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THESIS

This is to certifg that the

thesis entitled

An Investigation of the Applications
of Statistical Method to the Auditing of Sales

presented by

James Grafton Carter
has been accepted towards fulfillment

of the requirements for

Master's degree in Accounting

Maggi.

Major professor

Date May 16, 1951

0-169

 

A] I'VESTIGATIOI OF THE APPLIOATIOIB OF STATISTICAL METHOD
TO THE KUDITING OF SALES

By
Jane. 0. Carter
A THESIS

Submitted to the School of Graduate Studies or nichigen State
college of Agriculture end.1pplied Science in partial
fulfillment of the requirements for the degree of

[ASTER OF ARTS
Department of Accounting - Division of Business

1951

THEQIs

g.
/

{if “

ACKNOWLEDGMENT

Specific nention of the names of all those who have
contributed time, skill and advice to the development of
this study would entail considerably more space than it
is practical to. devote to this privilege. The author,
therefore, simply acknowledges his debt to the many persons
who have thus contributed.

Among the persons who, by reason of unusual assistance
must be mentioned by name are Joseph lewman, Comptroller
and Paul 31a, Chief Accountant, Rec Iotors, Inc. Without
the help of these men the author's theory would not have
been tested under actual conditions. Also of great help
was the advice of Dr. I. D. Eaten, Professor of Mathematics
at Itiehigan State College who helped clarify many points
regarding statistical concepts.

The greatest credit, however, should be given to my
major professor I. A. Gee, Head of the Department of Account-
ing at Michigan State College whose cooperation and encourage-

ment have been very beneficial in my graduate work.

$355916

TABLE OF CONTENTS
?&£0
1

I Introduction
II Present Testing Procedure in Auditing of Sales 3
III Review of Literature on Statistical Auditing 13
IV Statistical lethodclogy 23
Y .A Case Study of an.Aspect of Statistical Auditing

of Sales 30
71 Conclusion as
Appendix

A - Case study using sequential analysis in auditing #6

S - Table of Three Sigma Confidence Limits for
Binomial Distribution in.Percent #7

c - Summary of Computation of (p) is

D - Summary of Computation of Upper and Lower Limit
for aofiuSamples and‘Total of the Three Samples #9

I - Summary of Computation of Upper and Lower Limit
for 10$ Samples 50

Bibliography 51
List of tables
Table l - An Analysis of Typical Sampling in the

Audit of Sales 7 -
Table 2 - Probability of Encountering at Least One

False Item 1h
Table 3 - Action Points for Sequential Tests 19

Table # - Summary of Stratification of Sales Test Data 33

Table 5 - Summary of Three 20$ Samples and Total of
the Three Samples 35

Table 6 - Summary of Six 10$ Samples 36
Table 7 - Summary of Results - Upper and Lower Limits 39

IITRODUCTIOI

For some years auditors have been expressing opinions
based on tests of the accounting records. The reliability
of their confidence in the tests commonly employed has been
seriously opened to question by various authorities and
various attempts have been.made to develop standards of
auditing as guides for the exercise of judgment. This seems
to skirt the problem, so this studyttas designed to develop
a method of determining the limits of confidence whereby the
auditor's judgment of materiality or immateriality may be
attested and by which reasonable standards may result.

51.32»?

The problem wee-limited to sales because(the~eaee«study
involved anwaspect of sales, and further{)if the method and
arguments presented are valid for sales, they may be general-
ised to other pertinent parts of auditing with the exception
of out offs necessitating a testing procedure.H

The more recent researchers have been concerned with the
use of sequential sampling developed during the was by the
Statistical Research Group at Columbia University. Another
area of research not approached in.this study concerns itself
with an analysis of errors.

The methods of research involved reading all the avail-
able material on statistical auditing, reviewing various tests
and case studies for the auditing techniques relevant to sales.

- 2 -
. searching various statistics texts for clues and finally
making the case study. The most intriguing and at times
most disheartening problem was the finding of a suitable
statistical approach which omitted insignificant refine-
ments of theory, since the auditor requires practical
methods without theoretical elaboration. Quinn ltcltemar's
Psychological Statistics was found the most useful in the
development of this method.

PRESENT TESTING PROCEDURES IN AUDI'I'IIG:~

In any discussion of testing, particularly as it applies
to auditing, it weuld first be best to develop the concept ef
testing. we may well ask, what does the word testing mean?
The dictionary definition of the verb suggests that to test
is to prove, try the quality of, or examine. (1) As a noun,
the meaning connotes an examination. (2) This definition in-
fers that the testing will be in tote. However, as applied
to auditing, the transitive verb testing means to sample or
to determine the accuracy by selecting and studying repre-
sentative items or samples from a given cellectien or class
sf transactions or other data. (3) Another auditer, lcntgomery,
says that test checking is based on the mathematically-founded
assumption that an analysis of representative samples of a
group of items indicates the quality ef the whole. (3) Howb
ever, no attempt is made by lontgomery to particularise the
mathematics to which he refers. These definitions differ with

 

 

(U '1 Dictionary c? Imerfcan India's" Vol IV, Edited
' by Sir William A. Craigie and James at Hulbert,
Univ. of Chicago Press, Chicago (l94h) P. 2308

(2) 'er Standard Dictionary of the English Language“
Idited'bz Isaac 1. Punk, Funk a lagnalls 00., low
York (19 3) P. 2M9

(3) I. L. Iohler, AUDITING AND INTRODUCTION To the work
of the Public Accountant, 1st. ed., Prentice Hall,
Dew York, P. 21 (19“?)

(to Robert a. hontgcmery, AUDITING moon! inn rescuer,
6th «1., 11ch Press, New York, P. 36 (19,40)

-V‘"- —

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the dictionary meaning as stated above. Another accountant
defines testing, the noun, as a limited examination or
verification by sampling. This definition is in agreement
with the dictionary. (5) The dictionary makes a distinc-
tion between testing and sampling. As a verb, sampling means
to examine by use of a portion or specimen. The noun, sample,
is defined as a part of anything presented as evidence of the
quality of the whole. (6) The latter definition corresponds
with the use of the word as applied in statistics. A sample
is a part of the whole, the entire data, if available, or that
is to say, the defined population. (7) where we can not take
the aggregate or whole; we do the next best thing and try to
obtain a selection of members, which is called taking a sample.(8)
hontgcmery states that the basis of test checking may be all
items in a specified period, or all items over a certain mini-
mum amount of dollars for a period. As a guide,'cne may use
letters of the alphabet, or percentage of the total either in
dollars or amounts. The method of sampling will depend on the

wimractizenggblems, lstfred” Ronald Press, new

York, P. 6 (19118)

(6) 'lew Standard Dictionary of the English 33383080"
Op. Cit. P. 21“

(7) in itclemar, PSYCHOLOGICAL STATISTICS, lit. ed.
ohn Wiley 4. Sons., Inc. lew York P. #6 (l9lt9) and
Herbert Arkin, R. 0. Colton, AI ourun: OF STATIS-
TICAL nTHODS ltth ed. rev. Barnes 6: Noble, Inc. Dew
York P. 113 (1950)

(S) G. Udny Yule, l. G. Kendall, An Introduction to the
THEORY OF STATISTICS, 11th ed. Charles Griffin 8: 00.

