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PUBLICATIONS NISTIR 88-3889

Survey Sample Design for

Microfilm Inspection at the

National Archives

Keith R. Eberhardt

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Engineering Laboratory

Center for Computing and Applied Mathematics

Gaithersburg, MD 20899

October 1988

Sponsored by:

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Center for Computing and Applied Mathematics

Gaithersburg, MD 20899

and

National Archives and Records Administration

Seventh and Pennsylvania Avenue, NW

Washington, DC 20408

QC

100

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#88-3889

1988

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NISTIR 88-3889 ' f S

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>- Survey Sample Design for

Microfilm Inspection at the

National Archives

Keith R. Eberhardt

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Engineering Laboratory

Center for Computing and Applied Mathematics

Gaithersburg, MD 20899

October 1988

National Bureau of Standards became the

National Institute of Standards and Technology

on August 23, 1988, when the Omnibus Trade and

Competitiveness Act was signed. NIST retains

all NBS functions. Its new programs will encourage

improved use of technology by U.S. industry.

Sponsored by:

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Center for Computing and Applied Mathematics

Gaithersburg, MD 20899

and

National Archives and Records Administration

Seventh and Pennsylvania Avenue, NW

Washington, DC 20408

U.S. DEPARTMENT OF COMMERCE

C. William Verity, Secretary

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

Ernest Ambler, Director
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ABSTRACT

This report describes the statistical design of a sample survey to

monitor the condition of microfilm in a large collection maintained

by the National Archives. The design criterion developed for the

survey ensures that the number of rolls of film inspected will be

large enough to achieve a pre-chosen probability of detecting a

specified amount of damaged film that might exist in the population.

Tables and formulas are given to satisfy the design criterion under

a range of conditions. Other practical aspects of survey design are

discussed including the sampling frame, stratification, sample

selection procedure, pilot testing, and the use of replicated

sampling

.

Key Words and Phrases: Sample Size Determination, Sampling

Fraction, Sampling Frame, Sample Survey, Statistics, Stratification.

Note: This document was submitted to the National Archives and Records

Administration in 1987 as a letter report. This 1988 printing as an

Interagency Report of the National Institute of Standards and Technology

responds to requests for wider circulation of this material.

1
TABLE OF CONTENTS

ABSTRACT i

LIST OF TABLES iii

LIST OF FIGURES iii

1. Introduction: Goals of the Survey 1

2. Characteristics of the NUS Population, Sample Units and Frame .... 2

3. Selection of Sample Size: Probability of Detecting a Problem .... 4

4. Sample Size When Defective Rolls Belong to Several Logical Groups . . 8

5. Discussion of the Sample Size Criteria 10

6. Stratification 13

7. Sample Selection Procedure 14

8. Use of a Geographic Frame for Sampling 17

9. Other Practical Considerations for Sampling 18

10. Acknowledgements 18

11. References 19

Appendix 1 '. . . 20

Appendix 2 24

Appendix 3 26

11
LIST OF TABLES

Table 3.1. Sampling Fraction for Selected Values of and A, Under the

Assumption That All Defective Rolls of Microfilm Belong to a Single

Identifiable Group 7

Table 5.1. Probability of Detection for Various Values of A, f, and k . . 11

Table 5.2. Sampling Fraction for Various Values of A, k, and P^ (k) ... 12

Table 7.1 Hypothetical Population for Example 15

Table Al .Rough Count of Number of Rolls of Stored Microfilm at NUS . . 21

LIST OF FIGURES

Figure 1. Plot of probability of detection vs. sampling fraction for

various numbers of defective rolls 6

Figure 2. Plot of probability of detection vs. sampling fraction showing

dependence on number of groups 9

Figure 3. Sketch of the NARA underground storage facility at National

Underground Storage, Inc 23

iii
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1. Introduction: Goals of the Survey

Regular surveys of microfilm collections are necessary in order to

monitor the condition of the film and to detect any serious problems of

degradation or other damage to the records. This report deals specifically

with the design of a survey for the large microfilm collection maintained by

the National Archives and Records Administration (NARA) at National

Underground Storage, Inc. (NUS) in Boyers, Pennsylvania

The practical survey procedure envisaged consists of two separate phases.

The first phase has the purpose of detecting and identifying any parts of the

microfilm collection where serious degradation has occurred. After this

initial broad inspection is carried out, a follow-up investigation will be

conducted as the second phase of the survey. The purpose of the second phase

follow-up study is to evaluate the extent of any problems discovered in the

first phase and to recommend or initiate appropriate corrective action. This

report addresses the statistical aspects of designing the first phase survey

of the microfilm.

As thus described, the purpose of this survey is somewhat different in

character from many typical sample surveys. It is more common for a survey to

be conducted for the purpose of estimating the percentage or number of items

in a population that belong to a given category (e.g. damaged) than for the

purpose of detecting or locating the presence of damaged items.

This report describes a method of statistically assessing the merits of a

given sampling design in terms of the probability of detecting a group of

damaged microfilm rolls in a population of rolls. It is found that a rule

based on selecting a fixed proportion of the population, no matter how large

or small the population, is a statistically defensible approach. Some of the

statistical properties of this approach are characterized.

1
2. Characteristics of the NUS Population. Sample Units and Frame

The storage area at NUS consists of an underground vault in a tunnel of a limestone mine. The storage area covers about 5000 square feet and contains

about 40 rows of shelving, with six to ten compartments per row.

A summary description of the shelving and its contents is given in

Appendix 1 . The summary is based on notes taken by Alan Calmes when he

visited the site in August, 1985. A sketched map of the facility is included

in the Appendix.

