .12

9 1/16 to 9 9/16 1638 .77 467 .83

9 9/16 to 10 1/16 4 .00 4 .01

10 1/16 to 10 9/16 21 .01 5 .01

10 9/16 to 11 1/16 4 .00 1 .00

11 1/16 to 11 9/16 2 .00

11 9/16 to 12 1/16 1 .00

Total Sampl e Size 2808 1.00 644 1.00 2125 1.00 562 1.00

- 81 -
TABLE 35

Frequency (f) and relative frequeiicy (r.f.) by des ignated

Height of Cancel Short and Cancel Long mail at S.]P., L.A . , and

D . C. combined

il in Inches

Short Letters Long Letters

Intervs Regular Air Mail Regular Air Mail

f r.f. f r.f. f r.f. f r.f

2 1/16 to 2 9/16 2 .00 3 .01

2 9/16 to 3 1/16 54 .02 56 .09 1 .00

3 1/16 to 3 9/16 947 .34 202 .31 16 .01 7 .01

3 9/16 to 4 1/16 1481 .53 288 .45 460 .22 74 .13

4 1/16 to 4 9/16 192 .07 56 .09 1640 .77 478 .85

4 9/16 to 5 1/16 109 .04 25 .04 7 .00

5 1/16 to 5 9/16 16 .01 12 . 02 2 .00 2 .00

5 9/16 to 6 1/16 3 .00 2 .00

6 1/16 to 6 9/16 4 .00

6 9/16 to 7 1/16

7 1/16 to 7 9/16

Total Sample Size 2808 1.01 644 1.01 2125 1.00 562 .99

TABLE 36

Frequency (f) and relative frequency (r.f.) by designated

Thickness of Cancel Short and Cancel Long mail at S.F., L.A.,

and D.C. combined

inches

Short Lette;rs Long

Regular

f r.f.

Letters

Air

f

Interval in ] Regular

f r.f.

Air

f

Mail

r.f.

Mail

r.f.

to 1/8

1/8 to 2/8

2/8 to 3/8

3/8 to 4/8

4/8 to 5/8

Total Sample Size

2603

186

17

2

2808

.93

.07

.01

.00

1.01

574

67

3

644

.39

.10

.01

1.00

1340 .87

235 .11

44 .02

6 .00

2125 1.00

470

75

17

562

.84

.13

.03

1.00

- 82 -
TABLE 37

Frequency (f) and relative frequency (r.f .) by designated Length

of Cull mail at S,F. and L.A. combined

Regular Air Mail Regular + Air

Interval iri Inches f r.f. f . r.f. f r.f.

< 3 13/16 6 .00 6 .00

3 9/16 to 4 1/16 1 .00 1 .00

4 1/16 to 4 9/16 18 .01 18 .01

4 9/16 to 5 1/16 45 .01 4 .01 49 .01

5 1/16 to 5 9/16 296 .10 30 .06 326 .09

5 9/16 to 6 1/16 195 .06 41 .08 236 .07

6 1/16 to 6 9/16 686 .22 189 .36 875 .24

6 9/16 to 7 1/16 105 .03 19 .04 124 .03

7 1/16 to 7 9/16 232 .08 36 .07 268 .07

7 9/16 to 8 1/16 30 .01 1 .00 31 .01

8 1/16 to 8 9/16 19 .01 6 .01 25 .01

8 9/16 to 9 1/16 159 .05 6 .01 165 .05

9 1/16 to 9 9/16 1113 .36 154 .30 1267 .35

9 9/16 to 10 1/16 8 .00 6 .01 14 .00

10 1/16 to 10 9/16 31 .01 4 .01 35 .01

10 9/16 to 11 1/16 8 .00 2 .00 10 .00

11 1/16 to 11 9/16 3 .00 2 .00 5 .00

11 9/16 to 12 1/16 52 .02 11 .02 63 .02

> 12 1/16 50 .02 8 .02 58 .02

Total Samp

.e Size 3057 .99 519 1.00 3576 .99

- 83 -
TABLE 38

Frequency (f) and relative frequency (r.f.) by designated

Height of Cull mail at S.F. and L.A. combined

Regular Air Mail Regular + Air

Interval in Inches f r.f

f r.f. f r.f.

2 1/16 to 2 9/16 1 .00 1 .00

2 9/16 to 3 1/16 48 .02 48 .01

3 1/16 to 3 9/16 505 .17 29 .06 534 .15

3 9/16 to 4 1/16 1058 .35 247 .48 1305 .36

4 1/16 to 4 9/16 1212 .40 190 .37 1402 .39

4 9/16 to 5 1/16 63 .02 20 .04 83 .02

5 1/16 to 5 9/16 14 .00 5 .01 19 .01

5 9/16 to 6 1/16 9 .00 6 .01 15 .00

6 1/16 to 6 9/16 10 .00 1 .00 11 .00

6 9/16 to 7 1/16 2 .00 2 .00

7 1/16 to 7 9/16 19 .01 2 .00 21 .01

> 7 9/16 116 .04 19 .04 135 .04

Total Sample Size 3057 1.01 519 1.01 3576 .99

TABLE 39

Frequency (f) and relative frequency (r.f.) by designated

Thickness of Cull mail at S.F. and L.A. combined

Regular Air Mail Regular + Air

Interval in Inches f r.f. f r.f. f r.f.

to 1/8 2613 .85 449 .87 3062 .86

1/8 to 2/8 304 .10 41 .08 345 .10

2/8 to 3/8 77 .03 14 .03 91 .03

3/8 to 4/8 28 .01 6 .01 34 .01

4/8 to 5/8 13 .00 3 .01 16 .00

> 5/8 22 .01 6 .01 28 .01

Total Sample Size 3057 1.00 519 1.01 3576 1.01

- 84 -
TABLE 40

Frequency (f) and relative frequency (r.f.) by designated

- Thickness, Height, and Length of Cancel Regular mail

(includes both Long and Short letters) at S.F. and L.A. combined

Interval in

Inches Thickness

f r.f.