London, P. 9 (1937)

V-v...

I.-.

can-

- 5 -
type of item, volume tested, and the system of internal con-
trol. The auditing use of the word 'sampling' is not synony-
nous with the statistical word “sampling“ since its theoreti-
cal basis includes no probability theory as any statistical
method requires.

With the advent of big business, there developed the
necessity for making the audit a technique of analysis of
selected samplings of accounts rather than an attempt to exam-
ine all of the transactions for the period. (9) It has become
a custcn and, with few exceptions, has proved sufficient. (10)
To examine in detail all transactions requires a cost surpass-
ing all reasonable bounds of benefit or safeguards, and places
an undue burden on industry. However, the extent of such samp—
ling is left to the individual aocountant's judgment. (11)

This judgment is based on the client 's system of internal con-
trol. (12)
Consider, now, what' are the typical sampling procedures

  
  

   

   

  

  

er . 1 - ' . ' ' :' "1; ":I'
PRESIUT CENTURY: 1st. ed. Harvard Univ. Press, Cam-
bridge, lass. P. 10 (191R)

(10)IXTIIBIOIS OP AUDITIIG PROCEDURE, (Report Of hay 9,
1939, as modified and approved at the Annual hosting,
September 19, 1939) American Institute of Accountants,
lew York, P. 3 Oct. 18, 1939)

(ll)STATn[nTS OI AUDITIIG PROCEDURE lo. 1, Issued by the
Committee on Auditing Procedure American Institute
of Accountants, lew York, P. 5 (Oct. 1939)

(12)TIUTATIYI erirmsr OT AUDITING STAUDABDS - Their
Generally .Accepted Significance and Scope, Special
report by the Committee on Auditing Procedure American
Institute of Accountants, lew York, P. 25 (19‘?)

   

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- 6 -
that are used in auditing generally, and specifically as
related.to sales, that involve testing yet exclude cut-offs.
Iithout attempting to be exhaustive, representative methods
are presented in Table I on the next page.

As the reader may already have concluded after reading
Table I that there is great use of the word I'test,", but little
attempt appears to have been made to develop a technique of
representative sampling.

The following quotation typifies the auditor's approach
to testing: (13)

TEST—CHICIIUG’OR.TISTIIG - This means the complete veri-
fication of a.portion of accounting transactions. Testing
is common in audits to assure the auditor that transactions
are in order for the untested portion of the year, after
having verified all transactions for a certain limited
period of the year, or after having tested transactions

at random.

In order to test-check in a reasonable manner, account-
ing transactions and entries must be classified logically.
For example, they may be grouped as follows: sales records
footings, purchase record postings, purchase vouchers and
invoices, caeh receipts and disbursements postings, cost of
sales footings and postings, cancelled cheques, pay rolls,
‘balances of accounts receivable, etc. After proper class-
ification cf the items, the next step is the determination
of the number of items of each classification to be tested;

The majority of accounting entries are honestly and
correctly made; the purpose of the test check is to re-
view supporting evidences in order to detect errors and
fraud and in order to be in a.position to judge impar-
tially the accuracy of the accounts. The auditor must
be satisfied that the transactions are legitimate and
that the accounting for them is proper, so that the finan—
cial statements are not affected and so that an opinion
of the statements may be rendered. no definite rules can
be set forth for the amount of test-checking, as this is

 

 

(13) Arthur I; Helios IEDITIiG Principles :53 Procedure
2nd ed. rev., méurd n. Irwin, Chicago P. 101 (19k?)

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dependent upon the judgment of the auditor, his exper-

ience, the system of internal accounting control in

operation, and the conditions existing in each engage-

.onte " "' "'

Testing may take place for certain selected weeks

or months, or it may be at random. Random tests either

satisfy the auditor that the classification of items

sampled is correct to a high degree of probability or

the results of the test are unsatisfactory. If not eat--

isfactcry, additional tests must be made.

The reader's attention is .r'n. directed to the fact that
no theory of probability underlies the auditor's sampling methods.
In order for a ample to be random, each item must have an
equal chance of selection and each sample must include items
from the whole population. To base samples on one day, three
days, one month or three months does not qualify as random. It
is likely, at this point that the question will form in the
reader's mind, what is the statistical theory of sampling?

The object of sampling is to give the maximum information
about the population. (lit) Obviously these are estimates,there-
fore the next aim of sampling is to determine the degree of
confidence we can put in our estimates. (15) The accuracy of
the estimate will depend upon (a) the way in which the estimate
is made from the data of the sample and (b) the way in which
the sample was obtained. The process of sampling consists of
choosing a predetermined number of individuals from the parent
universe. This may be done in three ways; random, purposive,

or mixture of the two. Randomness exists when each member of

 

. U tile, I. . , . t.
(15) Ibid P. 335

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the universe has the same chance of being chosen, personal bias
must be eliminated. A method or code must be used which leaves
nothing to the observer's idiosyncracies. (16) such a method
may be developed by use of fippetta numbers. Purpoaive sampling
is to select the average in each group. The practical use of
random sampling lies largely in the fact that it allows us to
measure objectively, in terms of probability, errors of estima-
tion or the significance of a result obtained from a random
sample. The purposive methods have not as yet been able to do
so. (17) Usually as the random sample becomes larger, it be-
comes more representative, whereas owing to bias, the purposive
sampling in general does not. Further, the object of the sample
is to tell us about the parent population, purposive sampling
may tell us more about the mean, but will not tell us about the
extremes. (18)

Thus it can be seen that the auditors are constantly intro-
ducing bias into their samples which no hnewn methods can an-
press scientifically with any degree of confidence. i'rue enough,
the auditor is sampling but his methods involve purposive sampl-
ing. Iontgcmery has taken a step in the right direction but
seems to void his point by reference to the "auditcra'I sampling
methods. (19) for, as pointed out above, purposive sampling is

M
(17) Ibid P. 3”
(18) Ibid P. 3“?
(l9) 1!. 1.. Icntgcmery 0p. Oit. P. 36

 

vv

.. 11 -

not random.

(”The limitations of the present methods of auditing make
it impossible to substantiate objectively the auditor's judg.
ment. This becomes an important point if the auditor should
be forced to appear as an expert witness or as a defendant in
a court room. The question may well be put to him what ob-
jective measures of confidence does the auditor possess to
prove his right to an expression of an opinion, namely, the
unqualified certificate. Traditional methods will leave the
auditor at the mercy of the court because his measures normally
are subjectivepjyet there are objective measures available
which will relieve the auditor from such embarrassment.7

The auditor who samples two months of transactions
(16 2/3” and finds few errors concludes that the client's
system is working and that he has a right to rely upon it.
There are several points overlooked in this illustration of
typical field work. The first point overlooked is that the
auditor's sample is not random. By pieking two time periods
as pointed out previously under the definition of sampling,
the auditor is not in a position to draw a mathematically
(logical) conclusion with any limit of confidence. The second
viewpoint disregarded, is that few auditors ever actually rs—
late the total errors found in dollars with the total sampled
of a particular phase of the audit or accounting procedure such
as sales. tech error is compared with the auditor's concept
of materiality and not the sum of the errors in dollars. One

a.

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- 12 -
reason is that his emphasis is placed on individual entries and
not the total of the sample.

