Risk Based Audit Approach
Session 5.1
Statistical Sampling
Session Overview
Sampling is an important technique
since it enables the auditor to select
some transactions out of a large mass of
repetitive data in a manner that he can
draw valid conclusions about the entire
data after a thorough examination of the
selected
transactions.
Statistical
sampling may be used in different
auditing situations. The auditor may
wish to estimate how many departures
have occurred from the prescribed
procedures; or estimate a quantity, e.g.,
the value (amount) of errors in the
population. Based on whether the audit
objective is to determine a qualitative
characteristic or a quantitative estimate
of the population, the sampling is called
an attribute or variable sampling.
Attributes sampling estimates the
proportion of items in a population
having
a
certain
attribute
or
characteristic. In an audit situation,
attribute sampling would estimate the
existence or otherwise of an error.
Attribute sampling would be used when
drawing assurance that prescribed
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procedures are being followed properly.
For example, attribute sampling may be
used to derive assurance that procedures
for classification of vouchers have been
followed properly. Here, the auditor
estimates through attribute sampling the
percentage of error (vouchers that have
been miss-classified) and sets an upper
limit of error that he is willing to accept
and still be assured that the systems are
in place.
Variables sampling estimates a quantity,
e.g., amount of sundry debtors shown in
the balance sheet or the underassessment
in a tax circle. Variables sampling has
certain drawbacks which can be
overcome through monetary unit
sampling, which is an attribute sampling
which provides quantitative results and
is suited to most audit situations.
Learning Objectives
In this session we will understand the
rationale of statistical sampling, Sampling
methods, determining sample size,
selecting a sample and projecting of
results.
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At the end of this session, you will be able
to understand rationale of audit sampling,
Elementary concepts and approach to
statistical sampling
Basic Concepts
AUDIT SAMPLING
Audit sampling is the application of an
audit procedure to less than 100 per cent
of the items within a class of
transactions or an account balance to
enable the auditor to form certain
conclusions about that class or balance
as a whole. An auditor can apply
sampling in carrying out both
compliance procedures (to evaluate the
effectiveness of the internal control
system) and substantive procedures (to
obtain
evidence
regarding
the
completeness, accuracy and validity of
the data).
ELEMENTS OF STATISTICAL
SAMPLING
The theory of statistical sampling is used
in a large number of situations where a
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characteristic of a large mass of data is
to be evaluated. In auditing, the auditor
has to form his opinion about a large
mass of data. Therefore, it is possible to
apply statistical sampling techniques in
auditing. Actually, the application of
statistical sampling in auditing is merely
an improvement over test checking or
judgmental
sampling
which
has
traditionally been an important tool of
auditing.
Elementary Concepts of Statistical
Sampling
The statistical sampling theory basically
states that if a sample is selected at
random from a given population and
examined properly, a valid conclusion
can be drawn about the characteristics of
the entire population. In this context, let
us understand the precise meaning of
terms like 'sample', 'population', 'random
sample', etc., which we shall use
frequently in our discussion on statistical
sampling.
Sample:
A sample is the part of
an aggregate selected with a view to
drawing inferences about the aggregate.
Thus, if out of 6,000 debtor accounts, an
auditor selects, say, 349 debtor accounts
(or any number lower than 6,000) for
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obtaining confirmations, it can be said
that he is sampling the debtor accounts.
The debtor accounts selected by the
auditor constitute the sample.
Population: The aggregate or the
totality from which the sample is drawn
is called the population or the universe.
A population or universe may consist of
a definite number of elements (finite
universe) or may contain a limitless
number of elements (infinite universe).
In our example, 6,000 debtor accounts
constitute the population.
Stratification: For sampling to be
effective, population should be more or
less homogenous. In auditing situations
(as in many other cases), population can
seldom be homogenous in all respects.
For example, in a business enterprise,
some of the items of raw materials or
components may cost more than Rs.
10,000 per unit (e.g., a high capacity
hard disk of a computer) whereas others
may cost less than Rs 100 per unit (e.g.,
connecting cables in a computer).
Similarly, the sundry debtors in an
organisation may include some debts,
which are outstanding only for a day or a
week or a month whereas some others
may be outstanding for more than, say,
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three years. To make sampling more
efficient, the total population in such a
situation is divided into several subpopulations (each sub-population is
called 'stratum') each of which is, in
itself, more homogenous in nature, size,
importance or other characteristics than
the population as a whole. A sample is
then selected out of each stratum. This
process is called 'stratification'.