Height

f r.f.

Length

f r.f.

1978 .93

136 .06

18 .01

3 .00

to 1/8

1/8 to 2/8

2/8 to 3/8

3/8 to 4/8

4/8 to 5/8

> 5/8

1 9/16 to 2 1/16

2 1/16 to 2 9/16

2 9/16 to 3 1/16

3 1/16 to 3 9/16

3 9/16 to 4 1/16

4 1/16 to 4 9/16

4 9/16 to 5 1/16

5 1/16 to 5 9/16

5 9/16 to 6 1/16

6 1/16 to 6 9/16

6 9/16 to 7 1/16

7 1/16 to 7 9/16

> 7 9/16

7 9/16 to 8 1/16

8 1/16 to 8 9/16

8 9/16 to 9 1/16

9 1/16 to 9 9/16

9 9/16 to 10 1/16

10 1/16 to 10 9/16

10 9/16 to 11 1/16

11 1/16 to 11 9/16

11 9/16 to 12 1/16

s

> 12 1,'16

2 .00

29 .01

390 .18

885 .41 11 .01

765 .36 22 .01

51 .02 27 .01

6 .00 223 .10

3 .00 143 .07

4 .00 627 .29

118 .06

92 .04

7 .00

15 .01

148 .07

691 .32

1 .00

8 .00

.00

Total Sample Size 2135 1.00 2135 .99 2135 .99

- 85 -
TABLE 41

Predicted lower and upper limits for 95% of Cancel mail based on

data from S.F., L.A., and D.C. combined. Table gives 95%

tolerance limits with confidence coefficient .95.

Lower Tolerance Limit Upper Tolerance Limit

Regular Air Mail Regular Air Hail

Short Letters

(in inches)

Height 3 1/16 2 10/16 4 14/16 5 2/16

Length 5 1/16 5 2/16 (a) (a)

Long Letters

(in inches)

Height 3 10/16 3 9/16 4 9/16 4 9/16

Length (a) (a) 9 9/16 9 9/16

(a) By definition. Short letters are less than 7 9/16 inches in

length and Long letters are equal to or greater than 7 9/16

inches.

- 86 -
TABLE 42

Predicted lower and upper limits for 95% of Cull*

Regular, Cull Air, and Cull Regular and Air mail

based on data from S.F. and L.A. combined.

Table gives 95% tolerance limits with

confidence coefficient .95.

Lower Tolerance Limit Upper Tolerance Limit

Type of Mail Height Length

(in inches)

Height Length

(in inches)

Regular

Air Mail

Regular and

Air Mail

3 1/16

3 9/16

3 1/16

5

5

0/16

3/16

1/16

4 11/16 9 9/16

4 14/16 9 13/16

4 12/16 9 9/16

TABLE 43

Predicted lower and upper limits for 95% of Cancel mail

(Long and Short letters combined) based on data from S.F.

and L.A. combined. Table gives 95% tolerance limits with

confidence coefficient .95.

Type of Mail

Lower Tolerance Limit

Height Length

(in inches)

Upper Tolerance Limit

Height Length

(in inches)

Regular 3 1/16 4 12/16 4 12/16 9 9/16

* These limits have been determined by using the combined sample

distributions truncated as described on page 73.

- 87 -
LENGTH

J0 KEG. SHORT

9

6

>4-

-2

' ?S% OF"MAIL

r4 *-

4'M S'/k (,'/j6 7'/,e>

LZAl&TH

/0_, AIRMAIL SHORT

8-

•6-

•2-1 o

W# OP MAJC

I ta

-^K $'//* (,% f//*

INCHES

I

/.«-!

.8-

•4-

HEIG-HT

RtTGr. &UORT

'-* »J ?*%«>£- /W/£

Ao

8

.6

-4

O

iV'c 3fc +V* SV/i

/*ICH£S

J.o n 8

.4

4-

• 2-

HFIGHT

A/RMAiL SHORT

\J5% C MAIL

r*—

A0-,

• 8

• 6

2 4

*f//6 SVif, 4V/6 SV'6

/NC#£S

TH/CKA/BSS

A/R MA/L stfofiZT

J L WCHES

Figure 17

Relative frequency histograms for Length, Height, and

Thickness of Cancel Short mail for combined data of

S.F. , L.A. , and D. C.

- 88 -
LZNG-T/i

R£6. LONG-

/:o8-

6- +

f2-

o

»l fifcofr/HAJL

J L

^ -8-

a -6-

&//<> 9'/* /o'fii //'/«,

H£(&HT

i

1-

2-

95% *r#tAtL

8-

6-

-*-

•2-

*% 3% 4& 5*/,6

///oxers

_i L

/O-i

L£N&TH

A/K MA IL LO^Gr

8-

\J£%ofr*tA/C *-

2-

I o

|

—I—,_ »& ^'/'c /»'//*, ft

//*£#££

6.

A/R.A4A

<7

I

l 9S% of /V)AIL

I

I—. i i

4-

•2-

xV/c 3>//(. rk Wi,

/.Oj

T/¥/<r*A/£rss

A//^ MA/L tONGr

-8- —

6-

y4sYe

//yc//ss

% %%

Figure 18

Relative frequency histograms for Length, Height, and

Thickness of Cancel Long mail for combined data of

S.F. , L.A. , and D.C.

- 89 -
.f-l

-3-

.2

• /-I

LENGTH (R.SGULAR)

U

<?% ^7/6 *#4

* 7% 8& ?#6 ZO%, //fa >IZ'U INCHES

!

is

iHEIGHT (REGULAR)

r

n

4-

3-

2- —

•/- o LT —f r—

n

/.o

Z

9

//6 3

Jfa 4'//* S%6 <>*//<, I?'*

/A/c^/es

THICKNESS (REGULAR.

