Statistical Sampling in Audit

1. Introduction:

1.1 Our knowledge our attitudes and our actions are based to a very large extent on observations of few

samples. This is equally true in everyday life, in scientific research and also in audit. A person’s opinion of

an institution that conducts thousands of transactions every day is often determined by the one or two

encounters which he or she has had with the institution in the course of several years. In science and human

affairs alike we lack the resources to study more than a fragment of the phenomena that might advance our

knowledge. Sampling consists of selecting some part of a population to observe so that one may estimate

something about the whole population. For example to estimate the amount of recoverable oil in a region, a

few (highly expensive) sample holes are drilled .The situation is similar in a national opinion survey, in

which only a sample of the people in the population is contacted, and the opinions in the sample population

is used to estimate the proportions with the various opinions in the whole population. To estimate the

prevalence of a rare disease, the sample might consist of a number of medical institutions, each of which

has records of patients treated. Sampling is the science that guides quantitative studies of content, behavior,

performance, materials and causes of differences

1.2 Some obvious questions for such studies are how best to obtain the sample and make the observations and,

once the sample data are in hand, how best to use them to estimate the characteristic of the whole

population. Obtaining the observations involves the question of sample size, how to select the sample, what

observational methods to use, and what measurements to record. These are the issues, which are

scientifically addressed, in statistical sampling.

1.3 In the basic statistical sampling setup, the population consists of a known, finite number of N units - such as

transactions, households, people etc. With each unit a value of variable of interest is associated, may be

referred to as x-value of the unit .The x-value of the unit in the population is viewed as fixed, unknown

quantity. The units in the population are identifiable and may be labeled with numbers 1,2,…..,N. Only a

sample of the units in the population are selected and observed .The data collected consist of the x-value for

each unit in the sample, together with the unit’s label. The procedure by which the sample of units is

selected from the population is called sampling design. The usual inference problem in sampling is to

estimate some summary characteristics of interest of the population, such as the mean or the total of the x-

values after observing only the sample.

1.4 The basic statistical sampling view assumes that the variable of interest is measured on every unit in the

sample is without error, so that the errors in the estimates occur only because just part of the population is

included in the sample. Such errors are referred to as sampling errors. But in real survey situation non-

sampling errors may arise also. It may be due to non-response, measurement error, fatigue, detectability

problems etc.

2. Sampling in Audit.

2.1 In the early stages of the development of independent audit, it was not an uncommon practice for an

auditor to perform a 100% examination of the entries and records of the entities audited. However, as the

economy grew, it quickly became apparent that a 100 % examination of the tremendous volume of entries was

unwarranted and uneconomical. This developed into the test or test check approach, which is both widely

accepted and widely used in audit .It is quite obvious that such a method, involving examination of a portion of

a large quantity of entries in order to draw conclusions about the larger group, is a sampling operation, even

though the word “sample” is not generally used in connection with a test.

2.2 When sampling became a widely accepted tool in audit then another concept called ‘Risk Assessment’

came almost simultaneously, which allowed auditor to focus on risky areas through an objective analysis of

available information about the auditable unit. This is somewhat similar to auditor’s judgment based on

auditor’s experience and skill. The main idea of risk analysis is to identify risky areas in an objective way so

that the auditor can focus on more risky area and optimally use available resources to meet overall audit

objectives. Obviously sampling based only on risk assessment is non-statistical sampling. These non-statistical

samplings have been called by different names in literature as 1) Judgmental sampling, 2) Convenient

sampling, 3) Purposive sampling or, 4) Haphazard sampling.

2.3 Auditors may choose a non-statistical sampling plan–that is, they may want to rely on judgment or

specific knowledge about the population in selecting units for audit. Auditors cannot use the results from the

judgmental (non-statistical) sample to draw conclusions about the population in general. Alternatively, the

specific knowledge (or judgment) about the population units may be used to develop a statistically valid

sampling plan based on which one can draw conclusion about the population. One method could be to use this

knowledge to divide the population into several homogeneous sub-groups called strata, and from each stratum

some units may be sampled using some statistical procedure for audit. Such sampling plan will improve the

audit procedure.

2.4 The users of the audit report expect fairness in the selection procedure and more transparent reporting.

The auditors on the other hand are interested in commenting about the nature of problems in the population from

the audit findings in the sample. The statistical sampling, which provides estimates including the reliability of

the estimates of character of interest, is the scientific solution to these problems. The audit reports based on this

scientific approach are defensible. This enhances the acceptability and effectiveness of audit report.

3. Definition & advantages of statistical sampling.

3.1 The essential features of statistical sampling are:

(i) The sample items should have a known probability of selection- for example, by random selection.

(ii) The sample results should be evaluated mathematically – that is, in accordance with probability

theory.

Just because one of these requirement is met does not mean that the application is statistical. For example,

practitioners and others will sometimes state that they are using statistical sampling solely because a

random number method is employed to select the sample. However this is not statistical sampling as no

attempt has been made to evaluate mathematically sample findings (condition no. 2).

3.2 Statistical sampling allows auditors to calculate sample reliability and risk of reliance on sample. It

permits auditors to optimize the sample size given the mathematically measured risk they are willing to

accept. In this both - over auditing and under auditing can be avoided. It enables auditors to make objective

statements about the sample population on the basis of sample observations. In other words, the sample

finding can be projected to the population.

4. Various Statistical Sampling methods.

4.1 The simplest form of random/statistical sampling consists in selecting the sample unit-by-unit (or, item-by-

item), ensuring equal probability of selection to every unit at each draw. This technique of selection is

termed as Simple Random Sampling (SRS). SRS are of two types:

4.1.1 In Simple Random Sampling With Replacement (SRSWR) a unit is selected from the sampling

frame (list of units in the population); the unit is replaced back and the next unit is selected; the process is

repeated till a sample of the desired size is selected. As a result it is possible for a unit to be included more

than once.

4.1.2 In Simple Random Sampling With Out Replacement (SRSWOR) a unit is selected for inclusion

in the sample, it is removed from the sampling frame and the next unit is selected, therefore, a unit cannot

be selected again.

4.2. In Systematic sampling, the sample is chosen by selecting a random starting point and then picking every

Ith (sampling interval) unit in succession from the sampling frame. The sampling interval is the ratio of

population size to sample size, rounding to the nearest integer. Systematic sampling is less costly and

easier to implement than SRS, because random selection is done only once. Systematic sampling is of two

types (a) Linear Systematic Sampling and (b) Circular Systematic Sampling.

4.3. Stratified sampling is a two-step process in which the population is partitioned into sub-populations, or

strata. The strata should be mutually exclusive and collectively exhaustive in that every population unit

should be assigned to one and only one stratum and no population unit should be omitted. From each

stratum units are selected by any random procedure, usually following SRS. The population units in each

stratum should be as homogeneous as possible. A major objective of stratified sampling is to increase

reliability without increasing cost.

4.4 In Cluster sampling the target population is first divided into mutually exclusive and collectively

exhaustive sub-populations, or clusters. Then a random sample of clusters is selected, based on a

probability sampling technique such as SRS. For each selected cluster, either all the units are included in

the sample or a sample of units is drawn. Units within each cluster should be as heterogeneous as the

population i.e. heterogeneity within the cluster should be the same as that in population, but cluster

themselves should be as homogeneous as possible; each cluster should be small-scale representation of

population.

4.5 Probability Proportional to Size (PPS) sampling assigns higher inclusion probability of selection for

population units with higher sizes (size may be total expenditure, total population etc.). In other words, the

entities with higher sizes, based on some characteristics, will have higher chances of selection. Monetary

Unit Sampling (MUS) in audit is an example of PPS sampling with money value of transactions as size

measure. If repetition is allowed it is called Probability Proportional to Size With Replacement (PPSWR)

Sampling. MUS is actually PPS -Systematic.

4.6 Multi Stage Sampling: Sometimes, as in the case of cluster sampling, it is not possible to draw ultimate

units of interest, as the sampling frame of such units is not available. However a list of some suitable bigger

units or primary stage unit (psu’s ) or first stage units (fsu’s) each comprising several smaller units of

second stage units (ssu’s) may be available from which samples of psu’s may be selected. Instead of

completely testing all the ssu’s from the list of selected psu's some selected ssu's are then studied. This is

called two-stage sampling. If a sample of tertiary units is selected from each selected ssu’s the sampling

plan is called three-stage sampling. Similarly higher order multistage designs are also possible.

5. Estimation (Extrapolation) procedure & Sample size.

5.1 Unstratified Mean Per Unit (MPU): The unstratified MPU is used to project an estimated value of a

sample. After a sample is selected with SRS and a value is determined for each sample items, the sample

mean x

of sample values multiplied by the number of items in the population N to, produce an estimate of

total value of the sample population. Assuming normality the optimum sample size under SRS is

n=

2..

A

NSDZ r ,

Where rZ = confidence level coefficient [Refer table 1 pg. 5], A= margin of error and SD=

N

xnxjN

j

1

2

,

Standard Deviation, N=population Size.

Because MPU without stratification produces large sample sizes relative to other sampling methods, its use

in survey sampling is limited.

Z Score - Table

Confidence Level Z -value

75 % 1.15

80% 1.28

85% 1.44

90% 1.65

92% 1.75

94% 1.88

95% 1.96

96% 2.05

99% 2.58

5.2 Stratified Mean Per Unit: When the population is highly variable (large standard deviation), technically

called heterogeneous population, unstratified MPU may produce very large sample sizes. Stratification of

the population, as explained earlier, produces an estimate that has desired level of reliability with reduced

sample size. Using the following formula the sample sizes for each stratum may be optimally determined

as:

22 .)/(

).().(

iir

iiii

iSDNZA

SDNSDNn

Where, SDi = Standard Deviation of the ith stratum.

The estimated population total (for three strata) is

Total estimated value = N1 * 1x +N2 * 2x +N3 * 3x

5.3 Unstratified Proportion of audit objections (errors): The projected number of audit objections in the

population is the sample proportion of error multiplied by number of items in the population. The optimum

sample size under SRS is n=

2

2.)1.(.

A

PPZ r , Where rZ = confidence level coefficient, A= margin of

error and P= Proportion of errors that is expected in the population.

5.4 Stratified Proportion of audit objections (errors):

For three strata, the projected number of audit objections in the population is equal to

N

pNpNpN 332211 ... ,where pi is the proportion of audit objections in the ith stratum.

& N=N1+ N2+N3.

5.5 Estimation with PPSWR:

The estimate of population total of the character x = X̂ =

n

i i

i

p

x

n 1

1, where pi is the probability of selecting

the ith sample.

The estimate of population mean= Xˆ =

N

X̂

5.6 Estimation with two-stage sampling design:

Case-I: Samples are drawn with SRSWOR in both the stages:

Let yij is the measure of the characteristics of interest for the ith 1st stage unit and jth second stage

unit , where i=1,2,… n and j= 1,2,,…. mi.

Population Estimate =

im

j

ij

i

in

i

ym

M

n

NY

11

Case-II :1st stage sample are drawn with PPSWR and the 2nd stage samples are drawn with SRSWOR :

Let yij is the measure of the characteristics of interest ( e.g. completion of roads in k. m) for the

ith DPIU and jth Package , where i=1,2,… n and j= 1,2,,…. mi. xi is the size measure .X is total of all

size measures.

Population Estimate =

im

j

ij

i

in

i i

ym

M

x

X

nY

11

1

6. Concept of “Testing of Hypothesis” (Test of control / substantive testing in audit):

6.2 Some relevant terminologies:

6.2.1 Test of control sampling –Risk Matrix

Operating effectively Not operating effectively

Accept Correct decision Risk of assessing control risk

too low.(beta)

Reject Risk of assessing control risk

too high (alpha)

Correct decision

–Alpha risk: risk of incorrect rejection –Relates to Audit Efficiency :Auditee’s risk

–Beta risk: risk of incorrect acceptance – Relates to Audit Effectiveness

6.2.2 Audit risk model

OAR = IR x CR x AP x TD

The auditor assesses IR, CR and AP and the auditor chooses a desired level of OAR. Given OAR, IR, CR and

AP, the auditor can use the audit risk model to quantify how much beta risk he/she is prepared to tolerate:

TD = AR/(IR x CR x AP)

•Example:

OAR = 1%, CR = 35%, IR = 75%, AP = 80%

Beta risk = 5% (one tailed)

6.2.3 Tolerable rate:

The maximum deviation rate established that the auditor would be willing to accept. This

is used in compliance test .

.

6.2.4 Materiality:

The value of error that an auditor is willing to accept and still concludes that the audit

objective is achieved. The smaller the materiality, the larger is the sample size. This is used

in substantive testing

6.3Test of control /Compliance testing:

Assume that an auditor wants to test credit approval on 20,000 sales invoices processed during the year.