Based on the summary in Appendix 1, there are about 124,000 rolls of

microfilm at NUS, of which about 50,000 are 16 mm format and 74,000 are 35 mm

format

The standard inspection procedure for microfilm, as described in McCamy

(1964) in the context of inspection for aging blemishes, can be used for

evaluating the condition of individual rolls of microfilm. Since that

procedure is designed for evaluating the film on a 100 foot roll basis, the

appropriate sampling unit for this survey is the 100 foot roll. This means

that a workable procedure will be needed for inspecting a randomly chosen 100'

roll that is spliced together with nine other 100' rolls on a 1000' core.

A critical element in the successful execution of a survey of any

population is the existence of an adequate frame. A frame is a literal or

conceptual list that contains exactly one entry for each and every member of

the population to be sampled. In the present context, an adequate frame would

have to be capable of uniquely identifying every 100' roll of film in the NUS

storage facility. A number of finding aids and other forms of intellectual

control related to the contents of the microfilms exist. Unfortunately, none

of these forms a complete listing of the population of microfilm.

A computer listing of the microfilm by "Control Number" exists. - This

listing appears to be a useful starting point, but is not fully adequate as a sampling frame at present. One important deficiency of the Control Number

list is that there is apparently no simple and accurate way to figure out how

many 100' rolls of film are represented by each entry on the list. In

addition, it is not possible to manipulate the list (e.g. sort it, create

statistical summaries of the data it contains) owing to the inflexibility of

the computer hardware and/or software environment in which it resides.

An improved version of the computerized Control Number list would be a useful tool to aid in conducting periodic inspection surveys of the microfilm

population. A well-designed and maintained computer index to the population

could also be useful in other ways, including recording and monitoring the

results of the periodic inspections themselves. An adequate computerized list

should contain at least the following information:

2
• Identification number — ideally at the individual roll level

• Number of rolls represented by each entry, if more than 1

• Source of microfilm (NARA or Other)

• Film size (16 mm or 35 mm)

• Film type (camera negative, duplicate negative, duplicate

positive , etc .

)

• Storage location

• Year produced (at least grouped into the major categories

defined by: before 1955, 1956-1965, and after 1965)

For the purposes of conducting the inspection survey, it is not necessary to

include information on the intellectual content of the film rolls, but such

information is essential for other purposes.

A very important use for the information contained in the frame will be

in aiding the (second phase) follow-up investigations that will track down any

significant problems or deterioration detected in the (first phase) survey.

The identification of a problem in the survey sample would lead to follow-up

checks of other rolls of film that belong to the same Control Number, or are

housed in the same box, or were produced at the same time by the same

supplier, or have some other salient characteristic in common with the sample

roll(s) . For convenience in what follows, microfilm roils that can be

logically linked together through sharing some such characteristic will be

referred to as a "logical group" (or simply a "group" when the context is

clear.) To enable efficient use of the logical group structure in the survey,

it will be important that the computerized list, or sampling frame, contain

the necessary information for locating all rolls in the population that belong

to the same logical group as any sampled roll.

3
3. Selection of Sample Size: Probability of Detecting a Problem

An important consideration in the planning of a statistical survey is the

choice of the sample size, which in the present case amounts to the number of

rolls of microfilm from the population to be inspected. Clearly, as the

sample size increases, the precision of the inference from the sample data to

the population increases (as long as the quality of the inspection procedure

does not degrade with increasing sample size.) Therefore, the ideal of

inspecting the whole population must be balanced against the limited resources

available for conducting the inspection.

The data from a sample survey are typically put to many uses, all of

which are affected by the sample size. In order to treat the choice of sample

size mathematically, it is necessary to adopt a model for the population and

sampling procedure and to study the intended primary use of the survey data in

terms of that model.

Suppose that there exist in the population some number "A" of rolls of

microfilm that are significantly deteriorated, or are at substantial risk of

deterioration. In the statistical quality control literature related to this

problem, these rolls correspond to defective items in a lot of manufactured

goods. For brevity such rolls will be designated as "defective" rolls in this

report. If the individual defective rolls are isolated and randomly

distributed throughout the population, then a survey sample has a very small

chance of solving the problem that they represent, for this would require that

the sample identify all of the defective rolls in the population in order for

appropriate action to be taken in each instance. Fortunately, the practical

situation is more favorable, because a single defective roll identified in the

survey will typically lead to the discovery of others like it in the follow-up

process. It is fruitful to model this aspect of the survey.

As described in Section 2, the roils of microfilm can be categorized in

several ways and conceptually divided into "logical subgroups" that have

certain properties in common. As a starting point, consider the case in which

all of the defective rolls of microfilm belong to a single identifiable group.

It is not necessary to assume that the group consists exclusively of defective

rolls; the model and mathematical development to follow only depend on the

number of defective rolls contained in the group. For this reason, and for

simplicity, the following development will focus on only the defective rolls

in a group, often speaking as though the group consists entirely of defective

rolls

Conceptually, if the sample inspection procedure identifies at least one

defective roil of film, then the follow-up phase of the survey project can

examine the entirety of the logical group to which the defective roll belongs

and appropriately respond to the identified problem throughout that group. In

a sense, the main job of the survey, then, is to discover any such problems

that exist within the population. The sample size required to satisfy the

needs of the survey can be determined by specifying the probability that a

group containing at least a specified number of defective rolls will be

discovered

4
The following symbols are used to denote the quantities of interest:

n = sample size (number of rolls inspected)

N = population size (total number of rolls in storage at NUS)

f = n/N, the "sampling fraction"

A = number of defective rolls in the population

Pjj = probability of detection of the problem represented by the

group of defective rolls (i.e. the probability that one

or more defective rolls is selected into the random

sample)

Derivations of the formulas discussed in this section and in section 4 are

given in Appendix 3

.

Under the assumption of simple random sampling, the probability of

detection is

Pj = 1 - (1 - f)*. (3.1)

In this equation a key role is played by the proportion of the population

sampled, f = n/N, which is called the sampling fraction in statistical

literature

,

Figure 1 shows a plot of versus the sampling fraction for several

values of A. The plot illustrates that the probability of detection increases

as the sampling fraction increases and as the number of defective rolls

increases

.