•8-

6-

4

Z

/NCHEs

Figure 19

Relative frequency histograms for Length, Height, and

Thickness of Cull Regular mail for combined data of S.F

and L.A.

- 90
.*•- 1 FNGTH $R.£&ulak

4-

3-

Z-

/ -

i I—iHT-r-H i—i i - 1—1—

<*% 4

?A SVk 1% 7f* 8

9

//6 9& f<>V/6 // 9//6>/*%6

^ ; H£l&HT{%Tti?L

p

- P^

°

SLjk 3f/c 4% SV/6. t% 7%

THICKNESS {VJ&ZJ*

,-n

8-

6-

4-

•2- o H 3=L

Figure 21

Relative frequency histograms for Length, Height, and

Thickness of Cull Regular plus Air Mail for combined

data of S.F. and L.A.

- 92 -
, s LEN6TH fair mail)

>4-

5-

.1-

/-

j=Q ]—

n

XZL. T—

I—

t

<3%4% SV,6 £%£ Vh 8% P/,t 10% i,% »2'fo

1

Or

Vi

Hi

SI

HEIGHT (AIR ma/l)

•5-1

-*-

•5-

2-

./-

n J—

L

n

THICKNESS (AIR. MAIL)

•«-

.(I

2-

o

7a % %% %

Figure 20

Relative frequency histograms for Length, Height, and

Thickness of Cull Air mail for combined data of S.F.

and L.A.

- 91 -
LENGTH

5 - REGULAR

4-

3-

2

-

1 -

o L 1 1 ~1 r- 1 1 1 1 1 1 3%, 4%<. 0% 6% 7% S% f% fo\ ff 9U

I

•s. k

5-i

4Z1-

./_

H£/GHT

/2GG-ULAR.

O I—p=d I

I \=L L_l I I i I I 2% 3% 4%i &* (> 9

//6 yv«>

TH/CKNESS

3-

6

4-

ZJ i I

Figure 22

Relative frequency histograms for Length, Height, and

Thickness of Cancel mail for combined data of S.F. and.

L.A.

- 93 -
Short Regular mail

Short Air mail

Long Regular mail

Long Air mail

Proportion less than or equal to

2/8 inches in thickness

.983 to 1.000

.985 to 1.000

.967 to .987

.960 to .980

3.2.2 Color Characteristic . At the request of the

Post Office Department a fourth letter size characteristic,

color, was recorded during the San Francisco and Los Angeles

studies. This record was made at the same time the other

three characteristics were measured. Color indication was

made according to the following: White, Yellow, Red, Blue,

Gray, Brown, and Multi-color.

Tables 44 through 46 show the sample distributions

of color in Cancel, Cull, and Metered mail for San Francisco

and Los Angeles. The tables also list relative frequencies

according to color.

Statistical tests (Kolmogorov-Smirnov test used here)

show that there is no difference between Los Angeles and

San Francisco in the colors of Cull Regular mail. There

also is no difference in Cull Air mail between the cities.

There is no difference between cities in Cancel Regular mail

but there is a significant difference in Cancel Air mail.

Sufficient investigation has not been made to explain why such

a difference should exist. This difference may be due to the

- 94 -
TABLE 44

Frequency and relative frequency by designated Color of

Cancel Regular and Cancel Air mail at S.F. and L.A.

RegulLar Air Mail

Color Relative Relative

Frequency Frequency Frequency Frequency

SAN FRANCISCO:

White 613 .88 64 .66

Yellow 32 .05 1 .01

Red 5 .01 1 .01

Blue 17 .02 7 .07

Gray 13 .02 23 .24

Brown 17 .02 1 .01

Multi-color 2 .00

TOTAL 699 1.00 97 1.00

LOS ANGELES:

White 1131 .87 494 .82

Yellow 54 .04 10 .02

Red 5 .00 4 .01

Blue 53 .04 67 .11

Gray 40 .03 20 .03

Brown 15 .01 5 .01

Multi-color 3 .00 4 .01

TOTAL 1301 .99 604 1.01

- 95 -
TABLE 45

Frequency and relative frequency by designated Color of Cull

Regular and Cull Air mail at S.F. and L.A.

Regu lar Air Mail

Relative Relative

Color Frequency Frequency Frequency Frequency

SAN FRANCISCO:

White 795 .84 138 .72

Yellow 50 .05 4 .02

Red 4 .00 2 .01

Blue 17 .02 15 .08

Gray 18 .02 27 .14

Brown 68 .07 7 .04

Multi-color

TOTAL 952 1.00 193 1.01

LOS ANGELES:

White 1503 .84 218 .71

Yellow 84 .05 8 .03

Red 11 .01 2 .01

Blue 42 .02 51 .17

Gray 59 .03 9 .03

Brown 94 .05 17 .06

Multi-color

TOTAL 1793 1.00 305 1.01

TABLE 46

Frequency and relative frequency by designated Color

of Metered mail at S.F.

Regular

Color

Relative

Frequency Frequency

SAN FRANCISCO

White 548 .91

Yellow 5 .01

Red

Blue 3 .00

Gray 28 .05

Brown 16 .03

Multi-color 1 .00

TOTAL 601 1.00

- 96 -
small sample size taken in San Francisco, or to the fact

that San Francisco being a port of embarkation to the

Pacific may handle more Gray Air mail letters, because it is

suspected that a large percentage of foreign Air mail letters

are Gray, The data show that for San Francisco Cancel Air

mail, 24% of the letters were Gray as opposed to 3% for the

same category in Los Angeles. San Diego, New Orleans, and

New York may be of value in testing this hypothesis.

Since there appears to be little difference between

Los Angeles and San Francisco with regard to color of

envelopes, data from the two cities were combined. Tables

47 through 49 show the combined data and relative

frequencies expressed as percentages of the colors. Listed

also are 95% confidence limits on the percentages.