Random Sample: A random sample is
one where the elements constituting the
sample are so selected that all the items
in the population have an equal chance
of selection. There are various ways of
selecting a random sample.
The basic hypotheses of statistical
sampling theory are:
(a)
The population is a homogeneous
group.
(b)
There is no bias in the selection
of items of the sample. All items of the
population have equal chance of being
selected in the sample.
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Approach to Statistical Sampling
Statistical Sampling
Auditing Situations
in
Various
An auditor in carrying out both the
compliance procedures as well as the
substantive procedures can use statistical
sampling, like test checking.
Estimating
the
Qualitative
Characteristics of a Population
As we known that through compliance
procedures, the auditor obtains evidence
about the effectiveness of the operation
of the internal controls. The auditor
recognizes that even where some
deviations from prescribed controls have
taken place, he can place a certain
degree of reliance on internal controls.
If, however, the rate of deviations from
prescribed controls is higher, he has to
reduce the extent of reliance on internal
controls, thus, in carrying out the
compliance procedures, the auditor’s
objective is to ascertain the rate or extent
of deviations from prescribed controls.
The statistical sampling method to be
used for carrying out compliance
procedures has, therefore, to be such as
would enable the auditor to estimate the
rate of deviation from prescribed
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controls in the entire population based
on the results of his sample examination.
A statistical sampling method, which is
commonly used in carrying out
compliance procedures, is ‘attribute
sampling’. (The term ‘attribute’ refers to
a qualitative characteristic. In carrying
out compliance procedures, the auditor is
interested in ascertaining the presence of
a qualitative characteristic, namely,
deviation from a prescribed control.)
Attribute Sampling: Attribute sampling
is a statistical method to estimate the
proportion of items in a population,
which have certain attributes or
characteristics.
Suppose, the auditor wishes to evaluate
the internal control system relating to
preparation of sale invoices. For this
purpose, he would like to estimate the
number of invoices, which have errors.
The method of attributes sampling
would be of great assistance to him. By
taking a small sample, he can draw
reasonable conclusions about the
proportion of invoices in the total
population, which have errors. On the
basis of this estimate, he can reach a
conclusion as to whether the internal
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control system in this regard is effective
or not. We will discuss this method in
detail in session 6.1.
Estimating
the
Quantitative
Characteristics of a Population
We have already seen that substantive
procedures are such audit procedures as
provide evidence to the auditor
regarding the completeness, accuracy
and validity of the data under
examination. In respect of substantive
procedures, the auditor needs a statistical
sampling method, which helps him in
estimating the value (amount or
quantity) of the population. For example,
an auditor may wish to verify whether
the amount of sundry debtors, as shown
in the balance sheet, is correct.
Similarly, the auditor may wish to verify
the correctness of the value of various
categories of inventories shown in the
balance sheet.
The most commonly used methods of
estimating the value of a population are
monetary unit sampling (MUS) and
variables sampling.
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Monetary Unit Sampling: This is also
known as Probability-Proportional-toSize (PPS) sampling. This method gives
proportionately greater weightage to
those items, which have higher monetary
value. This method is particularly
suitable in the case of audit of financial
statements since in such cases, the
concept of material items often
corresponds with the monetary value of
the items. The main feature of monetary
unit sampling is that it treats the total
monetary amount of the balance under
examination as population and each
monetary unit (i.e., rupee) as an element
of the relevant population. For example,
if there are 342 items of inventory,
valued in aggregate at Rs. 16,61,478, the
population size is 16,61,478 and not 342.
Since, in random sampling, each element
of the population has an equal chance of
selection, it is obvious that the
probability of selection of an item is
directly proportional to its monetary
amount. For example, the probability of
selection of an inventory item valued at
Rs. 1,44,434 is 1,44,434/16,61,478.
The monetary unit sampling method is
easy to understand and generally simple
to apply since it does not involve
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complex mathematics. The method also
results in relatively small sample sizes
and is especially useful where the
auditor expects a low level of errors. It
may be noted, however, that monetary
unit sampling cannot be used in
situations where errors are expected to
be large, or where there are zero or
negative balances.
We will discuss this method in detail in
session 7.1.
Variables Sampling: This method of
sampling also gives an estimate of the
value of a population through a sample
and is, therefore, useful in auditing
situations. However, under this method,
the population size is the number of
items constituting a balance and not their
total monetary amount. In the example
of inventories given above, the
population size under variables sampling
would be 342 and not 16,61,478 as was
the case under monetary unit sampling.