Oranstcunfihae summarised the point well by saying: . . .

textbooks point out that the extent to which sampling

procedures are applied will vary under different condi-

tions and.that decision as to their extent must be a

matter of the auditor's judgment. Ihilc that is unp

questionably true, the fact remains that no standards
haye been set; the auditor therefore has no help in
forming his judgment, and there are no consistent
measures by which the reasonableness of his judgment

can be attested. (20)

This authority further’pcints out the need for standard
percentages or quantities that make up the sample. In other
words, he is attempting to arrive at standard sample sises in
order that the sample may be adequate, that cost of the audit
may not be excessive, and that the auditor may prove his work
before client or jury.

The purpose, than of this paper, is to develcp a.method
of determing limits of confidence in the auditing of sales
whereby the auditor's judgment of materiality or immateriality

may be attested and by which reaeonable standards may result.

 

 

(25} 'iITIam 5. Uranstoun, '1 law Cock at Eisiciluditing

Techniques' c J c A ccun anc Vol. 86:¥
P. 27kg} (cg—i. $55!

 

RIVIII'OF LITERATUEI OI STATISTIOLL KUDITIIG

The first article appearing in.print on the subject of
statistical auditing was that of Lewis Gunman published in
the‘Nmerican Accountant, Dec. 1933.(2l) Oarsan.was concerned
with the probability of discovering fraudulent entries in a
large group of entries. The idea was based on.the assumption
that the uncovering of a false item is the signal to stop
and reconsider the sampling process. For example, let us exp
amine (on the next page) one of the tables (Table 2) Carmen
used.

suppose there were 20 false items, then the most economi-
cal sample is 15$»and the possibility of uncovering one of the
false items is 19 out of 20 times used or 95$.ccnfidence. now-
ever, this does not go further than the uncovering of one false
item. The auditor is confronted with the necessity of continue
ing the sampling or extending the scope of his sampling.

The next article written was by Robert H. Prytherch. (22)
He points out that German must assume the number of false
entries. This, however, is not known. 8o Prythereh proceeds
to give an approach based on the following example involving
the testing of purehaees to compute a reasonable assumption.

 

(21) fewis 1.5arman, i‘fiolfficacyo? Eat a“ mum.
Oil AOOOUITANT, Vol. 1711! (Dec. 1933) IP. 360-366

(22) Robert a. Prythereh, “How Much Test Checki is
Enough?' TED JOUBILL OF LOOOUITAIOY, '01. 7 lo. 6

(Dec. 19h2) P. 525—530

H

TABLE 2
PEOBABILITY OF ”000!!!an A! war on PALS! IN!"

Assumed lumber lost Economical Probability of lucounto
of Pelee Items I‘landcm Sample cring at Leas} One Pelee

 

 

 

1! arm; x on re
:3 at 0:0: 95¢

i: ii = i3
10 22$ : 9o$

‘ 3%”: =3;
3 . : :5.
:i 53 - n1

 

‘ Lewis A. German 'The Efficacy of Tests" The American
Accountant Vol. XVIII (Dec. 1933) P. 362

 

a .m-h‘hu

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. - 15 '-
Total purchases 080,000

lumber of entries ll-OO
Average dollar amount 3 200
9‘ of total possible false entries ILOCK)
Average number of false entries 20

Referring to Table 2, it would require 15$ cf I1-00 entries
or that 60 entries be examined. The probability would be
19-1 of uncovering at least one false entry. Should a false
item be found, it is then necessary to extend the audit to
find other false items.

This sampling, however, does not mean one or two months
transactions, but that the samples be chosen on a random basis
of every 5th or 6th item until a total of 151» is obtained or
some other method satisfactory for the basis of random selec-
tion. (23)

The reasoning behind this is that if the errors are con-
centrated in six months, say 12 of them, the probability of
uncovering a single one by examining two months completely
(16 2/3” is 771», but if the errors are concentrated in one
month, the probability of uncovering any one of them is
16 2/31.. (at) Thus, one can see the tremendous risk the audi-
tor is taking under existing practices. Tet note that al-
though the concept was developed first in 1933 then expanded

(55) f513 F. 555 ___
(2!) Ibid P. 528

 

- 16 -

in 1942, eight years later, no change has yet been made in
typical methods. See Table 1

In 1947 Leo Rerbert and also Jerome Abrams came out in
the law Icrk GPA with articles on sampling. (25) (26) The
article by Eerbert reiterated the three classical rules for
sampling: the sample must be representative, adequate and
stable. By representative is meant that the sample must be
chosen at random in order for the theory of probability to be
operative. Adequacy has reference to avoiding small samples.
However, this seems to ignore the whole body of theory in-
volving small samples. Stability means that any increase in
the sample size develops no significant change in findings.

Abrams brings out an interesting point that German and
Prytherch based their findings on the normal distribution, but
he feels that the errors more nearly approximate the Pcissonian
distribution which is a greatly skewed curve. However, he does
not follow up his supposition which this study will do.

Reopening of the subject literarily was initiated by
John [star in lay 1949. He discussed the application of sequen—
tial sampling to auditing which this study will develop with

 

c or rac ic amp ng or Auditorsl H!
m roar cu’mnrn Pusuc account”, (Jan. 19137)
Vol. 1711 lo. 1 P. 58

(26) Jerome Abrams, 'Bempling Theory Applied to the Test-
Lndlt' THE am You CERTIFIED PUBLIC AOOOUITANT,

Vol. XVII, lo. 10 (Oct. 19fl7)P .6N-5-652

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- 17 -
Lawrence L. Vance, the next writer. (27) without solving it,
leter raised the practical problem of setting up a criterion
upon.which errors of transactions can be classified as to
their effect in.making the transaction as a.whcle acceptable
or unacceptable. In September of l9h9, Lawrence L. Vance pointed
out that statistical sampling is a tool and the method can be
used only upon the basis of some standard.pcpulation, ( a concept
which Vance failed to define), with which the actual sample can
be compared. (28) He also pointed.out: "that the method is
appropriate only for those areas of accounts where a relatively
large amount of detail work of homogeneous character can be
isolated.‘ (29) F;ince the'statistical method will not parti-
cularly uncover the isolated.error.hut/::lps the auditor to
evaluate the quality of the sizzner worku“#he approach is no
advance in uncovering the occasional fraudulent error except
that more adequate samples may result in raising the general
standard of auditing;1

In 1950 Vance published a book on sequential analysis as

 

(27) Join later, 'In.fnvestigation o? the UsefEIness of
Statistical Sampling hethods in.Auditing", THE JCURRAL
OP ACCOUNTANCY, Vol. 87, lo. 5 (lay 19M9) P. 390-398
and also John later, "The Application of Statistical
Techniques in.Auditinngrocedures" THE ll! YORK CERTI-
nm PUBLIC iooounriar, Vol. x11 Re. 6 (June 1919)

P. 35-350

(28) Lawrence L. Vance, "Auditing Uses of Probabilities in
Selecting and Interpreting Test Checks" THE JOURRLL
or ACCOUNTANT Vol. es, lo. 3 (Sept. 19%) P. zit—217

(29) Ibid P. 216

a.

‘n

v- .--.