He or she needs a statistical sample that will give 90% confidence that not more than 5% of the sales invoices

were not approved .The auditor estimates from previous experience that about 1% deviations (are not

approved).

Expected deviation rate = 1%

Tolerable rate = 5%

Confidence level = 90%

For SRSWOR sampling Hypergeometric distribution is appropriate. But statistical tables are not easily

constructed for this distribution. One can use Binomial tables as a close approximation for large sample. If the

expected rate of deviation is very low then one can even use Poisson table for large sample.

From the Binomial table (in Annex –III) it is observed that the required sample size is 77 and if the number of

deviation in 77 samples is 0 or 1, then the auditor can conclude with 90% confidence that the tolerable rate is

not more than 5%, in other words internal controls are reliable. Else, the tolerable rate is more than 5% and

confidence level will also be less than 90% and the internal control is unreliable.

6.4 Audit Hypothesis Model / substantive testing.

The audit hypothesis approach statistically discriminates between the hypothesis that the amount as

represented is correctly stated and the alternative hypothesis that the amount is materially misstated. This is also

known as substantive testing. An essential first step in planning a testing of hypothesis/substantive testing in

audit is to make a preliminary judgment about the amount that will be considered material to the account or

transactions being audited. This amount is called tolerable misstatement . Then the hypothesis to be tested is

that “the value of misstatement in the account balance is equal to the tolerable misstatement” as against the

alternative hypothesis that “the value of misstatement is greater than the tolerable misstatement”.

6.4.1The Audit Hypothesis Model can be categorized into four separate phases.

–Internal Control Assessment

–Substantive test panning

–Substantive test execution

–Substantive test evaluation

6.4.2 Steps in Audit Hypothesis Model : (Let the hypothesis is there is material correctness of the book value)

.Classical approach

Step 1 The internal control assessment is done to assign % to CR and subsequently used in the beta risk

equation.

Step 2. Appropriate variable sampling plan has to be selected based on audit objective and population

characteristics.

Step3 .If SRSWOR is the sampling plan then the sample size (n):

Step 4. Ur is determined based on acceptable alpha risk

Step 5. ‘SD’ may be estimated using a pilot sample of size 30.

Step 6. ‘N’ is the population size

Step 7. ‘A’ must be calculated based on desired or calculated beta risk as:

where ,A=precision, Ur=Alpha risk coefficient , Zβ = Beta risk coefficient (Ur& Zβ is obtained from Normal

Curve Area Table in Annex-IV)and T.M=tolerable misstatement

Step 8. Select the sample by SRSWOR

Step9. Perform a test of samples. The sample mean book value and the population mean book value should not

be substantially different. If so a new sample to be selected discarding the first one or the sample design should

be changed.

Step 10. Perform audit procedure on the sample items selected for substantive tests.

Step 11. Analyze misstatements noted in the sample to determine their cause, nature and whether systematic

pattern exists. A systematic misstatement is a recurring misstatement does not occur randomly.

Step12. Calculate SD of the sample observations

N

n

nn

asreducedfurtherbecan

sizesamplethethenhighisN

nif

A

NSDUn

o

o

o

ro

1

..2

ZU

UMTA

r

r

.

Step 13.Calculate Achieved Precision:

Step 14. If A’ not equal to A then calculate A”

Step 15. Calculate

Step 16. Calculate Estimated Audited Value (EAV) :

Step 17. Calculate Decision Interval:

Book Value (Adjusted for any systematic (nonrandom) differences) ± A”

Step 18. Rule:

** If EAV falls within this interval, conclude that the statistical evidence support the book value

** Otherwise, conclude that the statistical evidence does not support the material correctness of the

book value

6.4.2.1 It may be noted that the systematic misstatements (also called nonrandom misstatements) are excluded

from statistical evaluation (please see step 17)

6.4.3. Monetary Unit Sampling (MUS):

Sampling methods used by auditors have evolved over the years. The trend now is to use less rigorous

sampling techniques to reduce cost. However, it has been empirically demonstrated by experiments that

MUS is substantially more capable of detecting material error and can be used for both proportional test of

controls and substantive testing. The use of qualitative analysis that documents the nature and cause

of each misstatement found in a sample can mitigate some of the risk associated with sampling. The

use of a statistical approach such as PPS can further reduce this risk, and, at the same time, permit

the use of a smaller sample.

"The auditor has a responsibility to plan and perform the audit to obtain reasonable assurance about

whether the financial statements are free of material misstatement, whether caused by error or

fraud."

6.4.3.1 Caveats on MUS

MUS sampling may be especially useful in the audit of accounts receivable and inventory. However, it is

not appropriate for accounts receivable if there are a large number of unapplied credits or for inventory

where the auditor anticipates a significant number of audit differences, or where the detection of an

understated balance is an important consideration.

N

nN

n

SDUrA 1'

)'

1('"A

ATMAA

n

valueauditedeachofx

xNX ˆ

Moreover, MUS evaluation technique is so sensitive to any errors found in the sample that it tends to

overstate the allowance for sampling risk and thereby project a potential misstatement that could be two or

three times the actual misstatement. The reason PPS tends to exaggerate its projection of misstatement is

that it does not simply extrapolate the total error found in the sample. Instead, it looks at each erroneous

item individually, and projects a misstatement amount proportional to that item's percentage of error rather

than its amount of error. Thus, an item with a Rs.100 book value but an audit value of Rs.10 is considered

90% misstated and results in the same projected misstatement as a Rs.1,000 item 90% misstated. Though

this appears illogical, under the PPS selection method, the Rs.1,000 item has ten times more chance of

being selected for audit than does the Rs.100 item. So when errors are found among the relatively few

small items that have been selected, they are given proportionately more weight. But in so doing, PPS

subjects the auditor to a high risk of incorrect rejection, that is, the risk of rejecting an account balance that

is not materially misstated. To put it bluntly, PPS is prone to false alarms.

6.4.3.2 Finally:

MUS's job is only to warn us of a possible fire, not to assess the extent of the fire or estimate the damage.

This requires classical forms of statistical sampling and extracts the price of a much larger sample. The

auditor's response to the alarm is essentially the same regardless of the degree by which MUS's projected

potential misstatement exceeds tolerable misstatement (assuming the excess is more than trivial).

However, qualitative as well as quantitative analyses are equally important. The auditor should identify

and document the nature and cause of each misstatement found in the sample. It takes finding only one

misstatement of a particular type for the auditor to become aware that that kind of misstatement is

occurring, at which point the auditor can apply additional procedures to determine the extent of

misstatements of that type. One misstatement may indicate a breakdown in a control procedure that

suggests other errors of a similar nature, and might in fact have implications elsewhere in the audit. A

second misstated item might clue the auditor to an inappropriate accounting principle that probably affects

all similar transactions. By working with the client to identify and correct other similar errors, the potential

misstatement might be reduced to an acceptable level. If not, other kinds of tests that serve the same audit

objectives, such as appropriate analytical procedures, may provide the additional evidence needed to

support the corrected book value of the account. Of course, if the possibility of fraud is indicated, further

effort and more careful consideration are required.

6.4.3.3

The PPS sampling approach in auditing was developed to convert misstatement rates into money value

.Goodfellow .Loebbecke and Neter outline the method for PPS sampling evaluation of the maximum

misstatement rates found with the Poisson distribution. Poisson probabilities are obtained from an

idealized mathematical process generating occasional random event (in audit misstatement rate is small

less than 5%).

Let BV = Book Value ; TM=tolerable misstatement ; Sampling Risk=SR and RFx is the corresponding

reliability factor for X number of anticipated misstatements in the population ; N=Population Size.

Step1: Sample size=n= (RF0 * BV)/TM

Step 2: Draw a PPS-systematic sample of size n from the population.

Step3 : Audit the physical n units

Step4: Evaluate the sample and determine tainting t

Where t =(amount of misstatement)/(reported book value of the unit) .

arrange them from highest to lowest as t1, t2 ,….

maximum possible value of misstatement (MVM) with confidence (1- SR) % =

MVM = (BV * RF0/n)*1+ (BV *(RF1-RF0)/n)*.t1+(BV *(RF2-RF1)/n)*.t2+….

Step 5: Make a decision about the acceptability of reported book value by comparing MVM with TM

[RF values may be obtained from Poisson table as in Annex –V]

6.5 The statistical sampling described above may also be categorized into three broad categories: Attribute,

Variable and Probability–proportional to size sampling. Attribute sampling is used primarily to estimate

number of incidence or in test of controls .In contrast; variable sampling and PPS sampling are most frequently

used to estimate population average or total or to test monetary value of account balances.

6.5.1 There are some other types of attribute sampling that are being used in audit:

6.5.1.1 Discovery Sampling

Discovery sampling is a sampling plan which selects a sample of a given size, accepts the population if the

sample is error free, and rejects the population if it contains at least one error. With discovery sampling the

auditor may not be interested in determining how many errors there are in the population. Where there is a

possibility of avoidance of the internal control system, it may be sufficient to disclose one example to

precipitate further action or investigation.

6.5.2 Stop or Go Sampling (also called Sequential Sampling)

Involves sampling a universe in increments and examining each incremental sample before

deciding when to stop.

Is appropriate for preliminary sampling and survey audit testing.

Allows auditors to determine from the smallest possible sample size if an error rate exceeds a

predetermined level.

Provides assurance, within a fixed degree of confidence, that the error rate in a population is less

than a predetermined acceptable error rate.

Does not provide an estimate of actual error rate; however, it can readily be converted into

attribute sampling, which can be used to estimate actual error rate.

8. Glossary of statistical terminologies:

Alpha Risk Risk of incorrect rejection

Beta Risk Risk of incorrect acceptance

Bias Difference between the true value and the expected value of the estimate.

Cluster Partitioning of the population into sub-population, called cluster, in such a way that

within each cluster the variation is more. It is convenient but less efficient sampling

design often known as area sampling.

Coefficient of

variations (CV)

Ratio of S.D. to mean .It is unit free & generally expressed in percentage term. This

measure is widely used to measure the reliability of estimate in survey sampling.

Also see Standard Deviation and Mean

Confidence level The certainty with which the estimate lies within the margin of error.

Estimate Projected value to the population from the sample observation.

Estimation

sampling

Use of sample observation to estimate some characters of interest in the population

Expected rate of

occurrence of

error

The rate of error (audit objections) that are expected in the population.

Extrapolation Projected to the population from the sample .

Heterogeneity Variation in the in population is more. It is opposite to homogeneity.

Homogeneity Variation in the in population is less. It is generally measured by standard deviation

(S.D). Less SD indicates more homogeneity.

Materiality The value of error that an auditor is willing to accept and still concludes that the audit

objective is achieved. The smaller the materiality, the larger is the sample size.

Margin of error A measure of the difference between the estimate from the sample and the population

value

Mean Average of observations

Symbolically, Mean = X = N

XN

i

i1

Multi-Stage

Sampling

Statistical sampling at different levels that is capable of generating estimates at various

levels. Mostly used in a large-scale sample surveys.

Monetary Unit

Sampling (MUS)

MUS give transactions with larger recorded amounts proportionally more opportunity

to be selected than units with smaller recorded amount.

Non-random

error

Errors those are systematic in nature.

Non-sampling

error

Non-sampling error is generated due to failure of measurement of true

characteristic(s). Which may be due to, (i) non-response (ii) measurement error (iii)

fatigue etc. Which can be controlled my better training & management.

Non-Statistical

Sampling

Selection of units based on judgment of an individual and probability of selection of

any unit is not known. Results of judgmental sampling cannot be used to draw

statistically valid inferences about the population.

Population All the elements (units/transactions) under consideration. In other words the entire

transactions/entities on which auditor wants to draw conclusion.

Population Size Number of elements in the population.

Probability It is a branch of mathematics that measures the likelihood that an event will occur.

Probabilities are expressed as numbers between 0 and 1. The probability of an

impossible event is 0, while an event that is certain to occur has a probability of 1.

Probability

Proportional to

Size (PPS)

Sampling

Selection probability is more where the size measure is high. Selection of sample units

with unequal probability.

Reliability It is the probability that the value of the feature of interest in the sample is

representative of the entire population i.e. within the desired margin of errors..

Sampling Design It is an organized method of sample selection & plans for analyzing and interpreting

the results.

Sampling error Sampling error is generated due to failure of selecting the representative sample .It is

measurable under statistical sampling.

Sampling Frame

List of all elements in the population. Often list may include some more information

other tan the identity of the element.

Simple Random

Sampling (SRS)

Selection of sample units with equal probability.

Standard

Deviation (S.D)

Positive square root of variance

Statistical

assurance

It is the measure of reliance an auditor places on inferences drawn using the sample .It

is commonly expressed in a probability statement with margin of error and confidence

level.

Statistical

Sampling

Any sample for which the selection of transactions or units to be included is

independent of the sampler and probability theory can be associated with it.