Under variables sampling, each item
constituting a population has an equal
chance of selection, irrespective of its
monetary amount. Unlike the monetary
unit sampling method, variables
sampling method can be used in those
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situations also where errors are expected
to be large, or where there are zero or
negative balances. However, this method
is somewhat complex, and the sample
size for a given population is usually
larger than that under the monetary unit
sampling method.
Sampling methods
Although there are large numbers of methods of
selecting the sample, the four important methods, which
are frequently used, are discussed below:
(a) Random selection:
Random selection is a sampling method in which each
and every item has an equal chance of selection. For
example, if 5 items are to be selected out of a
population of 1000, the items selected could be 1,2,3,4
and 5 or 14,15,16,998 & 999. The numbers can be
selected by using random number tables or through the
computers.
(b) Systematic selection:
This is a selection method in which one or two items
are selected randomly, but other items are selected by
adding the average sampling interval to the item
selected randomly. For example, if 5 items are to be
selected and the average sampling interval is 200, the
items selected could be 14, 214, 414, 614 and 814. In
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this the first item (14) is selected randomly and
subsequent items are selected by adding the average
sampling interval (200).
The greatest advantage of this method is that
when it is used in monetary unit sampling, it
automatically ensures that all items greater
than the average sampling interval (e.g.
planned precision) are selected. However, this
method cannot be used when some fixed
numbers are assigned to various categories of
transactions, which make up the accounts, as
either all items of a particular category may be
selected or ignored totally. For example,
assume that voyages from destination ’X’ are
given numbers ending with digits 21-30 and
the average sampling interval is 100 and the
first item selected is 24. Then the items
selected would be 24,124,224,324 etc., In this
case all the voyages selected for audit will be
voyages originating from destination ‘X’ and
the sample would be biased.
(c) Cell selection:
In this method, the population is divided into a
number of cells and one item is selected from
each cell randomly. For example, if the
average sampling interval is 200, sample size
is 5 and the total population is 1000, the
population would be broken into cells of 1-
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200, 201-400, 401-600, 601-800 and 8011000 and one item selected from each cell.
This method overcomes the drawback of
systematic sampling when fixed numbers are
given to various categories, but retains the
advantage of systematic sampling of
automatically selecting all items bigger than
the average sampling interval. The drawback
of the above method is that it is much more
time consuming.
(d) Haphazard:
This method of sample selection is also called as
judgmental sample selection, in which items are
selected according to the auditor’s judgement. This
method is more subjective and susceptible to bias
compared to other methods.
There are various tools available for
selecting samples. The most commonly
used are the random number tables and
the computers. Random number tables
are very useful in case of simple random
selection and systematic selection. In
case of cell selection, computers can be
used both for segregating the population
and for selecting items from each cell.
Advanced auditing software like IDEA.5
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is of immense use in performing
complex sample selection.
more than this estimate. The difference
between the error in sample estimate and
the error in the actual population is the
precision level. The auditor has to
decide the precision he desires to
provide in his estimates. Tolerable error
being the maximum error that the auditor
is willing to accept (sample estimate +
precision level)
Designing a sample
Once the method of sampling is decided,
it is essential to design the actual sample.
The basic stages that are involved in
attributes sampling are mentioned
below:
(a)
size

The confidence level or the level of
assurance that audit needs to provide is
to be defined. When a risk assessment
has preceded the sampling process, the
confidence level would be (1-detection
risk). Confidence level states how
certain the auditor is, that the actual
population measure is within the sample
estimate and its associated precision
level.

The occurrence rate or population
proportion is the proportion of items in
the
population
having
the
error/exception that audit wishes to test.

The required sample size can be
calculated using the formula (Handout
4.2-A), or read off from standard
statistical tables (Handout 4.2-B) at the
required confidence level.
Determining the sample
(b)
Selecting the sample and
performing substantive audit tests on the
sample
(c)
Projecting the results
Determining the sample size:

The first step is to define clearly the
target population and the error/exception
(attribute) that audit wishes to test.

The tolerable error or the maximum
errors that the auditor is willing to accept
and still conclude that the auditee is
following the procedures properly.
Audit test on the sample will throw up
an estimate of error for the population.
The true error of the population could be
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The sample size would be larger, higher
the confidence level and precision
required. Also if the occurrence rate in
the population becomes larger the size of
the sample would increase. In case of
variables sampling, where the estimate
of a quantity is required, sample size
becomes a function of the standard
deviation in the population rather than
the occurrence rate.