5
PROBABILITY

OF

DETECTION,

f

Figure 1. Plot of probability of detection, P^j, vs. sampling

fraction, f, for various numbers of defective rolls, A. For a fixed

number of defectives, the probability of detection increases as the

sampling fraction increases. For a given sampling fraction, P^^

decreases as the number of defectives in the population decreases

6
If the values of and A are specified, it follows from equation (3.1)

that the sampling fraction is given by

f = 1 - (1 - (3.2)

The interpretation of equation (3.2) is that the appropriate sample size,

n, depends on the value of the population size, N. In fact, it says that n is

directly proportional to N, the proportionality factor being determined by

and A as specified by (3.2). Therefore, the values of P^ and A determine the

sampling fraction and so the required sample size, n, can be calculated as a

fixed fraction of the population size, N.

Table 3.1 gives the values of the sampling fraction for a selection of

values of P^ and A.

Table 3.1.

Sampling Fraction, f, for Selected Values of P^ and A, Under the

Assumption That All Defective Roils of Microfilm Belong to a

Single Identifiable Group

No. of Defective

Rolls, A

Probability of Detection, P^

50% 75% 90% 95% 99%

30 .023 .045 .074 .095 .142

50 .014 .027 .045 .058 .088

100 .0069 .014 .023 .030 .045

200 .0035 .0069 .011 .015 .023

300 .0023 .0046 .0076 .0099 .015

600 .0012 .0023 .0038 .0050 .0076

1000 .00069 .0014 .0023 .0030 .0046

3000 .00023 .00046 .00077 .0010 .0015

As an example, suppose we wish to insure that the probability of

detection is at least 95% for a group containing 200 defective rolls. From

Table 3.1, the appropriate value of the sampling fraction is f = .015, or

1.5%. Thus, if the population consists of a total of N = 100,000 rolls of

microfilm, the sample size should be 1.5% of 100,000 or n = 1500. Similarly,

for a population of N = 40,000 rolls, the sample size should be n = 600 (which

is 1.5% of 40,000.)

7
4. Sample Size When Defective Rolls Belong to Several Logical Groups

In practice, it is unlikely that the defective rolls of microfilm in the

population will all belong to a single identifiable group. If the defective

rolls are distributed across several groups, the follow-up phase of the

inspection plan will locate all of them only if at least one member of each

and every defective group is identified by the sample inspection. It seems

intuitively clear that a larger sample size will be needed to achieve a

reasonable probability of detecting all of the logical groups containing

defectives in this case.

One simple way to extend the probability model to analyze the situation

involving several groups is to assume that the groups are all equal in size.

Specifically, let

k = number of groups containing defective rolls

A/k = number of defective rolls per group

(k) = probability of detecting all of the defective groups

(i.e., the probability that one or more members of each

defective group are included in the sample)

In this case, it can be shown (section A3 .of Appendix 3) that the

probability of simultaneously detecting ail of the defective groups is given

(approximately) by

Pd(k) - [1 - (1 - f)^/^]^ •

Again the sampling fraction, f = n/N , plays a key role in determining the

probability of detection. The plot of P^ (k) versus f in Figure 2 illustrates

how, for a fixed number of defective roils in the population (A = 300 is

shown) , the probability of simultaneous detection decreases as the number of

groups increases. Putting it another way, the sampling fraction, f, must be

substantially larger if a given number of defective rolls are distributed

across several groups compared to the case of all defectives belonging to a single group.

As was true in the case of a single group, the equation for probability

of detection (4.1) can be solved for the sampling fraction in terms of the

other parameters. The solution is

f - 1 - [1 - Pd(k)^^^]^"^^ . (4.2)

Formula (4.2) can be used to determine the appropriate sampling fraction for

specified values of A, k, and P^j (k) . From the sampling fraction and the

population size, N, the sample size, n, can be found by multiplying the

population size by the sampling fraction; n = f x N.

8
PROB.

OF

SIMULTANEOUS

DETECTION,

100

80

60

40

20

0

0 0.02 0.04 0.06 0.08 0.1

SAMPLING FRACTION, f

Figure 2. Plot of probability of detection, P(j^(k) , vs. sampling

fraction, f, showing dependence on number of groups, k. The plot is

constructed assuming that a total of 300 defective rolls are equally

distributed among k groups. The probability of simultaneously

detecting all k groups in a single survey decreases as the number of

groups increases.

9
5. Discussion of the Sample Size Criteria

Current regulations relating to inspection of microform records (NARA,

1985, section 1230.22) derive from a rule that the sample shall constitute a 1% sample of the population. This rule, which was originally proposed on

intuitive grounds, and without a rigorous statistical basis, is consistent

with the model and theoretical development outlined in Sections 3 and 4 of

this report. Additional detail on the nature of these regulations is

contained in Appendix 2, where selected portions of a 1986 draft revision to

the regulations have been reproduced.

The theoretical development in this report provides a statistical

justification for a sample size rule based on setting a fixed sampling

fraction that will be used with populations of varying sizes. Equations (3.1)

and (4.1) provide a statistical interpretation of the merits of any sampling

plan specified by a given sampling fraction. Candidate sampling plans can be

evaluated in terms of the probability of detecting a set of A defective items,

where the number A is chosen to be relevant to a particular situation. Table

5.1 presents the probabilities of detection for a range of sampling fractions

and group sizes. The table contains separate entries corresponding to the

assumptions that all defective items belong to one group [using equation

(3.1)] or that the defective items are distributed across 5 or 10 groups

[using (4.1)].

Table 5.1 shows, for example, that if there are A=100 defective rolls of

film in a population, a 5% random sample has a 99.4% chance of detecting at

least one of them. If those 100 defectives all belong to a single logical

group, they could all be found (and presumably fixed) by an follow-up

inspection of the 100% of group to which the sample defective(s) belong. If

the 100 defective rolls are distributed in 5 distinct (but individually

identifiable) groups (of 20 each) , that same 5% random sample of the

population has only a 10.9% chance of detecting all 5 groups. Further, if the

100 defectives are situated in 10 groups (of 10 each) , then the 5% random

sample has only a 0.01% chance of locating at least one member of each of the

10 groups.