- u -
it applied to auditing outlining the technique in detail. (30)
The basis for sequential sampling is the likelihood

ratio. It requires the use of two hypothesis, H1 that the
population has P1 (percentage) defectives or fewer, and H2,
that is, has P2 (percentage) defectives or more. If the pro-
bability of drawing the sample from 31 is P1 and of drawing

it frem H2 is P2 then the likelihood ratio is P . The level

of risk we accept in making a decision is exprePsed in two
parts. We will designate the risk of accepting Hg when ‘1
is true as (a) or alpha and the risk of accepting 31 when a:
is true as (b) or beta and (I) as designating the sire of the
sample Vance 's table appears as Table 3.

The values tentatively suggested are P1 (percentage 4....
100th" 01‘ 10"") = .005 PM Pg (percentage defectives or
more) a .03. The meaning is that 1 error in 200 is acceptable
whereas 3 in 100 are not acceptable. (31)

Vance defines errors as consisting of two groups, sub-
stantive and procedural errors. Substantive errors include
errors in computations, errors in posting, errors in account-
ing principles and errors in emission. An example of a pro-
cedural error would be the failure to put a countersignature

Wamnoefmm 1st.
ed. Univ. of California Press, Berkeley and Los Angeles

P. 87 (1950)

(31) Ibid P. 29 Attention is called to the fact that
Vance changes his notion on standard population
stated on P. 19 of this paper to a standard per-

centage of error.

 

TABLE 3
£01310! POII‘I'S FOR SEQUHTIAL TESTS '
(Vhen a 2 0.0 b m O 10 and ~ and . have the values shown

Pu ow

 

 

 

 

n I; (accept) ya (rejegt)

2 0e. 2
29 ... 2
89 o 3
100 o a
160 1
171 1 t
2 9 2 5
2 3 2 2
3°Z ’

1 a 6
7% 7
85 h 7
t6 5 s
#5; 2 8
3%; 6 3
588 7 1o
519 1 is;

 

The symbol (...) means that the next larger sample siss should
be used unless the number of rejects found is sufficient to
reject sample.

'" Lawrence L. Vance, SCIENTIFIC nTHOD TOR AUDITIRG
'Cniversity of California.Press, Berkeley and Los

Angeles P. 91 (1950)

   

 

 

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- go -
upon a check otherwise properly issued. (32) Vance also feels
that the definition of error should be as bread as possible
since there will be a few errors uncovered. (33)

In applying the method to auditing of sales, Vance points
out the original records relating to sales invoices, shipping
orders, customer orders, and sales journal are proper areas
for the statistical method. (34)

One of Vance "a cases is given in Appendix A in detail
merely to acquaint the reader with his approach.

[There it is desired to raise the representation of the
sampling, stratification may be need. This method is avail-

' able where the data may be classed in mutually exclusive
strata. Vith reference to auditing, the strata may be based
either on the amounts in the transaction or the amounts in

the calculations. The procedure requires that proportions

of the population falling into each category be known and that
samples taken from each category be related in sise by the
same proportions. 93).]

There is an attempt by auditors to stratify the data for
ordinary auditing. For example, in examining the extensions of
inventory, material items may be examined completely while
small items may be only scanned.

W
(33) Ibid P. 32
(3t) Ibid P. u
(35) Ibid P. 72

C
s

Q

. -
i

' Q
A
.l.
. ,-.

-~

..-A

- 21 ..

As the reader by now can appreciate, there is quite a
difference between the sampling done by statisticians and
that done by auditors. The auditors have a tendency to
examine the material items and neglect the small ones.

There is no quarrel relative to examining the significant
ones but the smaller case must be included in at least a
proportion of a minimum sample. Sales records are almost
invariably sampled for periods of a week or more. The result
is a biased sample with its inherent difficulties of math-
ematical precision. Since at present, the auditor's samples
are so frequently biased and difficult to defend, the only
conclusion that is tenable is that by 'the use of random
sampling the auditor may defend his methods which is indeed
a great advance if no other end was served. (36)

In. D. Crans'tcun in 19% listed three limitations of
present statistical auditing as he views the subject;

1. lo probability ratio can be calculated for a
combination consisting of two or more procedures.

2. The statistical approach is directed only to
calculating chances of discovering a single false -
item, when actually the auditor is concerned with
their extent and sise.

3. Since probability is based on the number of items,
no distinction is drawn as to the importance of
items. (37)

Excluding the types of audits designed to uncover fraud only,

W

(37) Villiam D. Cranstoun, "A lsw Look at Basic Auditing

 

Techniques" ran Joumm. or acoouanacr Vol. 86, so. '1

(Oct. 19“) P. 273-283

a- a..-

_ r

..-.-o.

adv

- 22 -

this investigation will solve questions 2 and 3 and suggest
an answer for 1 above. For this purpose an actual case was
designed and a method developed which will be described

later and discussed in the conclusions.

, STATISTICAL IETEODOLOG!

[I/fThe problem of the auditor is one of uncovering errors
either of omission, commission, or of principle. The auditor
examines the transaction sampled for existence or absence of
error. lo attempt is made to classify the errors as bad,
very bad and terrible. The only classification is in terms
of dollars and.more precisely, in terms of whether the error
is material or immaterial. The concept of materiality is a
subjective determination by the auditor based upon.the relap
vent factors. The error itself, although measured in dollars
is an error in recording, posting, calculation or of principle
which alone is not subject to classification. (38)

The study of the existence or nonpexistence of a char-
acteristic, error, in the data is referred to by statisticians
as a study of attributes. As the result of a sample, those
items possessing the attribute are placed in one class while
those not possessing the attribute are placed in a separate
class.. The classes are mutually exclusive. The first class
is designated as (p) and the second class is (q). The relation
of 1 - p a q exists between the classes. It is advisable to
reduce the frequencies of the classes to percentages. (39)

     

 

wrenoe . “as, :e an; '— w =... i. - ._
lst. ed. Univ. of Calif. Press, Berkeley and Les
lEGOIOs, P. 12 (1950)

(39) Q11“ Iolemar, PSYCHOLOGICAL STATISTICS let. ed.
John Viley & Sons, Inc. lew York P. 62 1939)

. 2h. -

Suppose it is determined by randcm sampling that 53 of
2°C invoices were improperly extended or that 10% of 100
sales invoices were improperly priced. The mathematical
model, the binomial theorem, permits one to generalise from
these statistics the amount of the error. we are faced then
with the problem of making an inference from the sample value
to the population value, i.e., from (p) to (P) where (p) stands
for the observed percentage possessing the characteristic
studied (errors in extension or errors in pricing) and (P)
stands for the percentage in the defined population that show
the characteristic. If we were to take successive samples of
sins (n) and make a distribution of the observed percentages,
the distribution would center about (P) with a spread or stan-
dard deviation equal to the square root of P(lOO - P) In.
Since we do not know (1’), we must use the observed percentage
as a basis for determining its standard deviation. _ The stan-
dard deviation of a binomial distribution of a percentage will
be given approximately by 0F : W in which:

p 3 observed percentage or attribute expressed as a decimal

q - 100 - p '

n m the number of cases studied in the sample

If 10 invoices out of 200 chosen at random from a year's
sales invoices possessed errors in multiplying quantity times
price it may be inferred that the (P) population percentage
of such errors is likely to be between the limits (p t 30”)

or 5 j 3 (1.5!) La. .38$ and 9.621» approximately. By using

 

.
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. 25 -

a three sigma (3 0/) confidence limit it may be said that in
99.7 samples out of 100 samples the population percentage (P)
will lie between .381» and 9.62% approximately.

