Audit Sampling

APT Financial Consultants

AUDIT SAMPLING

1. Guiding standard ISA 530, Audit Sampling and Other Selective Testing procedures.
2. Audit sampling can be defined as an application of audit procedures to less than 100% of items within an
account balance or class of transactions such that all sampling units have a chance of selection. This will enable
the auditor to obtain and evaluate audit evidence about some characteristic of the items selected in order to form
or assist in forming a conclusion concerning the population from which the sample is drawn. Audit sampling
can use either a statistical or a non-statistical approach.

3. Formalized audit sampling procedures have been developed and become commonplace in the
majority of audit firms. The use of audit sampling, on all audit assignments, offers
innumerable benefits to all auditors. These include:

• developing a consistent approach to audit areas;
• providing a framework within which sufficient audit evidence is obtained;
• forcing clarification of audit thinking in determining how the audit objectives will be
met;
• minimising the risk of over-auditing; and
• facilitating more expeditious review of working papers.

3.0 Sampling risk is the risk that the sample is not representative of the population from which it
is drawn and thus the auditor’s conclusion is different to that which would be reached if the
whole population was examined.
This may result in:
(a) ‘The risk of incorrect rejection’ (also called Alpha risk) which arises when the sample
indicates a higher level of errors than is actually the case. This situation is usually
resolved by additional audit work being performed. This risk affects audit efficiency but
should not affect the validity of the resulting audit conclusion;

(b) ‘The risk of incorrect acceptance’ (also called Beta risk) when material error is not
detected in a population because the sample failed to select sufficient items containing
errors. This risk, which affects audit effectiveness, can be quantified using statistical
sampling techniques. Although it is possible that an unqualified auditors’ report could be
issued inappropriately, such errors should be detected by other complementary audit
procedures (assuming that the sample size is appropriate to the level of detection risk).

4.0 ISA 530 defines non-sampling risk as the risk, which “arises from factors that cause the
auditor to reach an erroneous conclusion for any reason not related to the size of the sample”.
Thus non-sampling risk can also arise, for example, if the auditor fails to recognize an error
in an individual item in a sample. The auditor seeks to minimize the risk of erroneous
conclusions by proper planning, supervision and review.
Examples of sources of non-sampling risk include:

• Failure to investigate significant fluctuations in relationships when placing
reliance on analytical procedures; and

Sako Mayrick, 2008 Page 1


• Placing reliance on management representations as a substitute for other audit
evidence that could reasonably be expected to be available.

5.0 When planning the audit procedures to be adopted, the decision to sample account
balances and transactions is influenced by:

• materiality and the number of items in the population;
• inherent risk (of errors arising);
• relevance and reliability of evidence available through non-sampling procedures; and
• costs and time involved.

To obtain the overall level of assurance required, a cost-effective combination of
sampling and non-sampling procedures should be determined. Audit sampling
procedures are effected in four stages:
1.Sample design;
2.Sample selection;
3. testing (i.e., performing the audit procedure); and
4.evaluation.
Sample design
Sample design, which may be set out in a sample plan, includes consideration of:

• Audit objective(s) of the test;
• Population from which the sample is to be drawn;
• Sampling unit;
• Results or conditions that will be regarded as errors or deviations;
• Sample size.

In normal sampling techniques the following steps are followed.

(a) Identification of the population
(b) Definition of an error
(c) Computation the appropriate sample size
(d) Evaluation of the results of sample.

Reasons for sampling:

Cost benefit of 100% and sample.
Size of the firm’s transactions
Produce high level of assurance in respect of completeness.

Definition of key terms: -
Population – is the entire set of data from which the auditors wish to sample in order to reach a
conclusion.



Audit Sampling – Is application of audit procedures (e.g.. Compliance or substantive tests) to
less than 100% of the items within an account balance or class of transactions to enable auditors
to obtain and evaluate evidence about some characteristics of the items. Selected in order to
form or assist in forming a conclusion concerning the population, which makes up that account
balance or class of transaction.
Sampling units – Are individual items that make up the population.
Error – Is an unintentional mistake in the financial statements.
Precision – Is the measure of how much the conclusion drawn by the auditor from the results of
testing a particular characteristic of a sample of items difference from known population
characteristics at a given level of sampling risk.
Tolerable error – is the maximum error in the population that auditors are willing to accept and
still conclude that the audit objective have been achieved.
Sampling risk – Is the risk that the auditors conclusion, based on a sample, may be different
from the conclusion that would be reached of the entire population was subject to the same audit
procedure: -
Non- Sampling risk – Is the risk that the auditors might use inappropriate procedures or might
mis interpret evidence and thus fail to recognize an error.