Stratification Partitioning of the population into sub-population, called strata, in such a way that

within each stratum the variation is less.

Testing of

hypothesis

A method to prove a hypothesis is statistically incorrect.

Variance It is the average of the square deviations of the observations from the mean.

Symbolically, Variance = N

XXN

i

i

1

2)(

= N

XnXN

i

i

1

2

8. Suggested reading:

1) Audit Sampling : An Introduction (fifth edition)-D. M. Guy, D.R. Carmichael and R Whittington

2) Statistical Sampling and Risk Analysis in Auditing :Peter Jones

3) Theory and methods of survey sampling -Parimal Mukhopadhdhyay :

4) Sampling Theory & Methods -M.N.Murthy

5) Survey Sampling -Kish

6) Sampling Techniques -Cochran W.C

7) http://saiindia.gov.in/cag/sites/default/files/Rupe_Trail_First_Edition_0.pdf

Annex-I

Methodologies of selection of samples for SRSWOR &PPSWR

(Using Random Number Table)

1. Simple Random Sampling With Out Replacement (SRSWOR).

Let there be N number of auditable units in a stratum from which n number of units to be selected .

.

Step 1: Prepare or get the list of units and associate serial numbers for each school.

Step 2: Open a page at random of the random number table

Step 3: Select a random number( let it be r) from the page of dimension N, starting from the left most top

corner of the table and proceed sequentially from left to right.

Step 4: If ‘ r ‘is between 1 and N, then the school corresponding to the rth serial number is selected. If not,

select the next random number in the sequence & proceed sequentially as per step 3.

Repeat Step 3 and Step 4 until we select n distinct schools (please note the sample selected more than once may be ignored)

2. Linear Systematic Sampling

Step1 : Calculate the Sampling interval I= [N/n]

Step2 : Open a page at random of the random number table

Step3 : Select a random number from the page of dimension I ,starting from left most corner of the

Table and proceed sequentially from left to right. Let it be R .

Ste 4 : Selected samples would be r, R+I, R+2I , R+3I ,….. until we get n samples or the sampling

frame gets exhausted.

Please note in this way we may not get exactly n number of samples.

3 Circular Systematic Sampling

Step1 : Calculate the Sampling interval I= [N/n]

Step2 : Open a page at random of the random number table

Step3 : Select a random number from the page of dimension N ,starting from left most corner of the

Table and proceed sequentially from left to right. Let it be R .

Step 4: Selected samples would be r, R+I, R+2I , R+3I ,….. until we get n samples

Step 5: If at any stage R+kI > N ,N may be subtracted from R+kI as R+kI-N and that is the next

sample and then proceed further i.e. in a circular manner.

Please note in this way we get exactly n number of samples.

4 Probability Proportion to Size With Replacement (PPSWR):

4.1 Cumulative total method:

A table of cumulative total of sizes of the units is made .Let Ti=x1+x2+x3+….+xi, where xi is the size measure of

ith unit. A random number, say R is drawn between 1 to TN (= Total size) .The unit ‘i’ is selected if Ti-1 < R ≤ Ti . The

process is repeated n (sample size) times

4.2 Let us take the following example, for selection of cities with PPSWR, with size measure as expenditure in the scheme

under review .Let the sample size be three.

(Expenditure figures are all fictitious)

List of districts in Punjab along with its expenditure,

Sr.No.

Name of the

Districts

Expenditure under

the scheme (00

‘000)

Cumulat

ive

Total

1 Amritsar 368 368

2 Bathinda 1095 1463

3 Faridkot 1009 2472

4 Fategarh Sahib 1536 4008

5 Firozpur 3419 7427

6 Gurdaspur 534 7961

7 Hoshiarpur 621 8582

8 Jalandhar 534 9116

9 Kapurthala 323 9439

10 Ludhiana 223 9662

11 Mansa 278 9940

12 Moga 660 10600

13 Muktsar 1474 12074

14 Nawanshahr 1613 13687

15 Patiala 1038 14725

16 Rupnagar 527 15252

17 Sangrur 2131 17383

Total 17383

Select random numbers of 5 digits between 00001 and 17383 (=Total Expenditure), (Random number selection procedure is the

same as indicated above)

Let the Page No. 1 is selected in the Random Number Table (in Annex-VI),

Table for sample selection

Random

Number

Decision District

selected

Reason

10097 Selected Moga Moga is selected as the selected random number ,

10097 is greater than the cumulative total against

sl.no.11 9940 but less than (or equal to) 10600.

32533 No selection X As the number is > 17383

76520 No selection X As the number is > 17383

13586 Selected Nawabnshahr Nawabnshahr is selected as the selected random

number 13586 is greater than the cumulative total

against Sl/No.13 i.e 12074 but less than (or equal to)

13687

34673 No selection X As the number is > 17383

54876 No selection X As the number is > 17383

80959 No selection X As the number is > 17383

09117 Selected Kapurthala Kapurthala is selected as the selected random number

9117 is greater than the cumulative total against

Sl/No.8 i.e 9116 but less than (or equal to) 9439

Hence the selected districts are Moga, Nawabnshahr and Kapurthala with selection probabilities are

(660/17383), (1613/17383) and (323/17383) respectively.

Annex-II

Other statistical sampling techniques

Two-stage sequential sampling

Step 1: A Simple Random Sampling Sample Without Replacement (SRSWOR) of size n ([Reliability Factor

(Zero deviation)] / Margin of error) is first selected for audit. Occurrences of deviations (audit objections) are

noted.

Step 2: Final size of the sample say, ‘m’ is calculated using Poisson distribution table on the basis of deviations

observed in step 1, as m= ([Reliability Factor (No. of deviations in step 1)] / Margin of error).

Remaining (m – 150) cases were selected with SRSWOR from the remaining cases in the population for audit.

RAO-HARTLEY-COCHRAN METHOD (RHC-1962) - It is PPSWOR(n)

Here in the beginning the entire population is partitioned into n random groups and from each group a unit is

selected independently by PPS method. It has certain advantages and an improved method. Please note the

estimation formulae are different.

Annex-III

Determination of Sample Size: Reliability, 90%

Binomial Table (Risk of Assessing Control Risk Too Low 10%)

(Allowable numbers of deviations are in parentheses)

Expected

Population

Deviation

Rate

2%

3%

4%

5%

Tolerable

6%

Rate

7%

8%

9%

10%

15%

20%

0.00%

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

4.00

5.00

6.00

7.00

114(0)

194(1)

194(1)

265(2)

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

76(0)

129(1)

129(1)

129(1)

176(2)

221(3)

*

*

*

*

*

*

*

*

*

*

*

*

*

*

57(0)

96(1)

96(1)

96(1)

96(1)

132(2)

132(2)

166(3)

198(4)

*

*

*

*

*

*

*

*

*

*

*

45(0)

77(1)

77(1)

77(1)

77(1)

77(1)

105(2)

105(2)

132(3)

132(3)

158(4)

209(6)

*

*

*

*

*

*

*

38(0)

64(1)

64(1)

64(1)

64(1)

64(1)

64(1)

88(2)

88(2)

88(2)

110(3)

132(4)

132(4)

153(5)

194(7)

*

*

*

*

*

32(0)

55(1)

55(1)

55(1)

55(1)

55(1)

55(1)

55(1)

75(2)

75(2)

75(2)

94(3)

94(3)

113(4)

113(4)

131(5)

149(6)

*

*

*

28(0)

48(1)

48(1)

48(1)

48(1)

48(1)

48(1)

48(1)

48(1)

65(2)

65(2)

65(2)

65(2)

82(3)

82(3)

98(4)

98(4)

160(8)

*

*

25(0)

42(1)

42(1)

42(1)

42(1)

42(1)

42(1)

42(1)

42(1)

42(1)

58(2)

58(2)

58(2)

58(2)

73(3)

73(3)

73(3)

115(6)

182(11

)

*

22(0)

38(1)

38(1)

38(1)

38(1)

38(1)

38(1)

38(1)

38(1)

38(1)

38(1)

52(2)

52(2)

52(2)

52(2)

52(2)

65(3)

78(4)

116(7)

199(14)

15(0)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

25(1)

34(2)

43(3)

52(4)

11(0)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

18(1)

25(2)

25(2)

sample size is too large to be cost effective

Source :AICPA ,Auditing Guide ,Audit Sampling (New York ,2001)

Annex-IV Normal Curve Area Table

Standard

Deviation

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

.0000

.0398

.0793

.1179

.1554

.1915

.2257

.2580

.2881

.3159

.3413

.3643

.3849

.4032

.4192

.4332

.4452

.4554

.4641

.4713

.4773

.4821

.4861

.4893

..4918

.4938

.4953

.4965

.4974

.4981

.4986

.4990

.0040

.0438

.0832

.1217

.1591

.1950

.2291

.2612

.2910

.3186

.3438

.3665

.3869

.4049

..4207

.4345

.4463

.4564

.4649

.4719

.4778

.4826

.4865

.4896

.4920

.4940

.4955

.4966

.4975

.4982

4987

.4991

.0080

.0478

.0871

.1255

.1628

.1985

.2324

.2642

.2939

.3212

.3461

.3686

.3888

.4066

.4222

.4357

.4474

.4573

.4656

.4726

.4783

.4830

.4868

.4898

.4922

.4941

.4956

.4967

.4976

.4983

.4987

.4991

.0120

.0517

.0910

.1293

.1664

.2019

2357

.2673

.2967

.3238

.3485

.3708

.3907

.4083

.4236

.4370

.4485

.4582

.4664

.4732

.4788

.4834

.4871

.4901

.4925

4943

.4957

.4968

.4977

.4984

.4988

.4991

.0159

.0557

.0948

.1331

.1700

.2054

.2389

.2704

.2995

.3264

.3508

.3729

.3925

.4099

.4251

.4382

.4495

.4591

.4671

.4738

.4793

.4838

.4875

.4904

.4927

.4945

.4959

.4969

.4977

.4984

.4988

.4992

.0199

.0596

.0987

.1368

.1736

.2088

.2422

.2734

.3023

.3289

.3531

.3749

.3944

.4115

.4265

.4394

.4505

.4599

.4678

.4744

.4798

.4842

.4878

.4906

.4929

.4960

.4960

.4970

.4978

.4984

.4988

.4992

.0239

.0636

.1026

.1406

.1772

.2123

.2454

.2764

.3051

.3315

.3554

.3770

.3962

.4131

.4279

.4406

.4515

.4608

.4686

.4750

.4803

.4846

.4881

.4909

.4931

.4948

.4961

.4971

.4979

.4985

.4989

.4992

.0279

.0675

.1064

.1443

.1808

.2157

.2486

.2794

.3078

.3340

.3577

.3790

.3980

.4147

.4292

.4418

.4525

.4616

.4693

.4758

.4808

.4850

.4884

.4911

.4932

.4949

.4962

.4972

.4980

.4985

.4989

.4992

.0319

.0714

.1103

.1480

.1844

.2190

.2518

.2823

.3106

.3365

.3599

.3810

.3997

.4162

.4306

.4430

.4535

.4625

.4699

.4762

.4812

.4854

.4887

.4913

.4934

.4951

.4963

.4973

.4980

.4986

.4989

.4993

.0359

.0753

.1141

.1517

.1879

.2224

.2549

.2852

.3133

.3389

.3621

.3830

.4015

.4177

.4319

.4441

.4545

.4633

.4706

.4767

.4817

.4857

.4890

.4916

.4936

.4952

.4964

.4974

.4981

.4986

.4990

.4993

Annex –V: Poisson Table

sampling error

Number of Deviations 10% 5% 2.5%

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

2.4

3.9

5.4

6.7

8.0

9.3

10.6

11.8

13.0

14.3

15.5

16.7

18.0

19.0

20.2

21.4

22.6

23.8

25.0

26.0

27.1

28.3

29.3

30.5

31.4

32.7

34.0

35.0

36.1

37.2

38.4

39.1

40.3

41.5

42.7

43.8

45.0

46.1

47.2

48.3

3.0

4.8

6.3

7.8

9.2

10.6

11.9

13.2

14.5

16.0

17.0

18.3

19.5

21.0

22.0

23.4

24.3

26.0

27.0

28.0

29.0

30.3

31.5

32.6

33.8

35.0

36.1

37.3

38.5

39.6

40.7

42.0

43.0

44.2

45.3

46.4

47.6

48.7

49.8

51.0

3.7

5.6

7.3

8.8

10.3

11.7

13.1

14.5

15.8

17.1

18.4

19.7

21.0

22.3

23.5

24.7

26.0

27.3

28.5

29.6

31.0

32.0

33.3

34.6

35.7

37.0

38.1

39.4

40.5

41.7

42.9

44.0

45.1

46.3

47.5

48.8

49.9

51.0

52.1

53.4

Source: Adapted from a table developed by Marvin Tummins and Robert H. Strawser, “A Confidence Limits Tables for Attribute Sampling,” Accounting Review

(October 1976), pp. 907-912.