One limitation to the use of the above method is namely:
that each individual member of the universe must be replaced
before the next sample is drawn. This means that each event
has the same chance of success. (l-O)

Another condition for this method is that the success of
different events, the existence or non-existence of error, are
independent in that the result of one event is not affected by
the results in prior events. (#1)

The assumption underlying this formula is that the ob-
served (p) will be a very close approximation of the population
(P). The smaller the standard deviation is the closer the
approximation becomesw

holemar states that the relative form is unworkable when
(p) is small. (#3) Iule and Kendall, however, point out that

the formula 0’? 2. \) 23- is the relative form of 0’ P W

h

‘def the/\same results will be obtained by both formulas. (ll-1+)
‘ / /

They—further point‘gout” by the use of the lat-tear formula jthat

 

y ue, e n are ctionto the
when! or STATISTICS, 11th e'd. Charles Griffin .1 Co.
London, P. 350 1 357-8 (1937)

(#1) Ibid P. 350
(‘12) Ibid P. 354.355

(#3) Quinn hose-ai- Op Cit P. 62

-26-
if (p) is small. ID that P2 as compared.with (p) may be neg-

100393; then.pq = P (1-?) a P - 92 m approximately p, and
consequently we have approximately; (a W = W,

That is to say, if the proportion of failure be small,
the standard deviation of the number of failures is the square
root of the mean number of failures and hence the standard I
deviation can be determined even if (p) is unknown except that
it is small. (#5) Thus it would appear that the limitation of
Holemar is unjustifiable particularly when (n) is large.

In auditing, the (n) will usually be large so that this
limitation will be of no practical significance to the prac-
ticing accountant; .-.4 ~

The experimental unit used in this study is the pricing
of a part. The part to be priced.was chosen on the basis of
every fifth item in the strata. Since the characteristics of
the data are not determinable other than by detailed examination
and further, the parts were chosen throughout the temporal per-
iod, the sample then may be considered random because the bias
of the investigator was eliminated. The limitation involving
the replacement is not a factor because no order of parts existed
and success of selection of a part depended upon whether the
investigator began counting to five with the first part found
in that strata or successive items. For randomness, the

investigator based the beginning number of each sample upon

 

a 611! a C, it e 3
(#5) Ibid P. 356

.. 27 -

a random number table using 2,3,1 for the 20$ (approximately)
samples and splitting the 20$ samples in half for six 10$
(approximately) samples. The three 20$ samples were added to-
gether to give one 60$ sample including some overlap.

The independence of each event was not prejudiced by the
results of prior events by virtue of the sampling technique.

The study introduced a variant in the method of stating
(p) (q) and (n). Although the experimental unit was concerned
with the act of pricing, the successes or failures by themselves
are not the primry concern of the auditor because he is con-
cerned with materiality and not the amber of errors. There-
fore, it seemed quite logical to measure the factors in terms
of dollars. The errors in dollars divided by the total dollar
value of the sample is used as (p).

for example, in sample one, five errors in 105 items were
found. Using the standard deviation formula for a percent

a? z \Iflg the resulting standard deviation is

2.7$ or a three sigma variation of 8.1$ adding this to (p) of
t.7$ the limits range from O to 12.8$. This appears to mis-
state the facts. Referring to Table 7, with a sample of
$2201.32, only in of errors were found assuming It . (p),
which is .18$ or very near perfect. The upper limit.is .‘l-5$.
To the common sense of the average auditor, the method involv-
ing dollars seems more meaningful and more usable in the exer-
cise of judgment.

The limits of .003 variation sf a percent of a sample with

.
I m
" ‘ i e
T.
N
M ‘-
- I.-
.

inn.

- 23 -
a large (n) and a (p) may be expressed by the formula F I 3%
Varying values of this are available in tables, excerpts from
one of which may be found in Appendix B. Referring to Appendix
B, suppose in the audit of 610,000 of sales invoices from a
total of a 81,000,000 sales, O.6$ total errors were found.
Following in the upper control limit the line marked 0.65 to the
column marked 10,000 is the figure 0.83$; in the lower control
limit the corresponding figure is 0.37$. The interpretation is
that the variations due to sampling are from 0.83 to 0.37 of
one per cent. Thus, based on a 3 ( confidence limit, this sample
could not have arisen from a population containing more than
0.83 of one per cent of errors or less than 0.37 of one per cent
errors in 99.7 chances out of 100. By using dollar values for
(n) this means that based on the results of the sample there
will be between $3700 and 88300 of total errors, in the popula-
ticn, but no more, in 99.7 times out of 100 samples.

The principle of stratification is to break up non-homo-
geneous data into more homogeneous groups. Within each group
random selection is employed. The sire of each group in the
sample should be proportionate to the relative importance of
the stratum to the total of all the strata. There the differ-
ences between strata are pronounced, a more accurate sample

results. (#6) Dollar values lend themselves easily to strati-

 

 

(fi) Traderick WI to
lccncmics and Business 2nd ed. rev. Henry Bolt d 00.,

low York P. lt62 (193s)

- 29 -
fication. They are mutually exclusive, that is to say, 32.50
does not appear in a class interval of five to ten dollars.
holemar introduces a correction of the standard deviation

_ PG 0'"
formula for stratification namely 07’ - “51" ' ,5 .

to
However, in this study the (K 's) are so small that the correc-
tion becomes theoretical and has no significant bearing upon
our results. However, should (p) be much larger, say 30$, than
it may be well to use the correction. (t7)

Vance uses a method of weighting stratified results for
which no authority could be found to support him and thus his

technique is excluded from use in this study. (I8)

"'"'('l7)"Qu'In'!El'en'ai-" CE'CR P. 33‘

(M) Lawrence L. Vance Op Cit P. 73

 

vu— ..-

A CASE STUDY OF A] ASPECT Ol' STATISTICAL AUDITIUG OP SALES

Vance pointed out that it would be desirable to summarise
results in terms of the money value of errors in the population,
but this did not seem to be a very practical objective. (#9)
Accounting errors arise almost entirely from human fallibility
and there is no regular or simple pattern of human fallibilities
on which probaulity calculations could be based.

This, the investigation set out to prove. The idea was to
use indirect argument that the results would be consistent with
the thinking of Vance. If they were inconsistent, or in other
words, a pattern adaptable to statistical study and as accurate
and as sensitive as his results were found, then his position is
erroneously taken.

A fairly large manufacturer agreed to a limited examina-
tion of a small segment of his sales. The system of internal
control existing is as follows: A parts order is written up
and the customer's credit is approved. Vith approval, a fee-
tory move order is made in triplicate. The first copy is used
as an acknowledgement copy, the third as a packing slip and the
second follows the parts from the warehouse to shipping depart-
ment and thence to the inventory control where it is priced,
extended, and posted to inventory records. From the second copy
Winches CWT-SWIM—

lst ed. Univ. of Calif. Press, Berkeley and Los
Angeles, P. 12 (1950)

 

- 31 -
cf the extended move order, the billings are made, proofed,
and the extensions are checked by a ccmptcmeter operator.
Pour copies of the billings are made: One to the treasurer's
department, second to the accounting department with the
third and fourth going to the customer. Prom the second the
receivables are posted, the order is listed, and a sales dis-
tribution is made.