Figure 23 shows relative frequency histograms for the

color data of San Francisco and Los Angeles combined. On

the average, more than 80% of the envelopes in the samples

were white.

4. Ratio of hand canceled to machine canceled mail for

Washington j D. C.

4.1 Introduction . Approximately 50% of the mail originating in the Washington, D. C post office is of the "stamped

mail'* variety. This mail must go through a canceling process

before undergoing sorting. The letters may be canceled by

- 97 -
TABLE 47 Percentages and their 95% confidence limits

by designated Color of Cancel Regular and Cancel Air

mail at S„F, and L,A„ combined

Regu la r Air Mail

Confidence Confidence

Limits for Limits for

Color Percentage Percentage Percentage Percentage

White 87% 86%, 88% 80% 77%, 83%

Yellow 4 3 , 5 2 1 , 3

Red 1 1 , 1 1 , 2

Blue 4 3 , 5 11 9 ,14

Gray 3 2 , 4 6 4 , 8

Brown 2 1 , 3 1 , 2

Multi-co lor 1 , 2

101% 102%

TABLE 48 Percentages and their 95% confidence limits

by designated Color of Cull Regular and Cull Air

mail at S,F a and L„A„ combined

Regu lar Air Mail

Color Percentage

Confidence

Limits for

Percentage Percentage

Confidence

Limits for

Percentage

White

Yellow

Red

Blue

Gray

Brown

Multi-co lor

84%

5

1

2

3

6

83%, 85%

4 , 6

1 , 1

1 , 3

2 , 4

5 , 7

72%

2

1

13

7

5

67%, 76%

1 , 4

, 2

10 , 16

5 , 10

3 , 7

101% 100%

98 -
TABLE 49 Percentages and their 95% confidence limits

by designated Color of Metered mail at S.F.

Confidence

Limits for

Color Percentage Percentage

White 91% 88%, 94%

Yellow 1 0,2

Red

Blue

Gray 5 3,7

Brown 3 2,5

Multi-color

TEOT

- 99 -
IPO -

3o70-

.6c- »-

.-tO- tt

20

•lo-\

&5G-UIAR CANCEL

\

T $0-

N

js

- 7e-

•&)-

-ifl- '4>-

•;*>- va>-

ll>-

isI

I

o X

1 <0

regular cull.

X Q

2

1

I

%

i

i

/.0O- A/KMJ/L CAA/CEL

•00 •70-

•#-

•30- -20-

•/<y- ^ r i -J 1 , , X

I

(S

?

ly t.OQ _ s .?H

^ .70-

50-

fo. y>-

J0-

./&

! i

V) << lb

!

o

AIR. MAIL CULL

\

I •J . Of I

8

/.O0—

.

? -SiMETERED *

5 •»-

1 1 1 1 I

1? ^ In x

o ^ ^ S ^1 S, -J ^

•4 $ CQ O

Figure 23

1

«0

I

Relative frequency histograms by designated Colors for

the combined data of S.F. and L.A.

* - Metered mail applies only to S.F.

- 100 -
machine immediately after leaving the facing tables or they

may be canceled by hand, if for various reasons, they

cannot pass through canceling machines.

The purpose of the present study was to determine the

ratio of hand canceled mail to machine canceled mail.

The data shown in this section were previously presented,

together with an initial analysis and conclusion, to the Post

Office Department in memorandum form by Dr. Marvin Zelen, NBS,

in 1956. Here we present the results of an alternative analysis

which, for all practical purposes, are the same as those given

earlier. An additional ratio, reflecting the all day period,

has also been obtained. No subsequent study of this nature

has been made in any other post office. However, the study is

included in this report because it represents a simple statistical sampling method for determining ratios previously obtained

by complete enumeration.

4.2 Sampling plan and results. At the First Class

Hopper, stamped mail which cannot go through the canceling

machine is hand canceled. The post office has estimated that

the strictly letter size mail which needs to be hand canceled

is 1.75% of the total machine cancellations. It was desired

to check this figure without making a total count at the

First Class Hopper, The sampling procedure was to make a

ten minute count of "letter size mail'* at the First Class

Hopper, extrapolate the average rate per hour , and divide

- 101 -
-><- hlh^s f< (' hki * !.-.;» rfhui ti V

*• number <>r letters counted

in usti it$utM*f;<te, 'A te ii>im ( iH»«»s.i-'r t>l machine cancellations

I'd! n>* h<«»u « lonta tn \w\\ thAt !«'! minute periodj then the ratio

J; m

iitV^cs ilw c:tl Hjlrt * «5ii tiii'M uf \\A\vi\ t HIU-OloU ilU 1 1 to HUlch.mC

gRtlQftl&e1 m*U I .

The* i i^itPtsKi $» Um.Uy Obtained; howovor, were not ior

i>tvtotiv unifcir g|jie. niii, Uk= people involved in the actual

».M»niti)ji, e@ttn$tt(J riOt <m.lv Letter -u/,v> mail but alsomiiil

wMt'-u wa& "Mlighi lv hi'^ov," This W&® motivated by their

ito^iiuB |nal Ihls "ftlighttly tarter sisse mail" could also be

&6£Q«IHl(H}&tieti n* ^a\ j.. oporod sorting machine, Thus, because

,>t iMb ap|>a.uut> ("t*o»a the *m 'khw! sampling plan, the

• » i <vcjii s

.-. i td$ ''aiuusi fee u.sw, s<* cheek the post office figure

01 I -">•• ftis t-.--'.4? -« Mini ju-.H.Hiuft" R^e Included though,

iM iwfciiw^ ai i m&Ue t {. » .Miui! wittilex1 type studies, Pahle

fbe ro^« \

-

. • .