Stratification—The process of dividing a population into subpopulations, each of which is a
group of sampling units which have similar characteristics (often monetary value).

Sampling Decision tree

Judgmental - Proportion- Estimation
(Non-statistical) Sampling for Discovery
Attributes sampling
Acceptance sampling
Sampling

Random Estimation sampling of
(Statistical) - Value -Variables
Monetary unit sampling.

Characteristics of acceptable audit sample:

Sufficient large population (e.g. more than 100).
Anticipated error rate must be reasonably low.
Population should be representative of all major transactions.
Choice of sample population should be directly related to the test objective.
The population tested should be reasonably homogeneous both in the type of balance
and origin. E.g. Low-value and high value transactions.
The population should be representative of the time period under review. E.g. not
only for first 3 months.
Every item in the population should stand an equal chance of selection.



Some testing procedures do not involve sampling example:

Transaction and balances which, though few in number are of great significance in
terms of size: e.g. land and building and extra ordinary/exceptional items.
Non-homogeneous population where sorting will have to take place before sampling
can be attempted.
Small population where statistical, theory will create unacceptable margins of error.
Testing 100% of items in a population.
analytical procedures;
Tests in total (also called proofs in total or logic tests) i.e., calculations of
reasonableness based on independently verified data;
‘Walk-through’ tests, i.e., tracing a few transactions in order to obtain knowledge
and understanding of the design and operation of accounting and internal control
systems; and
Other selective testing of specific items, e.g., high-value, key and unusual (but not
representative) items.

Sample Selection Methods
The principal methods of selecting samples are as follows:
(a) Use of a computerized random number generator (through CAATs) or random number tables.
(b) Systematic selection, in which the number of sampling units in the population is divided by
the sample size to give a sampling interval, for example 50, and having determined a starting
point within the first 50, each 50th sampling unit thereafter is selected. Although the starting
point may be determined haphazardly, the sample is more likely to be truly random if it is
determined by use of a computerized random number generator or random number tables. When
using systematic selection, the auditor would need to determine that sampling units within the
population are not structured in such a way that the sampling interval corresponds with a
particular pattern in the population.
(c) Haphazard selection, in which the auditor selects the sample without following a structured
technique. Although no structured technique is used, the auditor would nonetheless avoid any
conscious bias or predictability (for example, avoiding difficult to locate items, or always
choosing or avoiding the first or last entries on a page) and thus attempt to ensure that all items
in the population have a chance of selection. Haphazard selection is not appropriate when using
statistical sampling.
(d) Block selection involves selecting a block(s) of contiguous items from within the population.
Block selection cannot ordinarily be used in audit sampling because most populations are
structured such that items in a sequence can be expected to have similar characteristics to each
other, but different characteristics from items elsewhere in the population. Although in some
circumstances it may be an appropriate audit procedure to examine a block of items, it would
rarely be an appropriate sample selection technique when the auditor intends to draw valid
inferences about the entire population based on the sample.



Approaches to sampling:

(a) Statistical sampling – an approach to sampling which requires the use of random
selection and uses probability theory to determine the sample size, to evaluate on
quantitative basis the sample results and to measure the sampling risk.

(b) Non statistical sampling (Judgement sampling): is any approach which does not fulfil all
the conditions set out alongside for statistical sampling.

The main stages in the sampling process are:

(i) Determining objectives and population
(ii) Determining sample size
(iii) Choosing method of sample selection
(iv) Analysing the results and projecting errors.

Main approaches to audit sampling are

(i) Attribute sampling (numerical) – is a aimed at detecting what proportion of a
population either has or lacks, a specific characteristic or attribute.
.
(ii) Variable sampling – is aimed at ascertaining (monetary) The monetary
value/extent of an error, be it over or understatement.
The attribute sampling is more appropriate to compliance testing and variable sampling
is better invited to substantive testing.