Based upon the accounting system a list of possible errors

was developed: .

l. The credit may not be approved. This was considered
under control because of matching the treasurer "a
copy with the customer's statement.

2. The incorrect part number may have been recorded on
the move order, the incorrect item shipped or im-
proper posting to the inventory. Since shipments
were impossible to check and management did not de-
sire that the service department personnel be in-
volved in the investigation, these items were not
included in the study, even though no checks or
balances to the knowledge of the accounting depart-
ment existed with reference to these possible errors.

3. Another type of error which was not controlled was
the pricing of the parts. This could be reviewed
without involving the service department and further
this was the item interested in by management.

a...
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- 32 -

h. The billing errors due to proofing and comptcmeter
operations did not seem too fruitful for study.

5. Since the year end closings were in progress and
the public accountants were present, the accounting
department felt that a study of the recording
errors in accounts receivable, sales distribution
and costing were not practicable at this time.

Therefore, the study involved items locked up in the
pricing book, The experimental error was defined as the
correctness or incorrectness cf the price looked up and re-
corded as the unit price times quantity. (50) The error
occurred in the act of pricing and was measured in dollars.

The first procedure was to tabulate the sales of these
particular parts for the year. It was found that there were
526 items to be looked up and the sun of unit prices times
quantity for the year totaled 87055.82.

The seeond procedure was to develop a scheme of strati-
fication which at best was unwieldly, but 12 strata were made
as shown in Table I» on the next page.

It will be observed from the stratification and distribu-
tion that the curve is highly skewed similar to the Poisson-
ion distribution. It was then decided to take two sets of

 

 

(SC) Prom here on the figures are em disguised,
but with no effect on the results.

TABLE (I.
SUMMARY OF STMTIFICATION 0F SALES TEST DATA
For the Year Ended Dec. 31, 1950
Class Class hid-point Experiment- $ of experi- Amount of

 

no. (in dollars) al units mental Units class
(inpdollars)
l 3 .50 1“? 27.95 C 61.15
2 1.50 68 12.92 99.02
3 2.50 36 6.85 87.12
t 3.50 36 6.85 122.58
5 h.50 23 5.37 101.53
6 5.50 25 “-76 133.98
7 8.50 60 11.to t90.89
8 15.00 70 13.30 1018.#4
9 35.00 #5 8.56 1522.¥l
10 125.00 10 1.90 905.25
11 300.00 5 .76 1117.h5
12 700.00 ...—3... ...—a3§_. ..112§129_.
42m: 1216 M22.

 

 

an“- -

_-,

---;

- 3h -
samples based upon the above stratification. The first set
consisted of three samples approximately 20$ each of the ex-
perimental units based on taking every 5th item, except the
first sample, in each stratifieaticn where possible. However,
in those strata, insufficiently large, successive samples were
not included. The item chosen was priced and extended. Any
difference was noted as an error. A feature complicating this
process was that a price revision had been inaugurated after
the price book was published. Every part was first detensined
if it was included in the revised listing, if not, the part
was found in the main body of the catalog and then compared
with the price on the invoice. If the price on the invoice
varied from the revision if listed there, or the catalog if
not, an error was recorded. The unit price times the quantity
was then extended to get the total error. lo attempt was made
to segregate the errors dependent upon their increase or de—
crease of gross sales. It was felt that this was not a factor
in the study. The three 20$ samples added to give one 60$ sample
with some overlap which is included in the table.

The second set consisted of 6 samples approximating 10$
each of the experimental units using the same procedure as
outlined above.

The results of the sampling are shown in Tables 5 and 6.
Included in the samples discussed above, the first sample of
20$ of the population was taken in groups of five scattered

through the stratum. At least one item was used from each

TABLE 5
emails! or 111an 201» ammo AND row. or TH]: ram SAMPLES
Por the Year Ended Dec. 31, 1950

9}”! r rs in doll re rror n u t
1st and 3rd Total 60$ Ist
gm; 2221 am; 8221; gmle Tots;

l t .10 d .13 t .20 t .53 2 It
2 .05 .05 .00 i .10 1 2
3 1.15 .15 1.t5 2.75 1 3
t 00 co 00 cc 0 o
5 co 00 co co 0 0
6 co co co oo o o
7 2.70 .65 oo 3.35 l 2
8 00 00 00 oo o o
9 00 co co co 0 o
10 co co co 00 c o
11 co co co oo o c
12 __99. ..22. .99.. ..22_. ._9_. _9_.
ma 8.1m. Lu: the: am. .1. u.

 

Ho‘-

d “--

TABLE 6
some! or six 10$ SAIPLES
For the Year Ended Dec. 31, 1950

M Error in doll rs

um.» 2.12212 9.22922 Jena.» 1° 11329—10 1.22212
1 S 0 S .10 3 .13 Q 0 S .20 S 0
2 0 .05 0 .05 0 0
3 0 1.15 .15 0 0 1.55
1|» 0 0 0 0 0 0
5 0 o 0 o 0 o
6 0 O 0 0 0 0
7 o 2.70 .65 o 0 0
8 O O O 0 0 O
9 O O O 0 O O
10 O O O 0 0 0
ll 0 0 0 0 O O
12 ...L _.Q_.. ...9. ....Q. ...Q. .9—
Totals 0.3. 0% gig. 132.: afl- Lg.

 

sea..- 4

- 37 ..
stratum. This was taken to compare the results of sequential
analysis with the method developed in this study.

Based upon the sequential sampling of Vance, the first
sample would have been rejected since five errors are more
than the three errors allowed by the Table Vance uses for
samples of approximately one hundred. (51) In order for this
sample to have been accepted, no errors should have been found.
Rejection is predicated upon the finding of three or more errora
Yet by the methods of this investigation, the sample would have
been accepted as demonstrated below. Even from a total of 60$,
with some overlap, the sample would have been rejected based on
sequential analysis as eleven errors were found, yet only $6.63
of errors were actually found.

It was concluded, after much cogitaticn, that since the
errors and (p) percentages were low the standard of comparison
should be high. Thus a 99$ confidence was set for this experi-
ment. However, the confidence limit could easily have been set
before the experiment or the audit engagement as well. The con-
fidence percentage will vary only for areas of work and not
necessarily from company to company. This overcomes a tremendous
obstacle inherent in standard sample sixes which would of necess-
ity vary from concern to concern and then not prove conclusive
because of poor results necessitating additional sampling. The
method of this study eliminates such difficulties and in addi-

tion places the emphasis on the major factor, namely, the degree

 

See a

- 3g -
of confidence that may be expressed.

The following Table 7 gives the upper and lower limits
for the various stratified samples based on dollar errors.
Additional tables will be found in the Appendix 0, D, and E
giving more detailed information on the results shown in
Table 7. It should be noted that in the large samples, the
upper limits are still less than 1%. If we should desire to
be within 11» of accuracy, these samples show the work to be
well within our standard.