Table 5.2 is an extension of Table 3.1 and gives values of the sampling

fraction corresponding to various values of A, k and (k)As an example of

the use of Table 5.2, suppose it is desired to insure that the probability of

detection is at least 99% for simultaneous detection of 5 groups of

defectives, each of which contains 600 defectives. For this example, we wish

to find the sampling fraction, f, in Table 5.2 corresponding to k = 5 groups,

probability of detection (k) = 99%, and total number of defective rolls

A = 5x600 = 3000. The tabulated value of f is .010, which implies that a 1%

sample of the population will be required. If the population contains 124,000

rolls of microfilm, for example, the required sample size would be 1% of

124,000 or 1,240 rolls to be inspected.

10
No.

Gro

k

1

11

1

1

1

5

5

5

5

5

5

10

10

10

10

10

10

Table 5.1.

Probability of Detection, P^j (k) in Percent,

for Various Values of A, f, and k.

Number of Sampling Fraction, f = n/N

Defective

Rolls, A .001 .005 .01 .05 .10

30 3.0% 14.0% 26.0% 78.5% 95.8%

50 4.9 22.2 39.5 92.3 99.5

100 9.5 39.4 63.4 99.4 99.9+

300 25.9 77.8 95.1 99.9+ 99.9+

1000 63.2 99.3 99.9+ 99.9+ 99.9+

3000 95.0 99.9+ 99.9+ 99.9+ 99.9+

30 0.0 0.0 0.0 0.1 2.3

50 0.0 0.0 0.0 1.0 11.7

100 0.0 0.0 0.02 10.9 52.3

300 0.0 0.1 1.9 79.0 99.1

1000 0.02 10.2 48.7 99.9+ 99.9+

3000 1.9 77.6 98.8 99.9+ 99.9+

30 0.0 0.0 0.0 0.0 0.0

50 0.0 0.0 0.0 0.0 0.01

100 0.0 0.0 0.0 0.01 1.4

300 0.0 0.0 0.0 8.9 64.8

1000 0.0 0.01 1.0 94.2 99.9+

3000 0.0 8.1 60.5 99.9+ 99.9+

11
No.

Gro

k

1

1

1

1

11

1

1

5

5

5

5

5

5

5

5

10

10

10

10

10

10

10

10

r

Table 5.2.

Sampling Fraction, f = n/N,

for Various Values of A, k, and (k)

Number of Probability of Detection, P^ (k)

Defective

Rolls, A 50% 75% 90% 95% 99%

30 .023 .045 .074 .095 .142

50 .014 .027 .045 .058 .088

100 .0069 .014 .023 .030 .045

200 .0035 .0069 .011 .015 .023

300 .0023 .0046 .0076 .0099 .015

600 .0012 .0023 .0038 .0050 .0076

1000 .00069 .0014 .0023 .0030 .0046

3000 .00023 .00046 .00077 .0010 .0015

30 .29 .38 .48 .53 .64

50 .18 .25 .32 .37 .46

100 .097 .13 .18 .20 .27

200 .050 .070 .092 .11 .14

300 .034 .047 .062 .074 .098

600 .017 .024 .032 .038 .050

1000 .010 .014 .019 .023 .031

3000 .0034 .0048 .0064 .0076 .010

30 .59 .70 .78 .83 .90

50 .42 .51 .60 .65 .75

100 .24 .30 .37 .41 . .50

200 .13 .16 .20 .23 .29

300 .086 .11 .14 .16 .21

600 .044 .058 .073 .084 .11

1000 .027 .035 .045 .051 .067

3000 .0090 .012 .015 .017 .023

12
6. Stratification

Stratification refers to the process of dividing the population into nonoverlapping subpopulations, or strata, that will be treated separately in the

sample survey. Since the stratification is done before sampling, the frame

must contain enough information to unequivocally assign each sample unit to

one and only one stratum before sampling.

There are four major reasons for stratification that might apply to this

survey of microfilm: administrative convenience (including the need to

prepare separate summary reports for each stratum), statistical efficiency,

existence of strata having significantly different historical or monetary

value, and allocation of resources for future surveys.

Administrative convenience. It is often useful to stratify a large

population into smaller pieces so that survey work can be organized

independently in each stratum. An additional advantage is that summary

reports and any statistical analyses of the data from each stratum can be

carried out separately.

Statistical efficiency. The theory outlined in sections 3 and 4 implies

that the probability of detection depends on A, the total number of defective

rolls in the subpopulation (or stratum) , and on k, the number of identifiable

groups to which the defective rolls belong. This implies that an efficient

stratification will divide the population according to expected values of A

and k.

For example, one stratum might consist of roils for which the sizes of

the identifiable subgroups (possible values of A if the rolls turn out to be

defective) are all relatively large compared to another stratum consisting of

smaller group sizes. Organizing the population into strata in this way would

have the advantage of making it possible to sample a larger fraction of the

stratum consisting of smaller groups. Similarly, if a stratum could be formed

in which it was expected that at most k=l group could contain defective rolls,

a relatively small sampling fraction would suffice, saving a heavier sampling

effort for another stratum in which k>l is expected.

Strata of significantly different historical or monetary value. A

stratum consisting of relatively more valuable microfilm can be sampled more

intensively than a less valuable stratum. For example, only camera negative

film was sampled in the 1984 survey conducted at the National Archives.

Planning for future surveys. If experience with the population suggests

that one stratum is significantly more at risk than another, a stratified

approach provides for the possibility of sampling the lower risk stratum less

often or less intensively in a periodic inspection program. This approach was

described in the proposed revision to the Federal Property Management

Regulations on Micrographics Records Management that the author helped prepare

in November, 1986. Relevant portions of that document are reproduced in

Appendix 2

.