This table is particularly applicable when:

Population Size > 1000

And the estimated population deviation rate < 5 percent,

Annex-VI

Random Numbers 00000 10097 32533 76520 13586 34673 54876 80959 09117 39292 74945

00001 37542 04805 64894 74296 24805 24037 20636 10402 00822 91665

00002 08422 68953 19645 09303 23209 02560 15953 34764 35080 33606

00003 99019 02529 09376 70715 38311 31165 88676 74397 04436 27659

00004 12807 99970 80157 36147 64032 36653 98951 16877 12171 76833

00005 66065 74717 34072 76850 36697 36170 65813 39885 11199 29170

00006 31060 10805 45571 82406 35303 42614 86799 07439 23403 09732

00007 85269 77602 02051 65692 68665 74818 73053 85247 18623 88579

00008 63573 32135 05325 47048 90553 57548 28468 28709 83491 25624

00009 73796 45753 03529 64778 35808 34282 60935 20344 35273 88435

00010 98520 17767 14905 68607 22109 40558 60970 93433 50500 73998

00011 11805 05431 39808 27732 50725 68248 29405 24201 52775 67851

00012 83452 99634 06288 98083 13746 70078 18475 40610 68711 77817

00013 88685 40200 86507 58401 36766 67951 90364 76493 29609 11062

00014 99594 67348 87517 64969 91826 08928 93785 61368 23478 34113

00015 65481 17674 17468 50950 58047 76974 73039 57186 40218 16544

00016 80124 35635 17727 08015 45318 22374 21115 78253 14385 53763

00017 74350 99817 77402 77214 43236 00210 45521 64237 96286 02655

00018 69916 26803 66252 29148 36936 87203 76621 13990 94400 56418

00019 09893 20505 14225 68514 46427 56788 96297 78822 54382 14598

00020 91499 14523 68479 27686 46162 83554 94750 89923 37089 20048

00021 80336 94598 26940 36858 70297 34135 53140 33340 42050 82341

00022 44104 81949 85157 47954 32979 26575 57600 40881 22222 06413

00023 12550 73742 11100 02040 12860 74697 96644 89439 28707 25815

00024 63606 49329 16505 34484 40219 52563 43651 77082 07207 31790

00025 61196 90446 26457 47774 51924 33729 65394 59593 42582 60527

00026 15474 45266 95270 79953 59367 83848 82396 10118 33211 59466

00027 94557 28573 67897 54387 54622 44431 91190 42592 92927 45973

00028 42481 16213 97344 08721 16868 48767 03071 12059 25701 46670

00029 23523 78317 73208 89837 68935 91416 26252 29663 05522 82562

00030 04493 52494 75246 33824 45862 51025 61962 79335 65337 12472

00031 00549 97654 64051 88159 96119 63896 54692 82391 23287 29529

00032 35963 15307 26898 09354 33351 35462 77974 50024 90103 39333

00033 59808 08391 45427 26842 83609 49700 13021 24892 78565 20106

00034 46058 85236 01390 92286 77281 44077 93910 83647 70617 42941

00035 32179 00597 87379 25241 05567 07007 86743 17157 85394 11838

00036 69234 61406 20117 45204 15956 60000 18743 92423 97118 96338

00037 19565 41430 01758 75379 40419 21585 66674 36806 84962 85207

00038 45155 14938 19476 07246 43667 94543 59047 90033 20826 69541

00039 94864 31994 36168 10851 34888 81553 01540 35456 05014 51176

00040 98086 24826 45240 28404 44999 08896 39094 73407 35441 31880

00041 33185 16232 41941 50949 89435 48581 88695 41994 37548 73043

00042 80951 00406 96382 70774 20151 23387 25016 25298 94624 61171

00043 79752 49140 71961 28296 69861 02591 74852 20539 00387 59579

00044 18633 32537 98145 06571 31010 24674 05455 61427 77938 91936

00045 74029 43902 77557 32270 97790 17119 52527 58021 80814 51748

00046 54178 45611 80993 37143 05335 12969 56127 19255 36040 90324

00047 11664 49883 52079 84827 59381 71539 09973 33440 88461 23356

00048 48324 77928 31249 64710 02295 36870 32307 57546 15020 09994

00049 69074 94138 87637 91976 35584 04401 10518 21615 01848 76938

00050 09188 20097 32825 39527 04220 86304 83389 87374 64278 58044

00051 90045 85497 51981 50654 94938 81997 91870 76150 68476 64659

00052 73189 50207 47677 26269 62290 64464 27124 67018 41361 82760

00053 75768 76490 20971 87749 90429 12272 95375 05871 93823 43178

00054 54016 44056 66281 31003 00682 27398 20714 53295 07706 17813

00055 08358 69910 78542 42785 13661 58873 04618 97553 31223 08420

00056 28306 03264 81333 10591 40510 07893 32604 60475 94119 01840

00057 53840 86233 81594 13628 51215 90290 28466 68795 77762 20791

00058 91757 53741 61613 62269 50263 90212 55781 76514 83483 47055

00059 89415 92694 00397 58391 12607 17646 48949 72306 94541 37408

00060 77513 03820 86864 29901 68414 82774 51908 13980 72893 55507

00061 19502 37174 69979 20288 55210 29773 74287 75251 65344 67415

00062 21818 59313 93278 81757 05686 73156 07082 85046 31853 38452

00063 51474 66499 68107 23621 94049 91345 42836 09191 08007 45449

00064 99559 68331 62535 24170 69777 12830 74819 78142 43860 72834

00065 33713 48007 93584 72869 51926 64721 58303 29822 93174 93972

00066 85274 86893 11303 22970 28834 34137 73515 90400 71148 43643

00067 84133 89640 44035 52166 73852 70091 61222 60561 62327 18423

00068 56732 16234 17395 96131 10123 91622 85496 57560 81604 18880

00069 65138 56806 87648 85261 34313 65861 45875 21069 85644 47277

00070 38001 02176 81719 11711 71602 92937 74219 64049 65584 49698

00071 37402 96397 01304 77586 56271 10086 47324 62605 40030 37438

00072 97125 40348 87083 31417 21815 39250 75237 62047 15501 29578

00073 21826 41134 47143 34072 64638 85902 49139 06441 03856 54552

00074 73135 42742 95719 09035 85794 74296 08789 88156 64691 19202

00075 07638 77929 03061 18072 96207 44156 23821 99538 04713 66994

00076 60528 83441 07954 19814 59175 20695 05533 52139 61212 06455

00077 83596 35655 06958 92983 05128 09719 77433 53783 92301 50498

00078 10850 62746 99599 10507 13499 06319 53075 71839 06410 19362

00079 39820 98952 43622 63147 64421 80814 43800 09351 31024 73167

00080 59580 06478 75569 78800 88835 54486 23768 06156 04111 08408

00081 38508 07341 23793 48763 90822 97022 17719 04207 95954 49953

00082 30692 70668 94688 16127 56196 80091 82067 63400 05462 69200

00083 65443 95659 18288 27437 49632 24041 08337 65676 96299 90836

00084 27267 50264 13192 72294 07477 44606 17985 48911 97341 30358

00085 91307 06991 19072 24210 36699 53728 28825 35793 28976 66252

00086 68434 94688 84473 13622 62126 98408 12843 82590 09815 93146

00087 48908 15877 54745 24591 35700 04754 83824 52692 54130 55160

00088 06913 45197 42672 78601 11883 09528 63011 98901 14974 40344

00089 10455 16019 14210 33712 91342 37821 88325 80851 43667 70883

00090 12883 97343 65027 61184 04285 01392 17974 15077 90712 26769

00091 21778 30976 38807 36961 31649 42096 63281 02023 08816 47449

00092 19523 59515 65122 59659 86283 68258 69572 13798 16435 91529

00093 67245 52670 35583 16563 79246 86686 76463 34222 26655 90802

00094 60584 47377 07500 37992 45134 26529 26760 83637 41326 44344

00095 53853 41377 36066 94850 58838 73859 49364 73331 96240 43642

00096 24637 38736 74384 89342 52623 07992 12369 18601 03742 83873

00097 83080 12451 38992 22815 07759 51777 97377 27585 51972 37867

00098 16444 24334 36151 99073 27493 70939 85130 32552 54846 54759

00099 60790 18157 57178 65762 11161 78576 45819 52979 65130 04860

00100 03991 10461 93716 16894 66083 24653 84609 58232 88618 19161

00101 38555 95554 32886 59780 08355 60860 29735 47762 71299 23853

00102 17546 73704 92052 46215 55121 29281 59076 07936 27954 58909

00103 32643 52861 95819 06831 00911 98936 76355 93779 80863 00514

00104 69572 68777 39510 35905 14060 40619 29549 69616 33564 60780

00105 24122 66591 27699 06494 14845 46672 61958 77100 90899 75754

00106 61196 30231 92962 61773 41839 55382 17267 70943 78038 70267

00107 30532 21704 10274 12202 39685 23309 10061 68829 55986 66485

00108 03788 97599 75867 20717 74416 53166 35208 33374 87539 08823

00109 48228 63379 85783 47619 53152 67433 35663 52972 16818 60311

00110 60365 94653 35075 33949 42614 29297 01918 28316 98953 73231

00111 83799 42402 56623 34442 34994 41374 70071 14736 09958 18065

00112 32960 07405 36409 83232 99385 41600 11133 07586 15917 06253

00113 19322 53845 57620 52606 66497 68646 78138 66559 19640 99413

00114 11220 94747 07399 37408 48509 23929 27482 45476 85244 35159

00115 31751 57260 68980 05339 15470 48355 88651 22596 03152 19121

00116 88492 99382 14454 04504 20094 98977 74843 93413 22109 78508

00117 30934 47744 07481 83828 73788 06533 28597 20405 94205 20380

00118 22888 48893 27499 98748 60530 45128 74022 84617 82037 10268

00119 78212 16993 35902 91386 44372 15486 65741 14014 87481 37220

00120 41849 84547 46850 52326 34677 58300 74910 64345 19325 81549

00121 46352 33049 69248 93460 45305 07521 61318 31855 14413 70951

00122 11087 96294 14013 31792 59747 67277 76503 34513 39663 77544

00123 52701 08337 56303 87315 16520 69676 11654 99893 02181 68161

00124 57275 36898 81304 48585 68652 27376 92852 55866 88448 03584

00125 20857 73156 70284 24326 79375 95220 01159 63267 10622 48391

00126 15633 84924 90415 93614 33521 26665 55823 47641 86225 31704

00127 92694 48297 39904 02115 59589 49067 66821 41575 49767 04037

00128 77613 19019 88152 00080 20554 91409 96277 48257 50816 97616

00129 38688 32486 45134 63545 59404 72059 43947 51680 43852 59693

00130 25163 01889 70014 15021 41290 67312 71857 15957 68971 11403

00131 65251 07629 37239 33295 05870 01119 92784 26340 18477 65622

00132 36815 43625 18637 37509 82444 99005 04921 73701 14707 93997

00133 64397 11692 05327 82162 20247 81759 45197 25332 83745 22567

00134 04515 25624 95096 67946 48460 85558 15191 18782 16930 33361

00135 83761 60873 43253 84145 60833 25983 01291 41349 20368 07126

00136 14387 06345 80854 09279 43529 06318 38384 74761 41196 37480

00137 51321 92246 80088 77074 88722 56736 66164 49431 66919 31678

00138 72472 00008 80890 18002 94813 31900 54155 83436 35352 54131

00139 05466 55306 93128 18464 74457 90561 72848 11834 79982 68416

00140 39528 72484 82474 25593 48545 35247 18619 13674 18611 19241

00141 81616 18711 53342 44276 75122 11724 74627 73707 58319 15997

00142 07586 16120 82641 22820 92904 13141 32392 19763 61199 67940

00143 90767 04235 13574 17200 69902 63742 78464 22501 18627 90872

00144 40188 28193 29593 88627 94972 11598 62095 36787 00441 58997

00145 34414 82157 86887 55087 19152 00023 12302 80783 32624 68691

00146 63439 75363 44989 16822 36024 00867 76378 41605 65961 73488

00147 67049 09070 93399 45547 94458 74284 05041 49807 20288 34060

00148 79495 04146 52162 90286 54158 34243 46978 35482 59362 95938

00149 91704 30552 04737 21031 75051 93029 47665 64382 99782 93478

00150 94015 46874 32444 48277 59820 96163 64654 25843 41145 42820

00151 74108 88222 88570 74015 25704 91035 01755 14750 48968 38603

00152 62880 87873 95160 59221 22304 90314 72877 17334 39283 04149

00153 11748 12102 80580 41867 17710 59621 06554 07850 73950 79552

00154 17944 05600 60478 03343 25852 58905 57216 39618 49856 99326

00155 66067 42792 95043 52680 46780 56487 09971 59481 37006 22186

00156 54244 91030 45547 70818 59849 96169 61459 21647 87417 17198

00157 30945 57589 31732 57260 47670 07654 46376 25366 94746 49580

00158 69170 37403 86995 90307 94304 71803 26825 05511 12459 91314

00159 08345 88975 35841 85771 08105 59987 87112 21476 14713 71181

00160 27767 43584 85301 88977 29490 69714 73035 41207 74699 09310

00161 13025 14338 54066 15243 47724 66733 47431 43905 31048 56699

00162 80217 36292 98525 24335 24432 24896 43277 58874 11466 16082

00163 10875 62004 90391 61105 57411 06368 53856 30743 08670 84741

00164 54127 57326 26629 19087 24472 88779 30540 27886 61732 75454

00165 60311 42824 37301 42678 45990 43242 17374 52003 70707 70214

00166 49739 71484 92003 98086 76668 73209 59202 11973 02902 33250

00167 78626 51594 16453 94614 39014 97066 83012 09832 25571 77628

00168 66692 13986 99837 00582 81232 44987 09504 96412 90193 79568