Attribute (numerical) sampling:
Each error or deviation from a prescribed control procedures in treated (or weighed) equally
despite the fact the monetary values of the errors may be very different.
The results of an attribute sample in compliance content is normally expressed in terms of
“confidence level” or “precision”

The determination of sample size requires judgement of:

• assurance required;
• tolerable and expected error (or deviation rate); and
• stratification.

Absolute assurance cannot be achieved through sampling procedures. The lower the assurance
required, the smaller the required sample size. The tolerable error (or deviation rate) is also
called precision. It is the maximum error (or deviation rate) that can be accepted to conclude that
the audit objective has been achieved. (The combined tolerable error for all audit tests is
sometimes called gauge.)
For substantive tests, precision may be expressed as a monetary amount (which is less than
overall materiality) or a percentage of population value. For tests of control, precision is the



maximum rate of failure of an internal control that can be accepted in order to place reliance on it
(and is therefore likely to be small).
Errors increase the imprecision of results from sampling. Therefore, if they are expected, a larger
sample size is required.
In attribute sampling:
Sample size = Reliability factor (R-factor)
Precision.
Sampling risk is frequently expressed as a %. For example, 5% means that there is a 1 in 20
chance of material error going undetected (this is the risk accepted by many audit firms for any
specific audit tests). Risk can also be expressed in terms of confidence levels (assurance
required) and reliability factors.
A confidence level is the degree of assurance that material error does not exist; it is the converse
of risk.
Reliability (R-) factors are derived from the Poisson sampling distribution (a distribution of ‘rare
events’) and are related to risk percentages as shown in Figure 1. Note the ‘inverse’ nature of the
relationship between R-factors and risk and that a confidence level is the mathematical
complement of risk.

The R – factor is taken from statistical tables such as this one: -

Confidence
Risk R- factors R-factor one
Level/assurance
level No. errors error
required
99% 1% 4.6 6.61
95% 5% 3.0 4.75
90% 10% 2.3 3.89
85% 15% 1.9 3.38
80% 20% 1.6 3.0
70% 30% 1.2 2.44

The reliability factor is related to the amount of assurance the auditors wishes to draw from the
test.
Example:



Example 2

QUESTION 1:
Calculate the sample size, which should be, used in test control in the following circumstances: -

(a) No. Errors is anticipated in the sample; accept 5% risk that four or more items in 100 are
incorrect in the population.

(b) One error anticipated in the sample; accept 1-% risk that three or more items in every 100
are incorrect in the population solution.
(See class notes for answers)

An auditor is planning compliance tests and describes that he will accept the particular
population as correct as long as he runs no more than 5% risk that 2 or more items, in every 100
are incorrect. The degree of precision required is +-2%.
(i) If no errors are anticipated the sample
size would be;
(ii) If one error is anticipated the sample size rises to.



(iii) If in first example the auditor were prepared to accept higher error rate (say 4 items per
100) but no more) the sample size would fall:-
( See Class notes for this)

In practice some accounting firm find it preferable to offer their audit staff a set of approach to
statistical sampling for attributes; this minimizes both cost and the risk of variation between
auditors and audit clients.
The firms are confident that the samples size they recommended are representative of average
sample sizes which would be obtained by using the above attribute sample size formula.
The higher the sample size the higher the degree of reliance,

Other forms of attribute sampling.

Acceptance sampling: -
The number of errors found in the sample will determine whether or not a population is accepted
or rejected. E.g.: if error rate X% or less at a given confidence level the auditor may decide to
accept the entire population. If the error rate in the sample rises above X%
Then the population may be rejected as unreliable.

Discovery sampling:-
Called exploratory sampling is a secreening device for punching out populations which require
further investigation”. The method involves lying to discover of errors are occurring at over an
acceptable rate. Thus the auditor would sample until such an error was found.
Precision (materiality is determined by the auditor prior to the Commencement of sampling. R-
factors are arrived at as a result of compliance testing procedures. The value of population is
given in draft financial statements which the client prepares.

Variable Sampling and Monetary Unit Sampling ( MUS)
Variable sampling have difficulty in dealing with understatements of account balances.Why?
They can only test what is actually there – tests for understatements are generally carried out at
the compliance or attribute sampling stage.