-

- - ere that tl (

...<>.• ,t-,,M -• - .-' -. p.. . ; ; , pj •. fc&d canceled aai

!m.:., ,.>u> .is-.'. i) -> -.• Kill ' 0.&3 i^v a,h, end 3* §4%

• ,v %% im '' kg. M »

• oi '•- HXl

, v.\ ratio La

*b > t>, ;

. . v <.' • •

. v : .-

I ae was Ln which
o

m

COIS

©

ft

ftO

S3

CO

CO

bS iHo

+>

10 •H

EnSo

Sh <H

OS P

Q

>»

3

6*o

o

ti

P3

—

oo

pH

oS P

rH

ft5

G

•H

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- 103 -
5. Study of top and bottom Clearance Space of an addressed

envelope

5.1 Introduction. This section is concerned with the

Clearance Space on letters from the top of the top line of

intelligence to the top edge of the envelope, and from the

bottom of the bottom line of intelligence to the bottom edge

of the envelope. The data and analyses are presented in

memorandum form in Section 5.3 for Washington, D.C. and in

Section 5.4 for San Francisco and Los Angeles. These

memoranda are detailed and self-explanatory and were

submitted to the Post Office Department on April 22, 1957

for Washington, D. C, and November 15, 1957 for Los Angeles

and San Francisco. For Washington, D.C. the data were

obtained from a random sample of 634 letters selected from

the 4,000 letters available on film. For Los Angeles and

San Francisco the data were obtained from a random sample of

1285 letters selected from the 10,585 letters also available

on film. (Los Angeles 642, San Francisco 643.)

5.2 Sampling procedure. The procedures used in

gathering the sample are the same as those given in detail

in Section 3.2 of this report. Each letter in the sample

was microfilmed for a permanent record. A masking bar was

used to blot out the return address and the name of the

addressee (see Figure 24) . At a later time sub-samples were

taken from the microfilm samples and the clearance distances

measured and recorded.

- 104 -
or crtM j i*v>*rc- cctp

Figure 24

A letter illustrating the distance from the top of the

first line of intelligence of the address to the top

edge of the envelope, and from the bottom of the last

line of intelligence of the address to the bottom edge

of the envelope. Dotted line indicates outline of

masking bar.

- 105 -
5.3 Washington j D. C. memorandum.

To: C. W. Gray, Director April 23, 1957

Office of Research and Engineering

Post Office Department

From: N. C„ Severo and A. E. Newman

National Bureau of Standards

Subject: Study of Top and Bottom Clearance Space of an

Addressed Envelope

Information was requested by the Post Office Department

regarding the distribution of distances from the top of an

addressed envelope to the top of the first line of intelligence*

and from the bottom to the bottom of the last line of intelligence. This information, we understand, is to be used to help

in the design of an automatic stamp detecting and cancelling

machine

.

Data were obtained from a random sample of 634 letters

selected from the 4000 letters available to us on film. On

the basis of an analysis of these data, we recommend the

following:

1) A distance of 3/4 of an inch be used for the top

clearance space of an addressed envelope . Actually, the distance 0.757 inch turns out to be the lower 99% tolerance limit

with confidence coefficient .95 for the distance from the top

of the envelope to the top of the first line of intelligence

j

i.e., with fairly good betting odds of 19 to 1 it could be

expected that 99% of all measurements would be greater than

0.757 isch.

2) No tolerance limit is recommended for the bottom

clearance space because from 6% to 11% of all such measurements can reasonably be expected to be less than or equal to

0.2 inch . It turns out that 8.4% of the sample measurements

were less than or equal to 0.2 inch and the 95% confidence

limits for this estimate are 6% and 11%; i.e., with betting

odds of 19 to 1, one would expect the proportion of measurements which are less than or equal to 0.2 inch to be between

6% and 11%.

The original data are included in Tables 51 and 52

Frequency histograms, grouped in intervals of 0.2 inch, are

given in Figure 25

* Because of Security reasons, a masking bar was employed to

blot out the name of the addressee . Therefore, this distance

represents the minimum available distance between the top

edge of the masking bar and the top edge of the envelope.

cc: I. Rotkin

- 106 -
TABLE 51

Frequency tabulation of distance from the top edge of the

envelope to the top of the first line of intelligence.

Distance in

Inches

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

C

11

I1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2 9

3.0

Frequency

2

1

1

2

1

1

6

7

22

39

18

52

53

80

51

38

72

47

43

19

11

39

10

7

5

1

6

Average x = 1.842 inches

Standard deviation, s = .431 inches

k = 2.52 for P = .99

Lower tolerance limit x - ks = .7566

63 . Total

number

a = .95

With a reliability of .95, 99% of the mail will have no

intelligence at a distance of .75 inches from the top of

the envelope.

107 -
TABLE 52

Frequency ^tabulation of distance from bottom edge of the

enve >us r o the bottom of the last line of "Intelligence for

Washington, D„

Distance in

_inches_

0.1

0,2

0.3

0,4

0.5

0,6

0,7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2,4

2.5

2.6

2.7

2.8

2.9

3.0

Frequency

35

18

30

26

33

19

47

42

11

49

32

36

35

32

62

25

35

18

14

22

2

6

4

i

634 Total

number

Average, x - 1.062 inches

Standard deviation, s - .557 inches

Proportion less than or equal to .2 inches 8.36%

- 108 -
ISO

l%o

l/o

100

7080

b

70

Mj Go ? So

-

30

20

10 -

Top

CL£A/?£)/JCa

i [ i i 2 -4 .6

J L .& 10 /2 /A 1.6 1-8 2.0 2.Z ZA Z.(, 2-8 3

D/STANCE //V INCHES

100 -

Jo Bottom

80 Clea#i-)NC£:

70

60 \^ fo zJ~^--J

2o

10 -

o 1 1 1 1 1 1 1 1 1 1 I 1 ,—, . A • 6 S ho /.Z /.+ '& /8 2.o Z.tZA Z.& Z.S 3

Distance /n inches

Figure 25

Histograms showing top and bottom clearance space for data froi

Washington, D. C.