00169 44071 28091 07362 97703 76447 42537 98524 97831 65704 09514

00170 41468 85149 49554 17994 14924 39650 95294 00556 70481 06905

00171 94559 37559 49678 53119 70312 05682 66986 34099 74474 20740

00172 41615 70360 64114 58660 90850 64618 80620 51790 11436 38072

00173 50273 93113 41794 86861 24781 89683 55411 85667 77535 99892

00174 41396 80504 90670 08289 40902 05069 95083 06783 28102 57816

00175 25807 24260 71529 78920 72682 07385 90726 57166 98884 08583

00176 06170 97965 88302 98041 21443 41808 68984 83620 89747 98882

00177 60808 54444 74412 81105 01176 28838 36421 16489 18059 51061

00178 80940 44893 10408 36222 80582 71944 92638 40333 67054 16067

00179 19516 90120 46759 71643 13177 55292 21036 82808 77501 97427

00180 49386 54480 23604 23554 21785 41101 91178 10174 29420 90438

00181 06312 88940 15995 69321 47458 64809 98189 81851 29651 84215

00182 60942 00307 11897 92674 40405 68032 96717 54244 10701 41393

00183 92329 98932 78284 46347 71209 92061 39448 93136 25722 08564

00184 77936 63574 31384 51924 85561 29671 58137 17820 22751 36518

00185 38101 77756 11657 13897 95889 57067 47648 13885 70669 93406

00186 39641 69457 91339 22502 92613 89719 11947 56203 19324 20504

00187 84054 40455 99396 63680 67667 60631 69181 96845 38525 11600

00188 47468 03577 57649 63266 24700 71594 14004 23153 69249 05747

00189 43321 31370 28977 23896 76479 68562 62342 07589 08899 05985

00190 64281 61826 18555 64937 13173 33365 78851 16499 87064 13075

00191 66847 70495 32350 02985 86716 38746 26313 77463 55387 72681

00192 72461 33230 21529 53424 92581 02262 78438 66276 18396 73538

00193 21032 91050 13058 16218 12470 56500 15292 76139 59526 52113

00194 95362 67011 06651 16136 01016 00857 55018 56374 35824 71708

00195 49712 97380 10404 55452 34030 60726 75211 10271 36633 68424

00196 58275 61764 97586 54716 50259 46345 87195 46092 26787 60939

00197 89514 11788 68224 23417 73959 76145 30342 40277 11049 72049

00198 15472 50669 48139 36732 46874 37088 73465 09819 58869 35220

00199 12120 86124 51247 44302 60883 52109 21437 36786 49226 77837

00200 19612 78430 11661 94770 77603 65669 86868 12665 30012 75989

00201 39141 77400 28000 64238 73258 71794 31340 26256 66453 37016

00202 64756 80457 08747 12836 03469 50678 03274 43423 66677 82556

00203 92901 51878 56441 22998 29718 38447 06453 25311 07565 53771

00204 03551 90070 09483 94050 45938 18135 36908 43321 11073 51803

00205 98884 66209 06830 53656 14663 56346 71430 04909 19818 05707

00206 27369 86882 53473 07541 53633 70863 03748 12822 19360 49088

00207 59066 75974 63335 20483 43514 37481 58278 26967 49325 43951

00208 91647 93783 64169 49022 98588 09495 49829 59068 38831 04838

00209 83605 92419 39542 07772 71568 75673 35185 89759 44901 74291

00210 24895 88530 70774 35439 46758 70472 70207 92675 91623 61275

00211 35720 26556 95596 20094 73750 85788 34264 01703 46833 65248

00212 14141 53410 38649 06343 57256 61342 72709 75318 90379 37562

00213 27416 75670 92176 72535 93119 56077 06886 18244 92344 31374

00214 82071 07429 81007 47749 40744 56974 23336 88821 53841 10536

00215 21445 82793 24831 93241 14199 76268 70883 68002 03829 17443

00216 72513 76400 52225 92348 62308 98481 29744 33165 33141 61020

00217 71479 45027 76160 57411 13780 13632 52308 77762 88874 33697

00218 83210 51466 09088 50395 26743 05306 21706 70001 99439 80767

00219 68749 95148 94897 78636 96750 09024 94538 91143 96693 61886

00220 05184 75763 47075 88158 05313 53439 14908 08830 60096 21551

00221 13651 62546 96892 25240 47511 58483 87342 78818 07855 39269

00222 00566 21220 00292 24069 25072 29519 52548 54091 21282 21296

Source : 222 lines of random numbers from the random number table of RAND

Statistical Issues in Relation to Audit

1

Statistics are no substitute for judgment

Statistics are used much like a drunk uses a

lamppost: for support, not for illumination.

Pawan Dhamija

Statistical Advisor

What is Statistics ?

(a) Descriptive Statistics: Deals with

Collection, Organisation, presentation,

Summarisation and analysis of Data

(b) Inferential Statistics: In addition deals

with drawing of inference about a set of data

(Population) when only a part of data

(Sample) is observed.

2

3

Statistics

Descriptive Statistics Inferential Statistics

Collecting

Summarizing

Presenting

Analyzing

Collecting

Summarizing

Presenting

Analyzing

Generalizing

Draw conclusion about

the subjects studied

Draw conclusion about the

items or group which is bigger

than what has been observed

Why Statistics in audit

To develop an appreciation about averages and

variability.

For making data into information

Develop understanding of ideas of statistical

reliability/precision, probability, Risk/errors, etc.

Use these ideas to develop a proper sampling

design including decision about sample size and

for drawing valid inferences based on sample.

4

Population: Entire group of people or objects

(vouchers, bills, audit entities) to which the

researcher/auditor wishes to generalize the

study/audit findings.

Sample: A sample is a part of the population,

selected by the investigator/auditor to gather

information on certain characteristics of the original

population.

Population and Sample

5

Sampling: The Process of Selection of a sample from a

population to generate precise and valid estimates of

population parameter.

Census (100% enumeration): The process of collecting

relevant information/ data in respect of each and every

member/unit of the population.

Statistical Inference: Drawing Conclusions (Inferences)

about a population based on an examination/audit of

sample(s) taken from the population.

Sampling, Census and Statistical Inference

6

7

Describing Sample/Population Descriptive Statistics

Measures of Central Tendency

Measures of Variability/Dispersion

Other Descriptive Measures like

Minimum and Maximum Values

Sample size (n)

Percentile: 25 percentile ‘P’ is the value of the variable

X such that 25% of observations are < P.

Median -50th percentile; Q1-25th Percentile; Q3-75per.

8

Measures of Central Tendency (Averages)

Measures the “centre” of the data set

Single number that can be taken as a representative of the entire data set

Measures commonly used for averages are:

Mean

Median

Mode

Which measure to use depends on nature of data

It is okay to report more than one measure.

9

Measures of Central Tendency: Mean

The mean is given by the sum of the observations

divided by the number of observations. For e.g., mean

of 1,3,5,7,9 is

If the data are made up of n observations x1, x2,…, xn.

We can calculate mean as:

Σ is summation, ‘n’ is sample size and X is sample mean. Mean fulfills all the conditions of a good average.

However, it is largely affected by the extreme values so it must not be used for skewed (explained later) data

n

i

iXn

X1

1

10

Measures of Central Tendency: Mean Cont.

The population mean is usually unknown; so we try to

make inference about it.

According to statistical sampling theory, sample Mean can

be taken as a projection/estimate of the population mean.

For e.g. average misstatement in book values based on a

sample is an estimator (projection) of average misstatement

in book value in the population.

11

Median: “Middle observation” according to its rank in

data i.e. after arranging data in ascending order.

Better than mean if extreme observations are present i.e.

for skewed data.

If n is odd: Median = (n+1)/2th item.

If n is even: Median = ½* [(n/2)th + (n/2+1)th]

item after the data has been ordered.

For e.g. observations: 1, 3, 5, 7, 9, 13, 5, 7, 8, 2, 10, 5

in ascending order: 1, 2, 3, 5, 5, 5, 7, 7, 8, 9, 10, 13;

n = 12 (even); n/2 = 12/2 = 6

Median = ½(6th + 7th) item = ½(5+7) = 6

12

Good for Qualitative data like intelligence, beauty, honesty.

The value/observation with highest freq. gives mode.

For e.g. for observations: 1, 3, 5, 7, 9, 13, 5, 7, 8, 2, 10, 5

In ascending order: 1, 2, 3, 5, 5, 5, 7, 7, 8, 9, 10, 13;

Observation 5 occurs the maximum number of time (has

highest frequency) so mode = 5

However, if there are two or more observations with

highest frequency we use: Mode = 3*median - 2*mean

If data are symmetric (explained later):

mean = median = mode

Mode: Value that occurs most frequently

13

A tabular statement with two columns; first column describes

the variable category/values and the second column

represents the frequencies (i.e. number of times the variable

is taking that particular value). Example of freq. Dist.

No. of

errors (x)

0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40

No. of

vouchers (f)

328 350 720 664 598 524 378 244

Where N = Σf

Frequency Distribution :

It indicates there are 328 vouchers containing 0-5 (5 excluded)

errors, Arithmetic Mean for frequency Distribution is given by:

14

Mean, Median and Mode of Freq. Distribution

Xi fi (fi*Xi) c.f.

1 4 4 4

2 6 12 10

3 5 15 15

4 4 16 19

5 4 20 23

6 3 18 26

7 4 28 30

8 6 48 36

9 4 36 40

10 3 30 43

11 3 33 46

12 4 48 50

Total 50 308

Mean=308/50

= 6.16

N = 50 (even); n/2 =

25

Median =

1/2*(25th + 26th) item

Median= ½(6+6) = 6

Highest freq. = 6

occurring twice so

use

Mode=3*median-

2*mean

=3*6 –2*6.16

= 5.68

15

Dispersion: Measures of Variability

Measure the “spread or variation”

[Heterogeneity] in the data

Measures commonly used for variability are:

Variance

Standard Deviation

Range

Semi Inter-quartile Range

or Quartile Deviation (QD)

16

Measures of Variability: Variance

The sample variance (s2) may be calculated from the data. It is the average of the square deviations of the observations from the mean.

Where ‘n’ is the sample size and X is sample mean; For sample size n > 30 ‘n’ may be used instead of ‘n-1’

Population variance is denoted by S2. This is usually unknown.

s2 (Sample Variance) can be used as an estimator (projected value) of S2 (population Variance).

n

i

i XXn

s1

22

1

1

17

Standard Deviation (SD)

Square root of the variance

s = √s2 = sample SD

S = √S2 = population SD; Usually unknown

Merits of SD: Expressed in the same units as the

mean (instead of squared units like the variance)

Demerit: It is difficult to calculate

18

Range and Quartile Deviation (QD)

Range = Maximum - Minimum

QD = ½*(Q3 – Q1)

Where Q3 is third Quartile and Q1 first quartile

25% of observations are below Q1 and 75% below Q3

QD is powerful than range as range is based on just 2

items and QD is based on 50% of the items.

SD is the best and most useful measure of Variation;

however if there are outliers (i.e. if the data are highly

skewed) it should not be used.

19

Find Standard Deviation and Variance of no. of

errors in the vouchers [Solution on next slide]

No. of Errors Number of Vouchers

Xi fi

6 5

7 4

8 4

9 3

10 2

Total 18

20

No. of errors (A)Number of

vouchers (B)C: A*B D: x - m E: (x-m)2 F: B*E

x f f*x x-7.6 (x-7.6)2 f*(x-7.6)2

6 5 30 -1.6 2.56 12.80

7 4 28 -0.6 0.36 1.44

8 4 32 0.4 0.16 0.64

9 3 27 1.4 1.96 5.88

10 2 20 2.4 5.76 11.52

Total Σ 18 137 2 - 32.28

Mean (m) = Σf*x /Σf=137/18 7.6

Var. = Σf*(x-m)2 /Σf = 32.28/18 1.79

Std. Dev. = =√Var = √1.79 1.339

Solution

Skewness: Measures of symmetry of data set

Skewness measures lack of symmetry of data

Positive or right skewed: Longer right tail

Negative or left skewed: Longer left tail

21

22

Sampling: Some Facts

For very small samples (e.g., <5 observations),

summary statistics (mean, SD etc.) are not meaningful.