Invoices which not exists cannot be tested at substantive stage and thus a balances which are
materially understated will stand a lower chance of being selected for audit testing at the
substantive testing stage.
MUS gives a conclusion based on monetary amounts. Not rates of occurrence. It does this by
defining each Shs.1 of the population as a sampling unit of Shs.1. and an individual balance of
Tshs.300 is 300 sampling units of Shs.300.
From attribute sampling scheme:

Sample size = Reliability R- factor
Precision

The MUS sample size formula is slightly different: rather than stating precision as a rate of
occurrence; it is restated in monetary.



Substantive procedures: variable sampling,

Variable sampling is concerned with sampling units which can take a value within a continuous
range of possible values and is used to provide conclusions as to the monetary value of a
population.

The auditors can use it to.
(a) Estimate the value of a population by extrapolating statistically the value of a
representative sample items drawn from population.
(b) Determine the accuracy of a population that has already been ascribed a value
(generally described as “hypothesis” testing)
-The variable (Monetary) sampling is most suited for substantive testing situation.
-The true variable sampling involves the estimation of both the number of units m a
population and the standard deviation of a population a population. The above
might be too involving to construction of pilot (or preliminary) samples.

Monetary units sampling:

It is a technique developed to avoid traditional variables sampling approach and its time –
consuming methodology – there is a no need for the auditor to be aware of either the number of
units in the population or the standard deviation of those units.

The MUS focuses on materiality, the R-factor reliability and the total value stated in the
financial statements.

Terms as follows;
Precision (as in altribute = Tolerable error
Sampling formula above Population value

Therefore:

Sample size = Reliability factor x population value
Tolerable error

Sampling interval = Population value or Tolerable error
Sample size Reliability factor

Example:

Using the information in question 1(a), show the selection of sample items, given that.

(a) Tolerable error = 2,000,000
(b) Population value = Shs.50m.
(c) Random start item 100,000
(d) The first ten ledger balances on the ledger m question are: Shs.250,000, 272,500;



Shs, 751,000; Shs.84, 500; Shs99,000; Shs, 172,000;Shs.982,100;
227,190,Shs.35,900:shs.486,200.

ANSWER:

Sample size = ( See class notes)
Sampling internal = (See class notes)

The sample will be selected as follows:

Ladger Value Cumulative Selected
Balance Value Amount
Shs. Shs. Shs. Shs.
1. 250,000 250,000 100,000
2. 272,500 522,500 -
3. 751,000 1,273,800 760,666
4. 84,500 1,358,000 -
5. 99,000 1,457,000 1,433,320
6. 17,200 1,474,200 -
7. 982,100 2,456,300 2,099,980
8. 2,271,900 4,728,200 2,766,640
9. 35,900 4.764,100 -
10 486,200 5,250,300 4,766,620
Etc. Etc. Etc. Etc.

The sample in the above results will not be as great as 75 balance of item 8 on ledger (and other
balances of similar value will have the same effect0. This demonstrates one advantage of mus in
that all items larger than the sampling internal will be selected. This selection of larger items
gives a weighting, which makes a reduction in sample size acceptable.

Example 2:

An auditor, who is willing to tolerable errors tolling Shs.1, 000,000 in a population value of
Shs.20m. is working to precision of 5%. Substituting this into attribute sampling formula we get.

Sample size = Reliability factor x population value
Tolerable error
To extend the above sample, suppose the auditor has divided, on the basis of a full review of
inherent risk control risk and analytical review risk that the reliability factor should be 3.0. The
sample size will thus be.

Sample size = 3.0 X 20,000,000 = 60 items
1,000,000
Once the sample size has been calculated the auditor proceeds to actual Choice of sample items.



The first step in this process is to compute the sampling interval.

Sampling Interval = Population value = Shs.335, 330
Sample size
Or
= Tolerable error = 10,000,000
Reliability factor 3.0

= 333,330

As noted above, each Shs.1 of population is regarded as a separate sampling unit. Thus auditor
having selected a random starting point in (say) a list of ledger balances needs to select every
333,330rd sampling unit encountered, as the population is cumulatively added from zero at the
random staring point. Obviously every 333,330rd sampling unit is a part of particular debtor
balance. That particular balance is then selected for testing. This “systematic selection of
sample items is demonstrated below.