- 109 -
5. 4 Los Angeles and San Francisco memorandum.

To: C. Wo Gray, Director 15 November 1957

Office of Research and Engineering

Post Office Department

From: N. Severo and A. Newman

National Bureau of Standards

Subject: Study of Top and Bottom Clearance Space of an

Addressed Envelope

This report, which concerns top and bottom clearance

space for letters sampled in San Francisco and Los Angeles,

is a subsequent study to that reported by Severo and Newman

in a memorandum to Mr. C. W„ Gray on 23 April 1957. The

samples for this earlier study were obtained from a filmed

record of letters that had been sampled at the Washington

Post Office.

It was possible to make the present study more detailed

and, in one respect at least, more accurate than the earlier

one because the sampling procedure and photographing techniques

were designed specifically with the view in mind of answering

the questions concerning clearance space. These questions had

not been spelled out before the Washington samples were photographed and subsequently measured. Therefore, a detailed comparison of the results reported here for Los Angeles and San

Francisco with those reported for Washington is not possible;

however, sufficient agreement is manifest between Los Angeles

and San Francisco on almost all characteristics, and between

the West Coast cities and Washington on those characteristics

which are comparable to support the general conclusion that

there are no significant differences in the clearance measurements among the three cities thus far studied.

Data were obtained from a random sample of 1285 letters

selected from the 10,585 letters available to us on film.

(Los Angeles 642, San Francisco 643). Because of security

reasons, a masking bar was employed to blot out the name of

the addressee. Care was exercised to place the top edge of

the masking bar at the exact top edge of the top of the first

line of intelligence. Such care was not taken in the

Washington, D. C. study. The distances measured are the

distance from the top edge of the envelope to the top edge of

the masking bar and the distance from the bottom edge of the

envelope to the bottom of the last line of intelligence. On

the basis of the analysis of the data the following are

recommended:

- 110 -
1. A distance of 0.9 of an inch be used for the top clearance

space of an addressed envelope . Actually more detailed

tolerance limits, corresponding to different types of

mail, are listed in Table 53 . For example, it should be

noted that for Air Mail alone the corresponding tolerance

limit turns out to be .77 inches. These are 99% tolerance

limits with confidence coefficient .95 for the top

clearance space; i.e., with fairly good betting odds of

19 to 1 it could be expected that 99% of all measurements

would be greater than the tolerance limits listed.

2. No tolerance limit is recommended for the bottom clearance

space of an addressed envelope for Stamped Mail. A

tolerance limit of 0.4 of an inch is recommended for the

Bottom clearance space of an addressed envelope for Metered

Mail . A more detailed analysis is given in Table 54 ~ where for stamped mail, the estimates and confidence limits

for the proportions less than 0.2 of an inch are given.

The original data are included in Tables 55 and 56 . Frequency histograms of the data, grouped in intervals

of 0.3 of an inch given in Figures 26 and 27 .

- Ill -
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- 115 -
10 5 ANGELES

5TAMPE0 LONG

1.0 r- /S8 LETTERS

141-7

*4

3

SoLl l r-Kn I I I .6 /.i /.a z* J.o 3.6

STAMPED 6H0R.T _ J47 LETTERS

f

^kd h-i t^

.6 /.Z /.? ^4 5 3.6 INCHES

METERED (l + S)

Z38 LETTERS

ifl -6~ /.2 /.a J-4 J.O 3.6

3A/V FfiZANC/sco

/*rkl .

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f

L

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i i n LL

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t-rU ID

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Figure 26.

a

Histograms showing top clearance space for data from S.F. and L.A. The

vertical scale gives relative frequencies and the horizontal scale gives

the distance in inches from top of envelope to top of first line of

intelligence

,

- 116
LOS AN&ELES

Sj-8

N! -2 -

O

?? LETTERS

- x

l L

±d LtL

TOTAL

6+2 LETTERS

X,

id

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INCHES

SAN FRANCISCO

f.o \-9\-

x-2

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64-5 LETTERS

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LOS ANGELES 8c SAN ERA NCISCO

x

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if .*

I-3 ^ -2

TOTAL

/^sr LETTERS

K

=±d LL j .6 /.2 /.8 ^.4 3.0 3.6

/NCH£S

Figure 26.

Histograms showing top clearance space for data from S.F. and L.A. The

vertical scale gives relative frequencies and the horizontal scale gives

the distance in inches from top of envelope to top of first line of

intelligence

- 117 -
LOS ANGELES

STAMPED LONG*

/58 L£TT£R.S

t.o j—

•? .81

6

-5 - A -

•3 .2 i*L i 1 1 1 1 i .6 IZ 13 2.4 3.0 3.6

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t+7 LETTERS

Ik I I I I i

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METERED (L+S)

2.38 LETTERS

n u^ i i i i i •6 Ll 1.8 24- 3.0 3.6

X

1

l.o9

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n t-" ' » ' i ' ' i .6 1.2 IS 2.4 3.0 3.6 1l I I I l l l I .b /Z 1.8 2.4 3.0 J.6 LUCRES

304 LETTERS

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f 334-LETTBRS

m.6 A2 /.5 2.<f 3-<5 3.6///CH£S

.I..]—i. 'i,i i i i„i I.X It Af JO 3-6

Figure 27. a

Histograms showing botton clearance space for data from S.F. and

L.A. The vertical scale gives relative frequencies and the

horizontal scale gives the distance in inches from bottom

of envelope to bottom of last line of intelligence.

- 118 -
LOS ANGELES

ki .8

*.«

g./

AIRMAIL

J? LErrtrKs

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v*«

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TOTAL

642 L£rT£RS

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SAN FRANCI SCO

1.0 Total ^

6-? 3 Lerre/zs

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5-7

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Figure 27.