13
r

7. Sample Selection Procedure

In this section, it is assumed that a suitable list frame, as described

in Section 2, is available for use in sample selection. The sampling

procedure that will be described is a systematic sampling scheme which is

practical to implement and which can be used for list frames having entries

that represent more than one roll of microfilm. The details of the

recommended sampling procedure will be described and illustrated by working

through a hypothetical example.

Step 1. By referring to Table 5.1 and/or Table 5.2, choose the appropriate

sampling fraction, f , to achieve the desired probability of detection, , for

relevant values of A, the assumed number of defectives in the population, and

k, the assumed number of equal groups to which the A defectives belong.

Example: Suppose we want to design a survey so that there will be a

90% chance of detecting the presence of each of 10 groups that

contain at least 100 defectives, should such groups exist. In this

example we want = .90, for k = 10 groups and A = 100x10 = 1000

total defectives. From Table 5.2, we find that a sampling fraction

of f = 0.0446, or 4,46% of the population, will be required to

guarantee a 90% chance of detecting at least one defective roll from

each of the assumed 10 groups of defectives.

Step 2. Calculate the sampling interval, S, by the formula S = 1/f and round

down to the nearest whole number. The sample for inspection will be chosen as

"every Sth" unit in the population.

Example: Given that f = 0.0446, we find S = 1/(0.0446) = 22.42.

Rounding down to the nearest whole number yields S = 22, Thus the

sample will consist of every 22nd unit from the population.

Step 3. Choose a random starting number, R, between 1 and S. The first unit

selected will be serial number R and the rest of the sample will be chosen as

every Sth unit after that.

Example: Consulting a table of random numbers, we locate a "random"

starting point by placing a finger on the table without looking.

Then, scanning down the located column (or across the row or moving

diagonally — it doesn't matter since the table is completely

random) the first (two-digit) number encountered between 01 and 22

(=S) is 09, say. Thus the random starting number will be R = 9.

The sample rolls of microfilm will consist of serial numbers 9,

31 (=9+22), 53 (=31+22), 75 (=53+22), etc.

14
Step 4. Assign "serial numbers" to all the rolls of microfilm in the

population and select the sample rolls.

Example: Table 7.1 below represents a hypothetical population which

will be used to illustrate the sample selection procedure. The

population consists of 68146 rolls of microfilm represented in a

list frame with 5000 entries. The population is to be sampled using

random start R=9 and sampling interval S=22.

Table 7.1

Hypothetical Population for Example

Number of Serial Numbers

List Entry Rolls Per Serial Numbers for Selected

Number Entry Assigned Sample Rolls

1 23 1 - 23 9

2 11 24 - 34 31

3 7 35 - 41 - 4 19 42 - 60 53

5 6 61 - 66 - 6 12 67 - 79 75

7 14 80 - 94 - 8 28 95 - 123 97, 119

9 5 124 - 128 - 10 13 129 - 132 -

5000 12 68134 - 68146 68143

Total = 68146 Total number of rolls

roils in selected for the

population sample would be 3097.

Notice that two rolls (serial numbers 97 and 119) are to be

selected from list entry number 8. This will happen occasionally

for list entries that correspond to more than 22 roils (or, more

generally, S rolls) of microfilm.

In assigning the serial numbers, it is not necessary to actually number

every roll in the population. The serial numbers are used simply as a

conceptual device to keep track of the varying numbers of rolls per list entry

and, in effect, to insure that every roll in the population has a equal chance

of being selected into the sample.

15
Step 5. Select the individual microfilm rolls to^^ be inspected by making

random selections from the list entries identified in step 4.

Example: The procedure described in Table 7.1 results in selection

of a set of list entries corresponding to the sample of microfilm

rolls to be inspected. The final selection of the individual

microfilm rolls from the selected list entries is accomplished by

simple random sampling, as follows: First, physically locate all of

the microfilm rolls that belong a chosen list entry and count them.

It may be expected that the actual number of rolls for a list entry

will not always be exactly the number predicted. In the

hypothetical example, we may imagine that list entry #1 is actually

found to contain 26 rolls, rather than the 23 rolls that were

assumed when Table 7.1 was constructed. The sample selection is

completed by drawing one roll at random from the actual number of

rolls found, using a table of random numbers or a computerized

random number generator. In the example, a random number between 1 and 2^ would be drawn and the corresponding roll selected for

inspection. Continuing the example, note that two rolls would be

drawn at random from the group corresponding to list entry #8

because two serial numbers were identified as belonging to that

group

16
8. Use of a Geographic Frame for Sampling

The terra "geographic frarae" is used here to raean a conceptual frarae that

identifies each roll of raicrofiliii in the population with its physical storage

location. A geographic frarae can be used to obtain reasonably coraplete

coverage of the population, with the exception that if sorae group of film is

heavily used, iteras from that group would tend to be under-represented in the

saraple because iteras in use at the tirae of the survey would be unavailable for

inspection. This type of frarae is less desirable than a list because it is

less stable over tirae and does not lend itself to creating and raaintaining

records of data obtained from repeated surveys. However, if a list frarae is

not available and can not be constructed in a timely and cost-effective

manner, useful results still can be obtained from a survey conducted by a geographic frame

.

The procedure for sampling from a geographic frame follows essentially

the same steps described in Section 7 of this report, with appropriate

adaptations. In particular, the "list entries" are replaced by convenient

physical storage units, such as storage shelves or drawers.

To begin, the sampling fraction, sampling interval, and random starting

number are chosen following exactly the same methods described in steps 1, 2,

and 3 in Section 7. The number of rolls of microfilm on each shelf (using

shelf in place of list entry) must be determined and used to assign serial

numbers to the rolls as was illustrated in Table 7.1. The shelves are

selected systematically in the same way list entries were selected in Table

7.1. Similarly, the number of rolls that will be inspected on a given shelf

is determined by the number of selected serial numbers that happen to belong

to that shelf. Finally, the individual rolls are chosen at random from the

rolls on selected shelves following the method described in step 5 of

Section 7.