If further proof is needed, the null hypothesis theory
could be used, for say, the first sample. (52)

Where n m 2201.32 (the dollars)

Standard m P' a .01

Actual (p) e .0018

Standard Error 81/10]. x a? m .002

Difference between iotuslzand standard a .0082

008 s I$.10
Prom which it could be concluded that .0018 could not be

 

.01. In order for the study to be incorrect, the results must
be less than one.

Referring to Table 7, these values for upper and lower
limits may be found in published tables, a part of which is
included in Appendix B. Comparing this table with the results
of this study, the first sample is within .0002 of being .002.

(52) Quinn Holemar, PSICSCECCICIE CTITISTICS, 1st. ed.

John liley 1 Sons, Inc. law York, P. 63 (19‘19)

 

 

- 3g -
of confidence that may be expressed.

The following Table 7 gives the upper and lower limits
for the various stratified samples based on dollar errors.
Additional tables will be found in the Appendix C, D, and E
giving more detailed information on the results shown in
Table 7. It should be noted that in the large samples, the
upper limits are still less than 1%. If we should desire to
be within 11: of accuracy, these samples show the work to be
well within our standard.

If further proof is needed, the null hypothesis theory
could be used, for say, the first sample. (52)

Vhere n 3 2201.32 (the dollars)

Standard a P' a .01

lethal (P) e .0018

Standard Error ...-Vim x —‘m .002

Difference between actuhl and standard a: .0082

.0082 a 4.10

Prom which it could be concluded that .0018 could not be

 

.01. In order for the study to be incorrect, the results must

be less than one.

Referring to Table 7, these values for upper and lower
limits may be found in published tables, a part of which is

included in Appendix B. Comparing this table with the results

of this study, the first sample is within .0002 of being .002.

 

 

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- ho -
Using the sample else as dollars (2201.32) and using the .21.
column.the figure .0050 is found under sample also 2000 and
.00h4 is found under sample sise 3000. Since the 2201.32 is
the correct value, it would lie between .0050 and..00h4, and
also, since the percentage is .0002 less than .002 and.the
value for .1$ is between..003l and .0027 the figure of the
study of .0035 is very close.

This, then, is what.Abrams would have found, no doubt,
‘had.he developed his hypothesis. (53)

Thus a.simple procedure has been developed whereby the
accountant by relating his errors found in dollars to a perb
ccntage and.using an (n) of the total dollars of the sample
may find a factor which will give him the information necess-
ary to express or disclaim the client's work. lots that with
a l$ standard only, samples B and r would.have been rejected
which may be the chance fluctuation noted above cr'more likely
it may be that 10$ or less (in dollars) is too small a sample.
Assume for purposes of explanation that they are correct,
attention is called to the fact that the percentage of errors
is less than one half of one percent yet they are not within
our standards. In fact, the possible error is approximately
three times the error uncovered. Thus, just because the auditor
finds less than l$lerror is not necessarily sufficient evidence
to conclude that the possible error is less than l$. This method

 

 

erome'lbrams, 'SampIing Theory AppIiéd to the Test~

Audit", The He York Cert fiod Public Account t
V01. XVII Ho. IC F. 6153652 (Cot. I957)

- h1 -
then gives the auditor an objective measure of his reliability
based.upon.the facts he uncovers in.the audit. That embarrass—
ment the auditor*might experience in thinking that errors total-
ing 05000 of a.million.dollar sales (5 of l$) is all the error
whereas actually it may be $15,000. Particularly does this
become significant when.the standard is lower, such as 95$ or
90$ allowing 5 or 10$ error. As these errors are tripled, a
judge or jury may well ask what evidence did the auditor
possess to formulate his opinion or what confidence can.the
auditor justify.

The question arises why does this highly skewed.curve
(Poisson or Bernoullian) work for auditors. This study would
indicate that errors are normally low in.the average well run
business with an adequate system of internal control. In addi-
tion, this study showed.that few errors are made on large items
but occasional errors are made on small items. See Tables
5 and 6.

lo errors were found in the classes representing 90$ of
the total dollar value. Since therefore, the errors discover»
ed in this study were small, the distribution of the errors
is a.highly skewed.curve. This finding completely refutes
Vancs's contention that no pattern exists. (55)

Although the method.of this study is limited to errors
measurable in dollars, it has been proved as sensitive or more

sensitive than the sequential sampling method. Therefore, the

""—T557"Liireno3%ET'VEECC:_55_UIT_PT'I2*

 

.. he ..
statement by Vance noted above is inconsistent with the find-
ing of this study and a measure of mteriality for the pur—
poses cf auditing has been developed.

COHCLUSIOH

In conclusion, it would appear that Iontgomery's defini—
tion of testing in the auditing sense is more nearly correct
but that his bases for testing are not in agreement with
his definition but should include in all his bases for repre-
sentative samples the words "randomly selected. " (55)

The first limitation relating to two or more procedures
cited by Cranstoun on page 21 is generally valid as of today.
This study overcame such difficulty by defining the experi-
mental unit in terms of a specific job related to the validity
of original evidence of sales invoices. Yet this study gives
no evidence that the definition of error could not be expanded.

His second limitation regarding false entries seems too
narrow. Sequential sampling and.the technique of this study
have broadened the scope of statistical auditing much more than
merely fraud detection. In fact, these methods are the least
advisable for fraud.whioh, however, does not impair their use-
fulness since fraud detection today is not the most important
reason for an.audit, ranking eighth in a list of 12 reasons
for certified statements as given by Holmes. (56)

 

bert . Hontgcmery, I I)!
6th ed., Ronald Press, has York, P. 36 (19110)

(56) Arthur I. Holmes, AUDITING Principles and Procedure,
2nd ed. rev. Richard D. Irwin, Chicago, P. 3 (19117)

-..

.. Ni -

The third limitation concerning materiality cited by
Oranstoun is valid for the sequential type of analysis but
it is invalid with reference to the method of this study.

The reason is that dollars ( a common denominator) and not
the error itself can measure materiality as well as imnater-
iality. Therefore, a distinction is drawn because an error
of 8500 in a sample of a 81000 is 50% which changes the (p)
in the formula of this investigation. Although in sequential
analysis it is only one error. Thus the method presented in
this paper gives the auditor a procedure whereby his judgment
of materiality or imaateriality can be attested.

Rather than determining standard sample since as Oranstcun
has suggested, it would seem more logical to set standard pro.
eisicn estimates (probabilities). (57) That is to say, does
the auditor wish to be correct within 1“? hr in the final
analysis the auditor is interested in the confidence that he
may orpress in his report. To follow standard sample sizes
seems to place the accent on the wrong syllable. Thus, the
method used in this study results in standards rhich have more
universal application than sample sises. It is more signifi-
cant tc the profession to set a standard of 99$ confidence for
sales regardless of the sise of the business than to work out
sample sises for varying sise businesses.

Another allegation this study refutes is the notion of

[57} 53s F. 7 c? this paper

 

 

-ug-

Vanee's that a standard population is necessary. (58) The
method of this study does not require a standard populap
tion sines such criteria are unnecessary in studying attri-
butes as mentioned in the chapter on statistical method.

finally, although this paper has limited its discussion
to the auditing of sales, the method is applicable to many
areas of auditing such as vouchers, accounts receivable,
inventory and postings. Thus, wherever testing is used to
determine the reliance upon the system of internal control,
other than out offs, the method of this study may be eerb
icusly considered as the vehicle of accomplishment.