Selecting the sample and performing
substantive audit tests on the sample
There are a large number of methods of
sample selection. The most frequently
used method is random selection where
each item in the population has an equal
chance of selection. This could be done
by using random number tables or
through computers. In a systematic
selection, one or two items are selected
randomly, but the other items are
selected by adding the average sampling
interval. The greatest advantage of this
method is that when it is used in
monetary unit sampling, it automatically
ensures that all items greater than the
average sampling interval are selected.
However, this method cannot be used
when some fixed numbers are assigned
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to various categories of transactions,
which make up the accounts, as either all
items of a particular category will be
selected or ignored completely. In the
cell sampling method, the population is
divided into a number of cells and one
item is selected from each cell randomly.
This method overcomes the drawback of
systematic sampling when fixed
numbers are given to various categories,
but retains the advantage of systematic
sampling of automatically selecting
items bigger than the average sampling
interval.
Projecting the results
Once the audit tests are performed on the
sample, the test results need to be
projected to the population. Following
this, a conclusion has to be reached
whether the auditor can place an
assurance on the systems.
After the audit tests, the auditor obtains
the actual number of errors in the sample
selected. As the sample size and the
confidence level desired by the auditor
are known elements, the formula given
at Handout 5.2-A, can be used to solve
for the precision. The maximum error
estimate of the population would then be
obtained after loading the sample
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estimate with the precision. This is the
computed tolerable error. Instead of
solving the mathematical formula, it is
possible to read off the 'computed
tolerable error' straightaway from the
statistical tables for the desired
confidence (assurance levels). .
e.
In a case when the computed tolerable
error is less than the tolerable error, the
auditor can place the desired assurance
on the systems. When the computed
tolerable error is higher than the
tolerable error, the auditor cannot derive
assurance from the systems. The auditor
may, in such situations reduce the
assurance he derives from the control
and increase the assurance required from
substantive tests.
h.
Advantages of statistical Sampling Techniques:
a.
b.
c.
d.
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The sample result is objective and
defensible.
Objective evaluation of a test result is
possible.
The method provides a means of advance
estimation of sample size on an objective
basis.
The method provides an estimate of
sampling error.
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f.
g.
It may give more accurate results than a
100% examination for large population size
as failure to detect errors tend to increase if
the population size is very large.
It saves time and money.
It may be combined and evaluated even
though accomplished by different auditors.
It helps in bringing the observations in for
sharp focus as these can be analyzed for
each type of auditee.
Possible areas of application in
IA&AD
(i ) Audit of vouchers in CAP/CASS sections.
Results obtained through statistical sampling can be
used to qualify the accounts. In the present system if a
sample of 100 vouchers is studied from a given
population and mistakes/irregularities are observed in
10 vouchers we give our observations to the auditee and
if considered necessary get these corrected and certify
the accounts. No attention is paid to the fact there exists
a definite probability of presence of similar
mistakes/irregularities in the remaining part of the
population, which has not been subjected to audit. The
good thing is that this probability can be estimated and
a qualification made if the probability is more that a
predetermined minimum. By using this technique such
qualification can also be made DDO/department wisethus findings can be more focused and
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recommendations can be specific and direct. DDOs or
departments with high probability of occurrence of
certain types of irregularities (as inferred from the
sample) can either be picked up for intensive central
audit or can be selected for local audit. In other cases
suitable feedback can be provided to local audit parties.
(ii.)This technique can be used during local audit also.
It can also help in consolidating audit findings in a
department. A sample may be selected from similar
units and the audit findings can be generalized. For
example, if there are say, 1000 primary schools in a
state and the adequacy of financial systems being
followed in these schools is to be studied, a sample
can be selected studied and findings generalized
based on which specific recommendations can be
made.
iii. This technique can be used very effectively in
reviews. Presently samples are drawn for a review
but no effort is made to generalize the findings, as
we do not stick to the given sample.
iv. In revenue audit, we may use this technique for
determining with a certain degree of confidence the
range (in money terms) of under assessment of a
particular type of tax. Presently we state, “In so
many cases test checked, we found an under
assessment of so much.” This statement can be
made more meaningful with the use of statistical
sampling. We will discuss a case study on use of SS
techniques on CERA, in coming sessions.
This technique can also be used for audits under section
14 where the distribution of grants and irregularities
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notice during audit of a particular body/authority can be
studied for a department.
Summary
Statistical sampling provides a
scientific framework for auditing and
allows us to calculate a minimum
sample size needed to support a given
inference.
Attributes and Variable sampling can
assist an auditor to estimate a
proportion to a given accuracy.
Estimation sampling of variables can
be used to estimate a value. Monetary
unit sampling is best suited to the
external auditor. Sampling can be
applied in various auditing situations
with due consideration of sampling
risk.
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