Sampling Internal 333,330/=.

Ledger balance Value Cumulative Value Random Start
/Item Shs. Shs. Shs.160, 000
Selected Shs.
1. 100,000 100,000 -
2. 185,000 285,000 160,000
3. 200,000 485,000 -
4. 60,000 545,000 493,330
5. 65,000 610,000 -
6. 460,000 1,070,000 826,660
7. 96,000 1,166,000 1,159,990
8. 164,000 1,330,000 -
9. 1,267,500 2,597,500 1,493,320
10. 27,259 288,000 18,266,650
- - - 2,159,980
- - - 2,493,310
- - - 2,826,640
Etc. Etc. Etc. Etc.

In the example of systematic selection given immediately above the auditor will fail to achieve
the desired sample size of….items because of item 9 and any other items of a similar magnitude
which may occur later in cumulative addition process.

“Not only MUS gives higher – value items a greater chance of selection but it will also be noted
that all items whose value in is excess of the sampling internal are guaranteed to be selected. In
fact, such items. Could be selected more than once of two if course they need only be tested
once (on the assumption that high-value items have not been preselected and separately



evaluated). If balances larger than the sampling internal are present, the size of the sample
actually selected will be less than computed.

This shortfall is acceptable but should be reconciled to prove the accuracy of the sample by
reference to the number of items each 10% stratum item was selected.

Thus for example above, assuming that no further items exist which are equal to or more than
twice the sampling internal, the final number of items selected will be 57 (instead of 60) and the
reconciling balance will be three times that tem 9 was registered after initial selection.

Evaluation of MUS results
Where no errors are formed in the sample, then the “Prevision” achieved will be that predicted
in terms of “tolerable errors” by the auditors before the test was carried the conclusion that can
be drawn that the population from which the sample was drawn that the population from which
the sample was drawn was not overstated by more than the monetary precision specified here
(usually materiality).
-This conclusion cannot be drawn when errors are found.
-When an error is in item larger than the sampling interval, the auditors will be assured that the
absolute amount of error in this top “strata of balances is known, because all such items have
been examined in the test.

-When the error is found in an item is smaller than the sampling internal, then it will be
necessary to project the level of error on the rest of the population.

The projection has two aspects
(a) Estimating the probable error in the population
(b) Adjusting the value of each error by a prevision by a prevision gap widening factor to arrive
at the upper error limit. (The purpose of this is to estimate the error that may not have been
found because of the imprecision of the estimation technique).

STEPS:
(1) Calculate the most likely error – sort the errors into over and under statements.
E.g. in above example: suppose that there are three errors of over-statements.

Errors Item %error Sampling Expected
Shs. Value Internal Value of error
In sampling
Internal
Higher value items 13000 754000 - 666660 13000
Lower value items 123600 294800 8% 666660 533330
15000 3000,000 5% 666660 33330
Most likely error 99,660

Where an error is found in lower-value item (value of item & sampling internal), the error is
projected (or tamped) over the sampling internal (by Multiplying sampling internal by % error).



Where the error is higher value item (value item) sampling internal) the error is the exact amount
of error in the sample
(ii) Calculate the upper error limit:
This involves “Precision gap widening.” The errors are ranked in % terms and a precision
gap factor applied to them, based on the risk accepted.
Error No: Prevision gap To error Sampling Internal Precision
Widening factor (tempting gap widening
1 0.15 8% 666660 40000
2 0.55 6% 6666660 22000

The user error limit is calculated as follows:
Shs:
Sampling internal X reliability factor (666660 x 3) 2,000,000
Most likely error 99660
Prevision gap-widening 62000
Upper error limit 2,161,660

The implication of this result is that there is 5% risk that the error in the population will
exceed Shs. 2161,660.
If the upper limit is greater than the tolerable errir; the following guideline is available.
(a) As the client to adjust for any specific errors identified.
(b) Reconsider such aspects of process as risk levels, tolerable error and sample size
greater care should be taken before original audit judgements are revised.
(c) Consider the need for further adjustments of account balances concerned, e.g.
additional bad debt’s provision.
(d) Consider the eventual form of the audit report – is qualification or disclaimer
required.


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