 

(53} 53s F. 17 of‘this study.

APPENDIX A
A OAS! STUDY USIIG SEQUENTIAL ANALYSIS I] AUDITIHG"

The material of this case is the ran material
and merchandise inventory of a small manufacturer
of surgical appliances. The inventory consisted of
1,060 items, and the clerical work is the area tested.
Extensive examination of this inventory in the actual
audit, plus thorough scrutiny by the senior partner of
the firm (who did not participate in its preparation)
revealed only 5 errors. The examination of the cleri—
cal work was not complete, although 100 per cent of
footings and 25 per cent of extensions were checked
by the auditors. to may assume, however, that all
errors were found for the purpose of trying statis-
tical techniques. ‘Upon such an assumption, three inde-
pendent random samples were drawn. In the first two
samples no errors were found in the first 89 items
drawn, so that the population would have been accepted
with a.minimum sample if we use the values of Table 10*
(left section) as a basis for judgment. In the third
sample an error was found when item #5 was drawn; this
indicates an indeterminate result. A continuation of
the drawing was made and after 160 items were included
no other error had been found, so the population would
hays been accepted in the third case also without as
much effort as was expended by the auditors in practice.
In the event the reader is concerned about our failure
to run down.the 5 errors in this inyentcry, it should
be observed.at this point that where any error is conp
sidered likely to be very important and its discovery
essential, the sampling method is not appropriate,
whether applied scientifically or not. Such an atti-
tude requires complete examination of the area involved.

 

*Lawrence L. Vance, SCIENTIFIC IITHOD TOR.IUDITIHC
'University of California Press, Berkeley and Les
lngelos P. 51, (1950)

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summit or ran coma-non or (p)

 

Ancunt lrrcrs Amount divided
Couple (in dollars) (in dollars) by errors
‘ a! n rcent

I t “97.76 $6.63 .1t7$

1 2201.32 1.00 .18

2 1356.36 .98 .072

3 shows 1.65 .176

A 1396.93 0 0

s 336.39 Loo .tc

0 $0.33 .93 .115

D 715.h7 .05 .007

r steals .20 .037
.1 12M: 4&5 Jar

 

 

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- by -
APPENDIX D
coma! or oomunor or 01?an Am) 1.0er mar ron 20$
was up row. or m: mm was

5.3212 £229.]; 12212.3 We! 0 LP...“ 10
The decimal of sample

errors to sample (p) .0018 .00072 .00176 .0014?
The reciprocal (q) .9982 .9992! .99sah .9985}
Sigma ( cf ) .0009 .00073 .001! .0005?
Three sigma (3 o” ) .0027 .00231 .001»: .00171
Upper Limit (p + 3 o’ ) .0016 .00303 .00596 .00318

Lower Limit (p - 30’) .0000 .00000 .00000 .00000'

 

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BI SLI OGRAPSY

Bgckg

Arkin, Herbert and R. n. Colton, TABLES TOR STATISTICAIS,
Barnes 8: loble, Inc., low York (1950)

Byrnes, Thomas I. and I. L. Baker, 0. A. Smith, AUDITING
with Practice Problems, 1st ed., Ronald Press, New
York (1998)

Craigie, Sir Villiam A. and James a. Hulbert, A DICTION-
BY 01' “ERICA! “01.133 '01. IV, Univ. of Chicago Press
Chicago (1910; '

Punk, Isaac I. (Edited by) m STAIDARD DICTIONARY OF THE
ENGLISH LAIGUAGE, Funk 8: lagnalls C0., In York (19113)

Holmes, Arthur I. AUDITING PRINCIPLES AND PROCEDURE 2nd ed.
rsv., Richard D. Irwin, Chicago (19h?)

lohler, I. L., AUDITIIC an Introduction to the Iork cf the
Public Accountant, 1st. ed., Prentice Hall, low York

mm

lclemar, Quinn, Percaonocxcu. entrance 1st. ed. John
Wiley .1. Sons, Inc. lew York (19%)

hills, Frederick 0., STATISTICAL anDS Applied to Icono-
mics and Business 2nd ed. rev. Henry Bolt 1 00. new
York (1938)

Montgomery, Robert H. AUDITING THEORY AND PRACTICE, 6th Cd.
Ronald Press, low York (19%)

Staub, Walter A., AUDITIIG DWILOPIENTS DUBIBG THE PRESENT
Crazy, 1st. ed. Harvard Univ. Press, Cambridge, Mass.
1

Vance, Lawrence 1.. SCIENTIFIC “THCD TOR AUDITING let. ed.
Univ. of Calif. Press, Berkeley and Los Angeles (1950)

Yule, C. Udny and I. C. Kendall, An Introduction to the
THBORY or STATISTICS, 11th ed. Charles Griffin d 00.
London (1937)

- 52 -
BIBLIOGRAPHY - Contimed

W

Abrams, Jerome, "Sampling Theory Applied to the Test-Audit'
THE m YORK CERTIFIED PUBLIC ACCOUNTART Vol. XVII,
lo. 10 (Oct. 191$?)
American Institute of Accountants
Case Studies in Auditing Procedure - 'A Herspaper
Publisher"
Case Studies in Auditing Procedure I'A Public Utility"
Case Studies in Auditing Procedure - 'A Leading and
Hauling Equipment Hanufacturer'
Case Studies in Auditing Procedure - 'A Grain Company“
Case Studies in Auditing Procedure - I'A Steel Fabri-
cating Company“
Case Studies in Auditing Procedure - 'A Department Store'
Case Studies in Internal Control - I'The Hachine Hanufac-
turing Company“
I'thensions of Auditing Procedure' (Oct. 18, 1939)

"Statements on Auditing Procedure“ lo. 1 (Oct. 19 9)
'Tontative Statement of Auditing Standards“ (191$?

Carmen, Lewis A. "The Efficacy of Tests.” THE AHERICAR
ACCOUITAHT', Vol. XVIII (Dec. 1933)

Cranstcun, William D. "A low Look at Basic Auditing Techni-
qubs', rat JOURHAL or ACCOUHTAHCY V01. s6zh (Oct. 1968)

Herbert, Leo 'Practicsl Sampling for Auditors“ THE HEW YORK
CERTIFIED PUBLIC ACOOUHTAHT Vol..IVII n01 (Jan. 1997)

leter, John 'An Investi tion of the Usefulness of Statis-
tical Sampling Hath in Auditing" THE JOURNAL OF
Accounrinor, v01. 87, lo. 5 (Hay 19t9)

IIThe Application of Statistical Techniques in Auditing
Procedures' THE m YORK CERTIFIED PUBLIC ACOOUII'I'AJIT'I
Vol. III Ho. 6 (June 1949)

Prytherch, Robert H. “How Huch Test Checking is Incu ?"
THE JOURIAL or ACCOUHTANCY Vol. 7t lo. 6 (Dec. 19 2)

Vance, Lawrence L. I'Auditing Uses of Probabilities in Select-
ing and Interpreting Test Checks" THE JOURIIAL OF ACCOUHT-
AHOY, Vol. as, re. 3 (Sept. 19t9)

INTER-1188A” I
. LOAN
A ""
T ’ us& mu.
22 Aug J’ i

W5

 

 

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