17
9. Other Practical Considerations for Sampling

Pilot Test. Any survey that is planned will need a well-defined data

collection procedure and some sort of form or questionnaire for recording the

field data. These procedures should be tested by use in a realistic pilot

survey of some small portion of the population. This test need not be large — inspection of 10 or 20 rolls of microfilm should suffice — but it should be

conducted by one or more of the inspectors that will be involved in the fullscale survey. A pilot test will almost always lead to improvements in the

data recording form(s) and often uncovers serious deficiencies in the planned

survey procedures.

ReplicationThe easiest, and often most convincing, way to evaluate the

statistical uncertainty in a survey is to repeat it and compare results. In

fact, replication can be built into a survey by simply dividing the workload

into approximately equal pieces and conducting the survey in parallel and

independently (e.g. different inspectors, equipment, data summarization) on

each piece. This method of organizing survey work, using about 4 to 10

subsamples, has been used extensively and very effectively in practical work

in many fields. Examples are described by Deming (1960, Chapter 6) and Sudman

(1976, pp. 171-178).

As a concrete example, suppose that it is desired to take a 2% sample

(i.e. 1 in 50) of a population. The survey could be divided into 4 subsamples

by having each of 4 inspectors conduct a 1 in 200 sample (of 0.5% of the

population.) To make the 4 subsamples as comparable as possible, it would be

well to use so-called "interpenetrating" subsamples by having each of the 4

inspectors choose systematic samples with different random starts, but

counting from the same point of origin in the population.

10. Acknowledgements

The author is grateful for many helpful comments on various drafts of

this report which were received from Dr. Charles Hagwood, the late Mrs. Mary

G. Natrella, Dr. Joan R. Rosenblatt and Mr. Thomas C. Bagg, of the National

Institute of Standards and Technology, and Professor K. T. Wallenius, of

Clemson University.

18
1 1 References

Cochran, William G. (1977). Sampling Techniques . 3rd edition. New York:

John Wiley & Sons.

Deming, W. Edwards (1960). Sample Design in Business Research . New York:

John Wiley & Sons.

ISO (1974). Sampling Procedures and Tables for Inspection by Attributes (ISO

2859) . Geneva, Switzerland: International Organization for

Standardization.

McCamy, C.S. (1964). Inspection of Processed Photographic Record Films for

Aging Blemishes . National Bureau of Standards Handbook 96, United States

Department of Commerce. Washington: U.S. Government Printing Office.

Reprinted in Special Interest Package #3, Microspots and Aging Blemishes,

by the Association for Information and Image Management (AIIM) , Silver

Spring, Maryland, July, 1987.

MSTC (1981). Sampling Plan . Report prepared for National Archives and

Records Service under Contract No. NA 81-15. Falls Church, Virginia:

Micrographic Systems Technology Corporation.

NARA (1985). "Standards for the Maintenance, Use and Disposition of Microform

Records." National Archives and Records Administration. 36 CFR, Chapter

XII, Subchapter B, Part 1230 - Micrographics. Washington: U.S.

Government Printing Office.

Schilling, Edward G. (1978). "A Lot Sensitive Sampling Plan for Compliance

Testing and Acceptance Inspection." Journal of Quality Technology . Vol.

10 , No . 2 , pp . 47 - 51

.

Schilling, Edward G. (1982). Acceptance Sampling in Quality Control . New

York: Marcel Dekker, Inc.

Sudman, Seymour (1976). Applied Sampling . New York: Academic Press, Inc.

United States Department of Defense (1963). Military Standard. Sampling

Procedures and Tables for Inspection by Attributes (MIL-STD- 105D)

.

Washington: U.S. Government Printing Office.

19
Appendix 1

SUMMARY OF PHYSICAL LAYOUT AT NUS IN BOYERS, PA

This appendix is based on notes taken by Alan Caimes during an August, 1985,

visit to National Underground Storage, Inc. in Boyers, Pennsylvania. The raw

summary data are reproduced in Table Al . 1 . A sketch of the physical layout of

the facility is given in Figure 3.

Observations on Physical Layout at NUS

• There are about 49,654 rolls of 16 mm film (119 million frames)

+ 74,326 rolls of 35 mm film ( 89 million frames)

for a total of about 123,980 rolls or 208 million frames

in the entire storage area at NUS.

• These rolls take up only about 56 compartments (10 shelves per

compartment .

)

• The number of rolls per compartment is quite variable:

- many have 3000 rolls per compartment

- many have 1600 " " "

- many have 2112 " " "

- Max = 5416

- Min = 100

- Mean = 2200

• Most compartments have either 16 mm or 35 mm film, but not both.

20
Table Al .

Rough Count of Number of Roils of Stored Microfilm at NUS

(based on A. Calmes' notes from August, 1985, visit to Boyers, PA)

Aisle Compartment

No. of 100

16 mm

foot rolls*

35 mm

1 2 3000

3

- 3000

4 - 3000

5

- 3000

6

- 3000

7

- 3000

8

- 3000

2 3

- 3000

4 - 1660

5

- 2954

6

- 3780

7

- 3780

8

- 3780

3 3

- 3600

4 - 3600

5

- 3600

6 1700 1040

7

- 1600

8

- 1600

4 2

- 1600

3

- 1600

4 - 1600

5

- 1600

6 680 2200

7 544 1280

8

- 1600

5 3

- 1600

4 - 1600

5 1520 800

6 3040 - 7 3040 -

8 2702 -

9 2112 - 10 2112

21
Table Al . 1 , Continued

Aisle Compartment

No. of 100

16 mm

foot rolls*

35 mm

6 2 2112

3 2112 - 4 2112 - 5 2112 - 6 2112 - 7 2112 - 8 2112 - 9 5416 -

7 2 2396 -

3 2720 - 4 2176 264

5 304 1408

6 1976 480

7 2432 160

8

- 1440

9

- 100

Subtotals 49,654 74,326

Grand Total 123,980 rolls

* counts each 1000 foot roll as 10 x 100 foot rolls.