Histograms showing bottom clearance space for data from S.F. and L.A. The

vertical scale gives relative frequencies and the horizontal scale gives

the distance in inches from bottom of envelope to bottom of last line of

intelligence

,

- 119 -
6, Proportions of Long and Short letters

6.1 Introduction . The nature of handling and distributing letter mail in the post offices sampled was such that

separate studies were necessary for Long letters and Short

letters. In Washington, D. C.^Long and Short letters are

canceled on separate machines and hourly readings are made on

each machine. A complete enumeration is made each day and the

proportions of Long and Short letters are easily obtainable.

However, in both San Francisco and Los Angeles; Long and Short

letters are canceled on the same machine and no separate data

are available. Therefore, the sampling plan described in this

section addresses itself to the problem of determining the

proportion of Long letters to Long plus Short letters. Samples

were taken in San Francisco and Los Angeles and are summarized

and analyzed in this section.

6.2 Sample method and procedure . At a predetermined

time during the peak period a sample was taken at the stackers

adjoining the facing tables. A random selection was made of

the facing tables to be sampled. The stackers were first

swept clean, then all of the letters accumulating in the

stackers for a period of one and a half minutes comprised the

sample. Each such sample consisted of approximately 300

letters. The number of Long letters and the number of Short

letters contained in each sample were recorded. Samples were

- 120 -
collected during each of the two peak periods on each day

sampled.

San Francisco Los Angeles

Percent

Long

Confidence

Limits on

Percent Long

Percent

Long

Confidence

Limits on

Percent Long

A.M. Period 26% 21%, 31% 31% 27%, 36%

P.M. Period 51% 46%, 56% 48% 43%, 53%

All Day Period 46% 41%, 50% 45% 41%, 49%

6.3 Analysis. The data obtained by the method described

in Section 6.2 are presented in Table 57 . These data are

given on a daily basis so that each number represents a sum

of all the samples taken throughout the day. Summarized here

are the ratios of Long to Long plus Short letters for (a) the

A.M. period, defined from 8:00 A.M. to 4:00 P.M.; (b) the P.M.

period, defined from 4:00 P.M. to 11:00 P.M.; and (c) the All

Day period, defined from 8:00 A.M. to 11:00 P.M.

In order to calculate the All Day estimates of the ratio

of Long to Long plus Short, it was necessary to use certain

weighting factors which reflect the A.M. and P.M. volume of

mail. The weighting factors used are

V

A

a

1 Vv

p

where V. and V

p represent the volume of A.M. and P.M. mail,

respectively, and 1-a, * a„. Volume figures supplied by the

respective post offices indicated that

- 121 -
t-m

V)uPPP

r-i O

X! • 03 <; 3

iH 73 a c

ccj hfl c • O En J • co oP 5h

bfl«H ao eno O rH a&o

•H • p s

rH • o & O T3

ft.a

faO • A s

•H • 8"

en

+-> ci T3s

CO

>?Oo

r-i

X!

CO

J +

TJo (H H Cj

rH P

03

ft,

H

-po

COCo

to? oopoHpO

CO

i? fc? b? IS *? S§ i?

rH C?> 10 00 CM CO r-l 5D © m •* ^ CO Ifl O O 00 Gi CO iH rH CD CO CO CD 00 CO C7> rH O CO CD CO CD CO

rH rH iH rH iH 00

"* c* o rH o <tf

m

m rH "* CO t> lO o

rf CO CO co o CO iH

CO CO CO CO <* fr- CO O rH CM © O CM GO

C- t*

O CO 't> CO CM rH ^

ft? to? %S fc? to? *l? feS m o Ci ^ co cm co

CM CM CM CM CO CM CM

fafi ^ a tF o j o

COM<

CD P E H

CM

a X. a 55<

CO

CO

Ci CO

O G> CM CO CO

t>-m co co o co co

rH rH

m

m o co CO rH rH r-< Tf<

in co co o ^ co

m

CO rH

O TH t> <5i CO

rH en o ^ co cm

m

rH CO

in t* cm co

m ih

CO CO ^ CM 00O

CM rt< rH CO CM

rH ^ m co n co

CM CM CM CM CM \ \ \ \ \ CO CD CO CD CO

J

to? to? to? to? to? to? co cm m CO rH CO

•^

m

mm n< ^ m i> cm t> m co ^ co t}< co co t> 00 CO CM CO CO Oi

rH rH rH

m

CO CO CO C5 TF cm m o co Tf< cm ci o co m co m o

rH CO N ^ Ol 00 H ^

G> CO b- rH CO CO

t^ co co r* co oo

CM

CO

to? to? to? to? to? O rH CO CM rH CM CO CM ^ CO

CO CM 00 o CO Tf rH O r-i n< o CM r}< rH CO

rH rH rH rH tj<

CD

O

O rH t> ^ -* O ^ N

r> CO rH CD CO rH CO o cm oo cr» o t* o <o

CO CO CO Tf

cm co rt* r> co rH rH rH rH rH \ \ \ \ \ CD CO CO CD CO

C5

rHin

-3 - 122
a. .222 for San Francisco

,205 for Los Angeles

The 95% confidence limits given in the table were computed

by using the formulae given in 2.2c of the Appendix.

- 123
References

[1] R. L. Anderson and T. A. Bancroft, Statistical Theory

in Research , McGraw-Hill Book Company, Inc., New

York, 1952, pp. 313-337.

[2] W. G. Cochran, F. Mosteller, and J. Tukey, "Principles

of Sampling," Journal of the Am. Stat. Assoc,

Volume 49, (1954), pp. 13-35.

[3] H. Cramer, Mathematical Methods of Statistics , Princeton

University Press, Princeton, New Jersey, 1951, p. 353.

[4] C. Eisenhart, M. W. Hastey, and W. A. Wallis, Techniques

of Statistical Analysis , McGraw-Hill Book Company, Inc.,

New York, 1947, pp. 97-108.

[5] E. L. Grant, Statistical Quality Control , McGraw-Hill

Book Company^ Inc., New" York, 1946.