Simply list the data.

Beware that poor samples may provide a distorted view

of the population

In general, larger samples are better representative of

the population but they need more resources; so we

have to trade off b/w sample size and feasibility

(available resources).

Probability(P)

Measures the likelihood with which an event occurs

1

0

10

eventcertainaofyprobabilit

eventimpossibleanofyprobabilit

andbetweenliesP

casesofnumbertotal

casesofnumberfavourableP

23

24

Probability Distributions (P. D.)

A probability distribution describes the behavior of the

character of interest (called variable);

It identifies possible values of the variable and

provides information about the probability with which

these values (or ranges of values) will occur.

For e.g. in voucher sampling we have following P.D.

Important P.Ds are Binomial, Poisson and Normal

No. of

errors

0 1 2 3 4 More

than 4

Total

Prob. 0.50 0.20 0.10 0.06 0.04 0.10 1.00

Normal Curve and 90% value

1

Sampling and its types

P. K. Dhamija Stat. Advisor

2

Sampling

The Process of selection of some members of a

population to generate precise and valid

estimates of population parameters like

averages or proportions.

Sample

A sample is a part of the population, selected by the

investigator/auditor as its representative to gather

information on certain characteristics of the original

population

3

Sampling Terms Sampling unit (Basic sampling unit)

Example: vouchers, cheques, bills, districts, audit units

Sampling frame

List of all sampling units in the population

Sampling scheme

Method used to select sampling units from the sampling

frame

Parameter : Population characteristic like average, proportion

based on all the units in the population; it is constant/fixed.

Statistic: Sample characteristic like average, proportion based

on sample values; it varies from sample to sample.

4

Advantages of Survey Sampling

less expensive; Saves Time

The quality of information is maintained.

Possible to determine the extent of error due toSampling

Non Sampling errors are likely to be less

Even Census Results are verified by sampling

Law of Statistical Regularity lays down that amoderately large number of items chosen at randomform a large group are almost sure on the average topossess the characteristic of the large group.

5

Disadvantages of Survey Sampling

Results of a sample survey are subject to error due

to sampling.

A sample may not properly represent the various

subgroups of a population.

Sometimes the sampling methods may become

complicated requiring the services of an expert.

Note: If time & money are not important factors

and if population under consideration is not too

large, census is better than any sampling method.

6

Non-Statistical sampling

Statistical sampling

Types of sampling

7

Non-Statistical Sampling Units in the study population do not have a known

probability of being included in the sample

Subjective/Biased samples

Used when (i) the number of elements in the population is either unknown or units in the population can not be identified and (ii) there are time/ resource constraints

Advantages:

Practical and easy to conduct

Dis-advantages:

Not representative of the population

Not possible to (i) assess the validity of estimates (ii) Determine sample size

8

I Accidental/ Haphazard Sampling/

The auditor selects sample (audit units, bills, vouchers)

without any conscious bias ; the sample is expected to be

representative of the population. For e.g. avoiding first

and last voucher in a bundle.

II Judgmental /Purposive Sampling

The auditor selects sample (audit units, bills, vouchers)

which in his opinion contain maximum error say

vouchers with the highest values or vouchers of some

particular treasury.

Some Non-Statistical Sampling

Techniques

9

Statistical Sampling Each unit in the study population has a known probability

(may not be equal) of being included in the sample.

Advantages:

It provides estimates free from personal bias

It permits application of objective methods of minimizing error under the resource constraints.

Allows to draw valid conclusions about population

Dis-advantages:

Needs sampling frame

Compared to Non Prob. sampling it is difficult to apply

10

1. Simple random sampling (SRS)

The most commonly used Statistical sampling

Principle: Equal chance for each sampling unit to be included in the sample

Procedure1. Identify all sampling units in the population

2. Determine sample size (n) using appropriate formula/table

3. Draw (n) units using random tables

or computer programs like Excel or IDEA.

Advantages Simple

Sampling error easily measured

Disadvantages Need complete list of units

Not always best representative

SRS with replacement (SRSWR)

First unit is randomly selected from population

The sampled unit is replaced in the population

Then second unit is drawn; probability of selection of an element remains unchanged after each draw

The procedure is repeated until the requisite sample of size 'n' is drawn.

In practice SRSWR is not attractive; Same units can be selected more than once which may not add any value/additional information

But in mathematical terms, it is simpler to relate the sample to population by SRSWR.

11

SRS without replacement (SRSWOR)

Unlike SRSWR, once an element is selected as a sample unit, it is not replaced in the population

The selected sample units are distinct

SRSWOR provides two advantages:

Elements are not repeated

Variance estimation is smaller (efficiency is higher) than SRSWR with same sample size

12

13

2952 6641 3992 9792 7979 5911 3170

5624 4167 9524 1545 1396 7203 5356

1300 2693 2370 7483 3408 2762 3563

1089 6913 7691 0560 5246 1112 6107

6008 8126 4233 8776 2754 9143 1405

9025 7002 6111 8816 6446

Use the following Random Number Table to draw a

simple random sample (i) of 15 vouchers without

replacement and (ii) of 45 vouchers with replacement;

from a treasury having 500 vouchers.

Part of Random Number table

14

2. Systematic SamplingPrinciple:

In this method, first unit is drawn by random numbers;

thereafter, every kth (k = N/n is sampling interval) unit is

drawn. It gives equal chance of selection to each unit

Procedure

1. Prepare a list of all elements in the study population (N)

2. Decide the sample size (n)

3. Determine the sampling interval ‘k’ as the integer

nearest to N/n

4. Have the random start by choosing an integer ‘r’

between 1 and k.

5. Select every kth unit starting with the unit corresponding

to the number ‘r’.

15

Say, Target Population N= 54000 vouchers (Sampling Frame)

Sample size n = 6000

Sample Fraction (K)= Target Population / Sample size

=54000/6000 = 9

Number all vouchers of the population

Select 1 number between 1 to 9 (here K = 9) randomly

Say, number 5 is selected then 5th voucher is selected

Next 5+9=14th , 14 + 9 = 23rd voucher is selected and so on …

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35 ,…….

[Circular, multiple or random systematic methods also used]

Systematic Sampling contd..

16

Systematic Sampling (contd..)

Advantages –

Requires less time, sometimes less costly than SRS.

Ensures representativeness across list

Easy to implement

Disadvantages-

Works well only if the complete and up-to-date frame

is available and if the units are randomly arranged in

the frame; for this reason the units are arranged in

some order say alphabetical or in increasing/decreasing

order of value before selecting a sample.

17

3. Stratified sampling Principle

Classify population into homogeneous subgroups (strata)

Stratification may be done on the basis of income, age,rural-urban, Revenue-Capital, Treasuries, major heads, etc.

Draw sample (not necessarily equal) from each strata

Combine results of all strata

Advantages

More precise if variable associated with strata e.g. In MUS

sampling, variable is value of vouchers which is related to

strata so it is likely to yield better results than SRS

All subgroups represented, allowing separate conclusion about

each of them; say separate conclusion for each

state/District/treasury

Administrative convenience

18

3. Stratified sampling (contd..)

Disadvantages

Sampling error difficult to measure

Loss of precision if small numbers sampled in

individual heterogeneous strata

Example of stratified sampling:

(i) To select BPL households for a social audit; divide

the population of BPL into three categories (strata) say

top 25%, Middle 50% and Bottom 25% and select

separate samples from 3 categories/strata. (ii) Monetary

Unit Sampling (MUS) is also a case of Stratified

Sampling where the population is divided into 2 strata –

High value and low value vouchers/items

19

Allocation of Sample size in Stratified sampling

Proportional Allocation: ni = (n/N)*Ni

where ni is size of sample from ith strara, Ni is population of ithstrara; n is sample size and N is the population size

Optimum allocation ni’s are chosen so as to

(a) Maximise the precision for fixed sample size n; Neyman’sAllocation

(b) Maximise the precision for fixed cost

(c) Minimise the total cost for fixed desired precision

Disproportionate Allocation

No. of items selected from a stratum is independent of its size.

A large sample would be required from a stratum if

1. Stratum size Ni is large.

2. Stratum variability Si (variance/Std. Dev.) is large.

20

Exercise: Proportional Allocation Number of Vouchers coming from 3 treasuries are 300,

200 and 500 respectively. Draw a proportional stratified

sample of size 60 using the random number table given in

slide No.13.

Solution:

Here N1 = 300, N2 = 200 and N3 = 500; N = 1000

using ni = (n/N)*Ni; i = 1, 2, 3

n1 = (60/1000)* 300 = 18, n2 = (60/1000)* 200 = 12

and n3 = (60/1000)* 500 = 30.

Thus a sample of 18, 12 and 30 will be selected from

these three strata using random number table.

21

4. Cluster Sampling

The population is divided into non-overlappinggroups known as Clusters.

Clusters are commonly formed on the basis of geographical /administrative/political boundaries, e.g. GPs, Blocks, Departments may act as clusters.

Procedure

List all the clusters/groups of sampling units of the study population

Select Random Sample of clusters

Survey all or proportion of sampling units of selected clusters

For e.g. selecting some Districts from a state and auditing them leaving other Districts

22

Cluster Sampling (contd..)

Advantages

Simple: Complete list of units (sampling frame) is required only for clusters selected in the sample

Less travel/resources required

Disadvantages

Imprecise if clusters homogeneous (Large sample as compared to SRS is required for the same precision)

Sampling error difficult to measure

23

The two stages of a Cluster Sample

First stage: Probability proportional to size (PPS)• Find the number of clusters to be included

• Compute cumulative totals of the populations for each cluster with a grand total

• Divide the grand total by the number of clusters and obtain the sampling interval (k)

• Choose a random number less than k and identify the first cluster

• Add the sampling interval and identify the second cluster

• By repeating the same procedure, identify all the clusters to be selected

Second stage

In each selected cluster select a random sample of required number of units using a sampling frame of Basic Sampling Units in the cluster.

24

Let’s take treasuries as clusters/strata, the objective is to

select 30 clusters/strata i.e. 30 treasuries using PPS; size

being no. of vouchers in a treasury.

Procedure: List all Treasuries with number of vouchers in

them; find the cumulative totals of number of vouchers:

Treasury no. of vouchers Cumulative total

1 34 34

2 60 94

3 30 124

4 76 200

5 315 515 and so on

Total 4,715

Divide the cumulative total = 4715 by 30 – clusters to select

4,715 / 30= 157.1; Sampling Interval ‘k’ is 157

Selection of PPS Sample

25

Find a three digit random number [less than 157] say 123

Select the first cluster corresponding to 123 in cumulative Tot.

Select the remaining clusters from the cumulative distribution

by adding 157 (sampling interval) each time.

Treasury no. of vouchers Cum. total Cluster Selected

3 30 124 * selected

4 76 200

5 315 515 ** selected twice

(2nd 123+157=280)

(3rd 280+157=437)

Second Stage: In each selected cluster (treasury) choose required

number of vouchers by random or systematic selection.

Selection of PPS Sample contd…

26

• In both stratified and cluster sampling, the population is

divided into well-defined groups.

Stratified sampling is used when each group has small

variation (more homogeneity) within itself but wide

variation between the groups.

Cluster Sampling in used in the opposite case, when there

is considerable variation within each group but the groups

are essentially similar to each other.

In Stratified sampling estimate of each and every strata is

also available but not in cluster sampling.

Stratified Sampling Vs Cluster Sampling

27

Stratified Cluster Sampling - Cont.

• Suppose in a state there are 20 Districts;

• We take a sample of 15 villages in each of the 20

Districts to study the implementation of MGNREGA

• In all 300 villages are selected and studied

• This is an example of stratified sampling when

estimates of the desired characteristics for each of the

Districts (Strata) would also be available

• On the other hand let us select 5 districts out of 20 and

take a sample of 60 villages in each of the selected District

• In all 300 villages are selected and studied

• This is an example of Cluster Sampling

• In this case estimates of the desired characteristics for

each of the Districts (Cluster) would not be available

28

To obtain a sample of ‘n’ households in the country: the first

stage units may be states, the second stage units (SSUs) Distts.

from selected states, third stage units villages from selected

districts, ultimate stage units are households in the villages

Advantages Most feasible approach for large populations

No complete listing of units is required at various stages; second stage frame is required only for the selected first stage units.