22
LAYOUT OF NUS FACILITY

s 6 7 s

7

\

' 4

' T t 3

7 4- ] n r Cfi 7 7 \

3 —

biJ

Remainder is

empty shelving.

Aisle 1, Compartment 2 through

Aisle 7, Compartment 9 are in use.

Figure 3. Sketch of the NARA underground storage facility at Nat

Underground Storage, Inc., Boyers, PA. August, 1985.

23
Appendix 2

SELECTED PORTIONS OF THE NOVEMBER, 1986, DRAFT REVISION TO

36 CFR Part 1230, Subpart B

Standards for the Storage, Use and Disposition of Microform Records.

Section 1230.22

(a) Permanent Records

(1) Unstratified samples.

(i) Master films of permanent and unscheduled records microfilmed to

dispose of the original record shall be inspected on a 3-year cycle. At each

cycle, the inspection shall be made on a randomly selected sample consisting

of 1000 microform units, or 1% of the total number of microform units in the

collection, whichever is smaller. The term "microform unit" refers to a

single [100'] roll of microfilm, a microfiche, or similar appropriate unit for

inspection

(ii) To facilitate inspection, an inventory of microfilm must be

maintained, listing each microform series/publication by production date,

producer, processor, format, and results of previous inspections.

(iii) The elements of the inspection shall consist of (1) an inspection

for aging blemishes following the guidelines in the AIIM/MS HB96 [AIIM Special

Interest Publication #34, Association for Information and Image Management,

Silver Spring, Maryland]; (2) a rereading of resolution targets; (3) a remeasurement of density; and (4) a certification of the environmental

conditions under which the microforms are stored, as specified in sec. 1230.20

(a).

(iv) An inspection log shall be maintained. Information to be contained

in the log shall include (1) a complete description of ail records tested

(title; number or identifier for each unit of film; and inclusive dates,

names, or other data identifying the records on the unit of film); (2) the

date of inspection; (3) the elements of inspection; (4) the defects uncovered;

and (5) the corrective action taken. In addition, the log shall contain the

results of all archival film tests required by sec. 1230.14.

(v) The results of the inspection shall be reported to the Office of

Records Administration, National Archives (NI), Washington, DC 20408, 30 days

after the inspection is completed. Reports shall include (1) the quantity of

microform records on hand, i.e., number of rolls, number of microfiche, etc.;

(2) the quantity of microforms inspected; (3) the condition of the microforms;

(4) any defects discovered; and (5) corrective action taken. A copy of the

inspection report shall be stored with the microfilm and have the same

retention period as the microfilm.

24
(2) Stratified samples

(i) When the records required by sec. 1230.22(b) are maintained, it may

be possible to divide a microform collection into two strata, with one stratum

having an appreciably lower risk of deterioration than the other. This is

determined by analysis of the information derived from previous inspections.

One stratum consists of microform series/publications which showed no signs of

deterioration in past inspections. In such circumstances, it will be

sufficient to inspect the latter, lower risk stratum only in alternate

inspection cycles, i.e., every 6 years, while continuing to inspect the

former, higher risk stratum every 3 years. When the population is stratified

in this way, the sample size for inspection may, in some cases, be reduced in

alternate inspection cycles, as follows: when the inspection schedule calls

for inspection of only the higher risk stratum, the required sample size may

be computed as the smaller of 1000 units or 1% of the high risk stratum only.

(ii) When stratification is used in the inspection program, the stratum

definitions must be well documented, including the reasons used to determine

which microform records would be placed in the respective strata. This

information must be included in the inspection report submitted to the

National Archives.

(iii) This stratification of the collection shall not be used for an

inspection cycle until an inspection of the entire collection has been

conducted at least twice. In any case, the inspection procedures shall follow

the unstratified plan described in sec. 1230. 22 (a) above at least every 6

years

.

25
Appendix 3

MATHEMATICAL DEVELOPMENT OF FORMULAS FOR

A3 . 1 Derivation of Formulas for Probability of Detection

The formulas for the probability of detection, P^ , are derived as

follows

Define the random variable X to be the number of defective rolls of

microfilm drawn into a simple random sample of size n. In summary, the

essential quantities are denoted as:

N = population size (total number of rolls)

n = sample size

f = n/N, the sampling fraction

A = number of defective roils in the population

X = number of defective roils drawn in a simple random sample

(without placement) of size n

The probability law of the random variable X is the Hypergeometric

Distribution (Cochran, 1977, pp , 55-57). Thus P^ can be calculated as

Pd

^ N-A 1 (A3.1)

The practical range of values of A will typically be less than 10% of the

population size, or else a survey sample would not be needed to locate

defective roils. In this case, A/N is less than 0.1 and the probability in

(A3.1) can be well-approximated by use of the binomial distribution

(Schilling, 1982, pp . 64-65). Using the so-called f-binomiai approximation,

leads to the expression

Pj = 1 - (1 - f)^ . (A3. 2)

Using equation (A3. 2), one can solve for the sampling fraction, f = n/N,

in terms of P^ and A. The solution is

f = 1 - (1 - (A3. 3)

The theoretical model adopted here has been used previously by Schilling

(1978) in the development of a lot sensitive sampling plan for compliance

testing. As in the present application. Schilling's work deals with the

problem of trying to detect defective items in an isolated lot (or single

finite population) of items. Unfortunately, the tables provided in that work

26
are too specialized to be useful in the present application to microfilm

sampling. In particular, Schilling's main table applies only to the case = 0.90 and the smallest value of f given is 0.01.