[6] R. B. Murphy, "Non-Parametric Tolerance Limits," Annals

of Mathematical Statis t ics ^ Volume 19, (1948),

pp. 581-589.

[7] E. S. Pearson and H. 0. Hartley, Biometrika Tables for

Statisticians , Volume I, The University Press,

Cambridge, England, 1954, pp. 204-205.

[8] S. Siegel, Non-Parametric Statistics, McGraw-Hill Book

Company, Inc., New York^ 1956, p. 127.

- 124 -
Appendix

1. Tests

1.1 Control Charts

The limits here have been calculated by using

estimates of the standard deviation which are based on the

range of the sample averages. For detailed explanation of

control charts see Grant [5],

1.2 Analysis of Variance

An analysis of variance, weighted according to the

number of samples in each group, was used throughout. See

Anderson and Bancroft [lj.

1.

Kolmogorov-Smirnov Test

A straightforward application of the KolmogorovSmirnov test was used. See Siegel [8].

2. Formulae

2. 1 Tolerance Limits

a) Normal Theory . This was applied in the Clearance

Space study, Section 5 , following [4].

b) Non-Parametr ic-Order Statistics . This method

was used in the Letter size Characteristic Study, Section

following [6], Several approximations, involving a normal

approximation to the Beta distribution were used. These are

Given: x. < x„ < ... < x

1 — 2 — — n

Statement: 100p% of population lies between (x ,x _)

with probability a. r n~ r

Formula: For a . 95 (for large n)

m - n+1 - n£ - 1.64 ynp(l-p)

where ra « r+s

Statement: 100p% of population lies between (- oo , x )

with probability a. Formula: For a * .95 (for large n)

r - np + 1.64-/np(l-0)

- 125 -
2 .2 Confidence Limits

a) Confidence limits on proportions . These limits

were obtained by using the Clopper-Pearson charts available

in [7].

b) Confidence limits on ratios of hand to machine

canceled mail. Let (x

i;L , Y-n), where i=l,2, . . . ,n

hine canceled and hand canceled

-,, designate

the number of mac letters on

an hourly basis respectively for the A.M. period. Then the

ratio of hand canceled to machine canceled mail for the A.M.

period is

R.

n

l

2

i=l il

n

l 2

i-1 ll

Similarly, by characterizing the P.M. period with the subscript

2, the P.M. period ratio becomes

n

2

2

i=l

y i2

2y X

i=l i2

R.

Then by a straight forward application of the "propagation of

error" formula [3] the variance of R., j=l,2, may be estimated

by J

.

where, for j«l,2,

i=l

o

2

• J

J - X2.

2r

j V

A

a.

y x

n

• J

n . 2

J

i=»l ij

n . J

126-
A

0?.

n . Z

J

i-1

(y

±J

- y.3)

2

> n .- I

J

n . Z

J

i-1

(x ij- x

.j

)2

t v 1

3

n,

2

J (y. . - y ) (x. . - x .)

a a i=l 1J ,J 1J * J r. a o = ! J y • x

n .- 1

The variance formula for R. was derived under the assumption

that the numerator and J denominator of the ratio are functionally independent. Assuming approximate normality of the

sums comprising the numerators and denominators, the ^-conf idence

limits on the A.M. or P.M. ratio (i.e., corresponding to

j=l,2 respectively) are then determined by

j j

where z is the one-tail -^ (l~y) -percentage point of the

Gaussian distribution.

The variance of the weighted average (which is an estimate

of the All Day ratio)

R - aiR

x + a2

R

2

, where a-, and a„ are appropriate weighting constants determined

in a similar ' manner to that described in Section 6, may be

written as

Var R - a? Var R, + a£ Var R . 1 12 2

Here the Oj are assumed constant and their values are a1 =0.312

and a2~0.68S. Thus an estimate of Var R may be taken as

aR

= a

l aR

1 + a

2

aR

2

'

127-
The confidence limits on the All Day ratio of hand to machine

canceled letters are then determined by

R ± z ^r > where again z is defined as above.

c) Confidence limits on the ratio of Long to

Long plus Short letters . Let the number of Long and Short

letters collected in each A.M. minute and a half sample be

designated by (Li

-,, S.,), where i=l,2, . . . ,n, . Similarly

let (LjjO' s o^> i=1 > 2 > • • • > n o> designate the number of Long

and x Short letters collected in each P.M. minute

and a half sample. Furthermore let

N. . = L.. + S... i=l,2, . . . ,n . and j=l,2.

Then by a straight forward application of the "propagation

of error" formula [3] the variance of

n . I? L

i=l ij

J n.Z

J

i=l ij

may be estimated by

S

2 .a? + L

2 .al - 2S ,L ,r,at ac g2 m 1 J 1TLj .j Sj .j .j j Lj Sj

U- n . \ 4

where, for j=»l,2,

s

.j - 2

J S. . i=l 1J

n . J

n . 2

J L

i-1 1J

n .

-128-
N

.j = s

.j

+ L

.J <

n

i £2 i=l 1J ,J

\

^ '

2

J (L. - L .) 2

G

L. n. - 1

J J

a a i=l AJ ••* *«* ,J r

j

a

L.

a

S. = n~^HE —— •

(The formula for Var jjlj differs from that given in Appendix

2,2b for Var Ej because the numerator and denominator of p.. are functionally dependent.) The 7-confidence limits on 3

the A.M. and P.M. ratios (i.e., corresponding to j»l,2,

respectively) are then determined by

A

ix. + z a , 3 - »i

where z is the one-tail -^ (I-7) -percentage point of the

Gaussian distribution.

The variance of the weighted average (which is an

estimate of the All Day ratio)

W - a-j^ + a

2

|i2

,

where a, and a„ are defined in Section 6 and assumed here to

be constant, and the confidence limits for the All Day ratio are

obtained in a similar manner to that discussed in the above

Appendix 2.2b.

-129-

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