This leads to great saving in operational cost.

Disadvantages

Several sampling lists Sampling error difficult to measure May be less efficient compared to a suitable single stage

sampling of the same size.

Multistage sampling – an example

1

Sampling in AUDIT - III

P. K. Dhamija Stat. Advisor

2

Sampling

The Process of selection of some members of a

population to generate precise and valid

estimates of population parameters like

averages or proportions.

Sample

A sample is a part of the population, selected by the

investigator/auditor as its representative to gather

information on certain characteristics of the original

population

3

Sampling Terms Sampling unit (Basic sampling unit)

Example: vouchers, cheques, bills, districts, audit units

Sampling frame

List of all sampling units in the population

Sampling scheme

Method used to select sampling units from the sampling

frame

Parameter : Population characteristic like average, proportion

based on all the units in the population; it is constant/fixed.

Statistic: Sample characteristic like average, proportion based

on sample values; it varies from sample to sample.

4

Advantages of Survey Sampling less expensive; Saves Time

The quality of information is maintained.

Possible to determine the extent of error due toSampling

Non Sampling errors are likely to be less

Even Census Results are verified by sampling

Law of Statistical Regularity states that amoderately large number of items chosen at randomform a large group are almost sure on the average topossess the characteristic of the large group –Forms the Basis of Sampling.

5

Disadvantages of Survey Sampling

Results of a sample survey are subject to error due

to sampling.

A sample may not properly represent the various

subgroups of a population.

Sometimes the sampling methods may become

complicated requiring the services of an expert.

Note: If time & money are not important factors

and if population under consideration is not too

large, census is better than any sampling method.

6

Non-Statistical sampling

Statistical sampling

Types of sampling

7

Non-Statistical Sampling Units in the study population do not have a known

probability of being included in the sample

Subjective/Biased samples

Used when (i) the number of elements in the population is either unknown or units in the population can not be identified and (ii) there are time/ resource constraints

Advantages:

Practical and easy to conduct

Dis-advantages:

Not representative of the population

Not possible to (i) assess the validity of estimates (ii) Determine sample size using statistical methods.

8

I Accidental/ Haphazard Sampling/

The auditor selects sample (audit units, bills, vouchers,

Districts) without any conscious bias; the sample is

expected to be representative of the population. For e.g.

avoiding first and last voucher in a bundle.

II Judgmental /Purposive Sampling

The auditor selects sample (audit units, bills, vouchers,

Departments) which in his opinion contains maximum

error say vouchers with the highest values or vouchers of

some particular treasury.

Some Non-Statistical Sampling

Techniques

9

Statistical Sampling Each unit in the study population has a known probability

(may not be equal) of being included in the sample.

Advantages:

It provides estimates free from personal bias

It permits application of objective methods of minimizing error under the resource constraints.

Allows to draw valid conclusions about population

Dis-advantages:

Needs sampling frame

Compared to Non Prob. sampling it is difficult to apply

Audit Sampling

Application of audit procedure to less than 100

% of the items/transactions for the purpose of

evaluating some characteristic of the items/

transactions under audit.

Use of audit sampling may not be possible/

advisable in auditing procedures involving

scanning accounting records for unusual items

(outliers), inquiries (Satyam case), most

analytical/detailed procedures, etc.

10

Need of Statistical Sampling in Auditing

No auditor can check 100% of auditable entities

because of resource constraints

To provide assurance based on test checks

Audit methodology is under increasing scrutiny

so auditor need to use scientific/statistical tools

& techniques

Conclusions based on statistical sampling can

stand scrutiny of auditee/ other professionals.

11

Advantages of Statistical Sampling Offer a means of estimating errors/misstatement in

quantifiable and reliable manner

Takes into account risk and materiality for determining

sample size and cost.

Offers a means of arriving at an optimum sample size to

avoid under or over auditing

Properly designed sampling estimates are unbiased and

transparent.

Helps in forming an opinion about the extent of audit

objection/ value of misstatement [non-levy/short-levy of

taxes] in the population with specified sampling risk (say

5%) - for e.g. with 95% confidence we can say that errors

in vouchers are b/w 2.6 – 3.0%

. 12

13

Sampling Error

No sample is a perfect mirror image of population

The estimates obtained vary from sample to sample.

The sampling variance is the measure of variability of a

sample estimator like variance of average or proportion.

The square root of the variance of the sample estimator

is called the standard error of the estimator.

The lesser the value of standard error, the more efficient

would be the estimator.

Use of Proper sample design and sufficient sample size

reduce sampling error and increase the efficiency of the

estimate(s) obtained by sampling.

14

include any misjudgement or mistakes by the auditor that may lead

to incorrect conclusion(s) based on audit. They occur even if full

population is examined.

By careful planning & supervision and by using appropriate audit

technique non sampling errors can be reduced but they can't be

eliminated. Some of the cases of non-sampling errors in audit are:

Selecting inappropriate audit procedures to achieve specific

objective. For e.g. an auditor checks controller’s signature on

voucher & not disbursement approval.

Auditor may fail to recognize misstatements (errors) included in

documents that he examines – Can you think some examples?

selecting inappropriate population for e.g. selecting only BPL

households for audit of a scheme involving payment of subsidy.

Auditor makes an error in evaluation (say totalling mistake or

skipping some vouchers containing error) of the results.

Non-Sampling Errors in Audit

15

Factors Affecting Sample Size Variability in the target population – measured by Standard

Deviation of the characteristic under audit

Margin of error or precision: a measure of possible difference

between the sample estimate and actual population value;

Feasibility/Resource Constraints - trade off between ideal

sample size and survey cost (cost of sampling per unit, time,

money, human resources)

Importance of the decision - extent of penalty for making

mistake

Nature of the analysis/audit intended (exploratory - small

sample or conclusive – large sample)

Sample sizes used in similar audits in the past

Expected Incidence/Deviation (error) rates – larger error

rates means larger sample size

Statistical Sampling Plans used in Audit

Attribute Sampling Plans or Attribute Sampling

Variable Sampling Plans or Variable Sampling

16

Attribute Sampling Plans or Attribute Sampling

An attribute is a qualitative characteristic which can not be measured quantitatively. However, the population may be classified into various classes w. r. t. the attribute

Attribute Sampling is used in Tests of Controls (TOC) i.e. to find out no. of deviations, proportion (%) of deviations etc. –it deals with ‘How Many’

Examples of attribute sampling:(a) Whether the Financial and Accounting system of Canteen Stores Department (CSD) adheres to the laid down standards & procedures?

(b) If the system of identifying targeted beneficiary in Social Security scheme was in place and was working effectively?

(c) Verifying signatures or approval stamp on a bill/voucher

(d) Entries posted in the correct account/Head?

17

Attribute Sampling - Types

Fixed sample size attribute sampling - objective is

to perform test of control to estimate the

deviation/error rate of a population.

Sequential (stop or go) attribute sampling used for

not so common cases; it prevents oversampling.

Discovery/Exploratory Sampling: observing at least

one deviation - very rare cases.

Block Sampling: It includes all items in a selected

time period/group say all vouchers of January or

all vouchers of a particular department/treasury.18

Actual Extent of Operating

Effectiveness of the Control Procedure is

Adequate Inadequate

The Test of Controls

Sample Indicates:

Extent of Operating

Effectiveness is

Adequate

Extent of Operating

Effectiveness

Inadequate

Sampling Risks -- Tests of Controls

Correct

Decision

Incorrect Decision

(Risk of Assessing

Control Risk

Too Low) - (β)

Incorrect Decision

(Risk of Assessing

Control Risk

Too High) - (α)

Correct

Decision

19

Practical Illustration of Attribute Sampling

Objective: To find out if the controls are operating

effectively or not?

We Assume

Risk of Assessing Control Risk Too Low (β) - 5%; it means 95% reliability

Tolerable Deviation Rate — 9 %

Expected Population Deviation Rate —2%

It means we assume: (i) 95% reliability i.e. 5% statisticalchance of concluding that the control is operatingeffectively when it is not (ii) we are ready to tolerate amaximum deviation/error rate of 9% - it is similar tomateriality (iii) Estimate of population deviation rate is 2%(based on prior knowledge like an audit conducted in thepast or based on some pilot) 20

Statistical Sample Sizes for Tests of Controls at 5 Percent Risk of Assessing Control Risk Too Low

21

Sample Size and Evaluating Attributes Sampling Results

Sample size using the table (previous slide) is 68 (2)

It means auditor should select a sample of 68 items.

Bracketed number - (2) means; if 2 or lessdeviations are observed in a sample of 68, we mayconclude that audit objective has beenaccomplished. We may conclude like:

“I believe that the deviation/error rate in the

population is less than 9 percent.” We will be wrong

5 % of the time when the deviation is exactly 9 %.

If the deviation rate is in excess of 9 % we will be

wrong even less than 5 % of the time.

Planned assessed level of control risk is achieved.22

Evaluating Attributes Sampling Results

Case 2: more than 2 deviations identified - Since the

bracketed number was (2), audit objective has

not been met. We may conclude like:

“The achieved deviation rate is higher than 9

percent.” Accordingly, auditor may not “rely” on

internal control to the extent planned.

23

Variable Sampling Plans Variable (or quantitative) sampling is used when

the objective is to estimate a quantity (like

amount of loss to government , average loss per

transaction, etc.); it deals with “How Much”

It is used primarily for substantive testing. Most

commonly used variable sampling plan is

Probability Proportional to Size (PPS) or Monetary

Unit Sampling (MUS)

MUS: It is a hybrid plan combining the

characteristics of attribute and variable sampling.24

Sampling Risk/Errors for Substantive Tests – Variable Sampling

Correct

Decision

Incorrect Decision Risk of Assessing

ControlRisk Too Low (β)

Risk of Assessing Control

Risk Too High (α)

Correct

Decision

Sample Results Indicate

No material Misstatements

material misstatements

No material misstatements

material misstatement

Actual Conclusion based on 100% check

25

Risk of incorrect rejection’ (Alpha risk) or

Risk of Assessing Control Risk Too High

Risk that sample supports the conclusion that the accountbalance is materially misstated when it is not.

Corresponds to risk of rejecting a correct null hypothesis.

10% Alpha risk means there is a prob./chance of 0.1

(10%) of concluding that there is a misstatement while

actually there is none.

Arises when the sample indicates a higher level of

errors/risk than is actually the case.

This situation is usually resolved by additional audit work

being performed i.e. large sample

affects audit efficiency but should not affect the validity of

the resulting audit conclusion26

Risk of incorrect acceptance’ (Beta risk)

Risk of Assessing Control Risk Too Low

Material error is not detected in a population because the

sample failed to select sufficient items containing errors.

Corresponds to risk of not rejecting a false null hypothesis

– in audit which is more serious – alpha or beta

It affects audit effectiveness,.

This risk is 10% means prob./chance of concluding that

there is no misstatement while actually there is a

misstatement; it also indicates 100 – 10 = 90% reliability.

To control this risk we increase precision and hence

sample size.

27

28

Sample Size required for Un-stratified MPU*

Ur = reliability coefficient: depends on confidence/reliability reqd.

SD= Standard Deviation; A = Precision i.e. materiality

n0 = sample size, N = population size;

SRSWOR: Simple Random Sampling without Replacement

*Un-stratified Mean Per Unit (MPU): Statistical sampling tech. (not involving stratification) whereby sample mean is calculated and projected as an estimated total.

N

n

nn

asSRSWORforreducedfurther

becansizesamplethehighisN

nif

A

NSDUn

o

o

o

ro

1

;

..2

29

Exercise -steps of un-stratified MPU

ABC Limited desires 95 per cent reliability and

plans to use unrestricted (un-stratified) random

sampling without replacement to estimate the

value of inventory of a subsidiary. To estimate

mean and standard deviation (SD) of the

inventory population, a pilot sample of 30 items

from the total population of 2000 items was

selected. The pilot sample produced an

arithmetic mean of Rs. 4000 and a (SD) of Rs.

150.

30

Exercise cont.

95 per cent reliability means firm is willing to

tolerate 5 per cent chance of sampling error.

That is, 5 % of the time, projection plus/minus

precision may not include the true population total.

Based on 95 % reliability, reliability coefficient (UR)

is 1.96 – based on normal tables.

Precision (A) is judgmentally set equal to Rs. 60,000

(termed as materiality) – the amount considered

material for this application.

Using the Formula, we have the sample size as:

For SRSWOR sample size reduces to

31

32

Sixty- two additional sample items are added to

the pilot sample of 30 to yield the total sample

of 92. The 62 additional sample items are

selected using SRSWOR.

•A standard deviation based on 92 items is calculated. Assume that standard deviation is Rs. 136.

Revised precision will be calculated as:

33

Exercise cont. Calculated precision A’ is less than predefined

precision (materiality) A = Rs. 60000; so the sample size is adequate. What happens if revised precision A’ is greater than materiality?