It can be shown (Schilling, 1978) that the approximation used in equation

(A3. 2) is conservative in that the exact value of , from equation (A3.1), is

actually greater than or equal to the value calculated by the approximate

formula, (A3. 2). This means that the simple formula (A3. 3) for the sampling

fraction, and Tables 3.1 and 5.2 which were calculated from it, tend to yield

recommended sampling fractions that are slightly larger than would be found if

an exact calculation based on (A3.1) were performed.

A3 . 2 Effect of Distributing a Given Number of Defective Rolls Across Several

Groups

The formulas in section A3 .

assume that all the defective microfilm

rolls belong to the same identifiable group. If instead the A defective rolls

are assumed to belong to several logical groups, it would be necessary to get

at least one representative from each group in the sample in order to be able

to locate all the defective rolls in the follow-up phase of the inspection

program.

Formally, we consider the case in which the A defective roils are

distributed equally among k groups, each of size A/k. The relevant

probability of detection is the probability that at least one member of each

group is drawn in the random sample. Let

= number of members of group i in the sample

( i = 1 ,

. .

.

,

k) , and

P^ (k) = probability that > 1 for all i, 1 < i < k.

Treating the as approximately independent random variables, it follows from

(A3. 2) that

Pd(k) = [1 - (1 - . (A3.4)

Formula (A3. 4) was used in the construction of Figure 2.

As was the case for a single group, equation (A3. 4) can easily be solved

for the sampling fraction in terms of the other parameters. The solution is

f - 1 - [1 - Pd . (A3. 5)

27
A3 3 Relation to National and International Standards on Sampling

The author is aware that it would have been desirable to refer to

national or international standards for rules on choosing appropriate sample

sizes. The following discussion explains why this was not possible.

The sampling standards that are most relevant to the microfilm inspection

problem are MIL-STD-105D (U.S. Department of Defense, 1963) and its

international adaptation, ISO 2859 (ISO, 1974). Both are discussed in detail

in Schilling (1982) .

In parallel with this report, the tables and rules given in the standards

are based on the hypergeometric probability model, and on binomial and Poisson

approximations to that model (see, for example paragraph 11.1 of ISO 2859.)

Further correspondences in notation and concepts between those standards and

the present report are as follows:

This Work

ISO 2859 and

MIL-STD-105D

Pd

A/N

n

N

1—Pg , (Pg = probability of acceptance)

LQ, Limiting Quality

n. Sample Size

Lot or Batch Size

An important difference between this microfilm inspection problem and the

problems addressed by MIL-STD-105D and ISO 2859 is that the sampling standards

are designed to control the proportion of defective items in a long series of

lots or batches. In contrast, there is only one "lot" of microfilm to be

inspected in the problem considered here.

These standards do describe, as a secondary application, the use of their

tables for sampling "isolated lots," a situation which more nearly matches the

microfilm inspection problem. Unfortunately, the values of P^ ( = 1 — P^ ) in

the standard tables are only available for a very limited selection of values

of n and N. Hence the usefulness of the standards is similarly limited in the

present context. In particular, the plots and tables given in sections 3 and

5 of this report could not have been derived from the sparse values given in

the tables in the standards.

28
A3. 4. Comparison With Textbook Formulas for Sample Size

Equations (A3. 3) and (A3. 5) are significantly different from the formulas

for sample size usually given in textbooks on sampling. Those sources use a

different criterion based on specifying the desired length of a confidence

interval for the estimated proportion of defective items in a population,

rather than the criterion used in this report based on . Specifically, the

usual formula for sample size (Cochran, 1977, section 4.4) is:

(A3. 6)

1 + (n„-l)/N

t^ PQ/d^

population size

proportion of defective items in the population ( = A/N ) 1-P

1.96 for 95% confidence level (for example)

desired bound on the error of estimation (half-width of

a 95% confidence interval)

For a large population (N large relative to n) the value of n is wellapproximated by simply calculating n^ . The formula for n^ is widely used for

planning surveys (e.g. MSTC, 1981) to good effect when the goal of the survey

is to estimate the proportion of defective items in the population.

The effect of the population size in formula (5.1) is significantly

different in character, as well as detail, in comparison to the recommended

formula (3.2). Specifically, the textbook formula (5.1) is only weakly

dependent on population size whenever N is large compared to n (Cochran, 1977,

page 76). In contrast, the recommended formula (3.2) shows that n should be

chosen to be directly proportional to N.

In summary, the reason the sample size formulas in this report are so

different from the usual formulas is that a different criterion has been used,

based on , to determine the required sample size. The sample size formulas

given is textbooks are derived by considering how large a sample is needed to

estimate the proportion of defective rolls with a 95% confidence interval of a

given length. Our concern here is not with estimation , but rather with

detection of a group of defective rolls of microfilm.

n =

where

and

n.

N

P

Q

t

d

29
NBS.n4A (REV- 2.80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See i nstruction s)

1. PUBLICATION OR

REPORT NO.

NISTIR 88-3889

2. Performinj Orsan. Report No. 3. Publication Date

October 1988

4. TITLE AND SUBTITLE

SURVEY SAMPLE DESIGN FOR MICROFILM INSPECTION AT THE NATIONAL ARCHIVES

5. AUTHOR(S)

Keith R. Eberhardt

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

I. Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP)

NIST

10.

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

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

bi bliography or literature survey, mention it here)

This report describes the statistical design of a sample survey to monitor the

condition of microfilm in a large collection maintained by the National Archives,

The design criterion developed for the survey ensures that the number of rolls of

film inspected will be large enough to achieve a pre-chosen probability of

detecting a specified amount damaged film that might exist in the population.

Tables and formulas are given to satisfy the design criterion under a range of

conditions. Other practical aspects of survey design are discussed including the

sampling frame, stratification, sample selection procedure, pilot testing, and the

use of replicated sampling.

12.

KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) sample size determination; sample survey; sampling fraction; sampling frame;

statistics; stratification

13. AVAILABILITY

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Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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