The mean of the 92 inventory items is calculated as follows, assuming the sample totals Rs. 370977

Estimated Population Total Value (EV) = Rs. 4032.36 * 2000 = Rs. 8064,720.

34

Exercise cont.

ABC Limited is thus 95 per cent certain that the true inventory balance of all 2000 inventory items is within Rs. 8,064,720 +Rs. 54,273. (Calculated Precision)

ABC Limited should book Rs. 8,064,720 as inventory.

35

Attribute Sampling: Sample Size

36

2

2.)1.(.

A

PPZ r

Where Zr = confidence level coefficient = 1.96for 95% level of confidence, A= margin of error(we are prepared to accept normally 10 or 20%)P= Proportion of errors expected in thepopulation.

Objective is to estimate the number (proportion) of

audit objections (errors); estimated number of audit

objections in the population is sample proportion of

error multiplied by number of items in the population.

The optimum sample size under SRSWR is:

37

Estimating P: The formula requires the

knowledge of P, expected proportion of errors in thepopulation. However, this is what we are trying toestimate and is unknown

Ways to estimate proportion :

A pilot or preliminary sample. Observations usedin the pilot study can be counted as part of thefinal sample

Estimates may be available from previous auditreports and the upper bound of P can be used inthe formula

If impossible to obtain a better estimate, set p = 0.5 in the formula to yield maximum value of n

Reporting the Results

When reporting the results of a Sample it is important to cover the following key factors:

The Sample Size

The Sample selection methodology

The Estimate(s) resulting from the Sample

The precision (Std. Error) and confidence intervals for the Estimate(s)

38

Monetary Unit Sampling MUS MUS is nothing but Probability Proportional to

Size (PPS) - systematic sampling, where oneassigns high inclusion probability to thetransactions having high value.

In MUS method the sampling unit is not an invoice or any other physical unit, but an individual rupee. However, when the individual rupee is selected, the auditor does not verify just that particular rupee, but the rupee acts as a hook and drags the whole invoice with it.

Difficult to apply/understand manually and can be explained using IDEA

39

Control Measures for Non Sampling Errors

1

Sampling Errors arise due to:

• POPULATION SPECIFICATION ERROR—when the researcher does not understand who (s)he should survey. For e.g. in a survey about breakfast cereal consumption - who should be surveyed? The mother makes the purchase decision, but the children influence her choice.

• SAMPLE FRAME ERROR—when the wrong sub-population is used to select a sample. For e.g. if the sample frame is from car registrations and telephone directories. The results may be wrongly predicted.

• SELECTION ERROR—This occurs when only those that are interested respond. It can be controlled by pre-survey contact requesting cooperation, actual surveying, post survey follow-up if a response is not received.

• NON-RESPONSE—Non-response errors occur when respondents are different than those who do not respond. The extent of this non-response error can be checked through follow-up surveys etc.

• SAMPLING ERRORS—These errors occur because of variation in the number or representativeness of the sample that responds.

• These errors can be controlled by (1) careful sample designs, (2) large samples and (3) multiple contacts to assure representative response.

2

Non Sampling Errors: Types Conceptual Errors:

• Lack of qualified and suitable enumerators

• Lack of proper training of field staff to make them thorough with the concepts and definitions involved

Errors of Recording/ Transcription: Due to carelessness and negligence of the auditor

Errors of Inaccurate Measurement: Due to erroneous figure of measurement given by the informant/auditee

3

Errors in Totalling:

When there are many items to be totalled up

Totalling of subtotals may quite often lead to such errors

Errors of Omission:

When field worker fails to ask certain questions in the block

Due to non availability of required information

Non Sampling Errors: Types Contd…

4

Bias of the Interviewer:

Due to inadequate training or partial understanding of instructions

Putting a question in a specific way or telling suggestive answers

Errors of Inconsistency: When data are inconsistent with the similar information collected in some other part of the same schedule

Non Sampling Errors: Types Cont.

5

Response Error: Due to wrong notion present in the mind of

respondent Due to some kind of fear Due to wrong understanding of questions Due to Illiteracy Due to lack of clarity in questions Due to deliberate poor response.

Error due to Prestige/ Self interest: Due to prestige, pride or self interest, informant may introduce bias by upgrading education, expenditure and downgrading age, income, etc.

Non Sampling Errors: Types Contd.

6

Errors due to recall lapse: If the recall period is longer, answers may be based on guess or averages Error due to absence of right informant Error due to incorrect identification of sampling

units (say wrong marking of boundaries)• Boundaries not correctly identified due to lack of

adequate effort or due to misguidance by some person

Errors due to longer reference period: • Inclusion of information pertaining to period

out of reference period • Exclusion of information pertaining to the

period within the reference period .

Non Sampling Errors: Types Contd…

7

Methods of Controlling Non Sampling Errors

Recruitment of proper primary field worker who has got:

• Aptitude for field work

• Good knowledge of the survey area/local language

• Proper academic qualification

• Tactfulness and resourcefulness

Training: Required for facilitating and understanding the sampling design, various concepts & definitions, schedules of enquiry and procedure of data collection

8

Methods of Controlling Non Sampling Errors - Cont.

• Purpose of training/workshops is to bring uniformity in concepts and procedures

• Active participation by primary field workers and supervisors

Inspection/ Supervision:

• On the spot verification

• Instant feedback to the investigating staff

• Inspection norms

Probing:

• Probing questions should be simple

• Should not create any sort of bitterness

Cross checking

9

Methods of Controlling Non Sampling Errors - Cont.

Scrutiny and Super Scrutiny

Monthly Meetings

Feedback Reports

Role of experienced field staff in improvement of quality

Amendments in the Schedules and Clarifications: Pilot Survey

10

Risk Based Audit Approach Session 7.1

Note 7.1 1

Monetary Unit Sampling Session Overview This session is the last of the sessions on audit sampling. Session 5.1 was on the basic concepts of sampling and Session 6.1 on the application of attribute sampling for control procedures. This session will cover the basics of another type of sampling, the Monetary Unit Sampling and its application for substantive test of details. Actually, Monetary Unit Sampling (MUS) is used widely both for • Test of Controls • Account Balance In this session, we will discuss MUS for Substantive Test of Details covering the following important points: • Monetary Unit Sampling (MUS),

advantages and limitations of MUS and it’s application to substantive test of details

• Determining the sample size for substantive test of details; and

• Evaluation of results of substantive test of details.

. Learning Objectives At the end of the session, you would be able to apply MUS for substantive test of details, to the extent that, the steps are followed correctly and keeping in mind the advantages and limitations of MUS. The session is expected to provide only a broad overview of MUS and is not meant to impart expertise on MUS. Participants may read additional study material suggested in the bibliography for further knowledge.

Basic Concepts Some of the new concepts that would be discussed in this session are explained below: (a) Monetary Unit Sampling: Monetary Unit Sampling (MUS) is a sampling method in which the sampling unit is not an invoice or any other physical unit, but an individual rupee. However, when the individual rupee is selected, the auditor does not verify just that particular rupee, but the rupee acts as a hook and drags the whole invoice with it. For example, if as a result of sample selection, Rs.365 is selected for testing and if that rupee falls in voucher number 14, then that voucher will be audited and its quality assigned to the sampling unit. Let us assume that there are 6 items out of which 2 items are to be selected. The value of the 6 items are 100, 200, 300, 400, 500 and 1000. If attribute sampling is used to select the 2 items, then all the items have equal chance of selection, as the sampling unit would be individual item. On the other hand, if MUS is used, then the total value of 6 items works out to Rs.2500, i.e., there are 2500 sampling units. As 2 items are to be selected, the sampling interval would be 2500/2 (Rs.1250). This means that one rupee out of every 1250 rupees would be selected. In such a case, the chances of 1000 rupee item getting picked up is 10 times more than the 100 rupee item getting picked up. Thus MUS has a bias towards high value items. (b) Most Likely Error: Most Likely Error (MLE) is an estimation of the error in the population. Initially MLE will be estimated based on past experience and used for determining the sample size. After carrying out substantive test of details, the MLE will be projected based on actual sample results and used for drawing audit conclusions.

Risk Based Audit Approach Session 7.1

Note 7.1 2

(c) Basic Precision: Basic Precision (BP) is the allowance for errors which exist but no evidence of that is found in the sample. It is dependent on the confidence level and the size of the sample and is present even where no errors have been found in the sample. (d) Precision Gap Widening: Precision Gap Widening (PGW) is the additional allowance that must be made in the Precision as a result of errors in the sample. (e) Planned Precision: Planned Precision (PP) is the difference between Upper Error Limit and the Most Likely Error, i.e., Planned Precision = Materiality - Most Likely Error = Basic Precision + Precision Gap Widening. (f) High value and key items: The Auditor may decide that all items/ transactions above a particular monetary value are to be audited 100%. These items are called high value items. For e.g., the auditor may decide that all items above Rs.100,000 are to be audited fully. Similarly the auditor using his judgement may decide that some items due to their nature are prone to error. Such items are called key items. For e.g., if there is a complete breakdown of controls in a particular division in one account area, he may treat all items relating to that account area from that division as key items and check all these items. It is to remembered that all high and key items are to be deducted from population to arrive at representative population for sampling. (g) Maximum Possible Misstatements or

Upper Error Limit: The Upper Error Limit (UEL) is the maximum possible error estimated in the population as a result of the substantive test of samples. If the Upper Error Limit is above the materiality limit, then the auditor will either perform further substantive tests

to check whether there is a material error or conclude that there is a material error. (h) Tainting: Tainting is the percentage of error found in monetary terms in a sample item. For e.g., if the accounts receivable balance of ‘x’ as per financial statement is Rs.10,000 and its value as per auditor’s findings is Rs.8,000 then the tainting would be (Rs.10,000-8,000)/10,000 = 20%. MUS – Advantages, Limitations and Relevance to Substantive Tests As described previously, substantive tests are those tests of transactions and balances that seek to provide evidence as to the completeness, accuracy and validity of information in the financial statements. The objective of substantive testing is to obtain reasonable assurance that financial statement assertions individually and together correspond to established criteria within limits not exceeding materiality. Thus, substantive tests are intended to determine the monetary effect of errors in the financial statements. For example, the aim of substantive tests of accounts receivable could be to check the extent to which the balance is overstated. For substantive test of details (audit of individual transactions), we must use sampling to select individual items for examination. In MUS, as explained earlier, each rupee is treated as a sampling unit and acts as a hook for the physical unit in which it occurs; conclusions on the physical unit in monetary terms can be reached. The results from the tests of sample are then used to project the most likely error and the upper error limit in the population. As MUS helps in arriving at audit conclusions in monetary terms with quantification of the degree of confidence in the result, it is the preferred method of sampling for substantive test of details.

Risk Based Audit Approach Session 7.1

Note 7.1 3

Advantages of MUS The important advantages of MUS are: (i) It normally produces smaller sample

sizes than other substantive sampling plans.

(ii) There is no difficulty in expressing a conclusion in monetary terms.

(iii) The application of rupee unit sampling is not contingent on knowledge of the population size. This permits sample selection to be started before the total value of the final population is known. As will be explained later in the session, the average sampling interval can be worked out without details of population size.

(iv) No rupee stratification is necessary, as this will be accomplished automatically, thus avoiding problems of determining optimum strata boundaries and allocation of sample size among strata.

(v) It is relatively easy to apply compared to other sampling plans.

(vi) The problem of detecting the large but infrequent errors is solved, since all items greater than the sampling interval will be selected.

Disadvantages of MUS The main disadvantages of MUS are the following: (i) Accounts/items with nil balances will

have no chance of selection. (ii) The more an item is understated; the

less likely the item has a chance of selection. Hence, MUS is less useful for finding understatements.

(iii) A large percentage error in a small transaction can significantly increase the computed error limit.

(iv) It is very difficult to use MUS in a non-computerized environment as totaling the sample items in the population for the purpose of finding

out the particular item in which the dollar falls, is an onerous task.

(v) MUS is more time consuming than other sampling plans, as the number of sampling units (rupee) is higher than in attribute sampling (physical unit like voucher, cheque).

Steps involved in MUS The stages in MUS are: (a) determining the sample size; (b) selecting the sample for performing

substantive test of details; and (c) evaluation of sample test results. Steps involved in performing these stages are explained below: (a) Determining the Sample Size The steps involved in determining the sample size are as follows: (i) Set Upper Error Limit (UEL) (mostly

equal to materiality). (ii) Subtract the estimated Most Likely

Error (MLE) (usually based on prior knowledge, i.e. the results of last year’s audit; in the absence of sound prior knowledge approximately 15-20 percent of materiality is a good rule of thumb)

(iii) Subtract Precision Gap Widening (PGW) (approximately 1/2 materiality is a good rule of thumb, but can vary depending upon the number and magnitude of errors anticipated and the assurance level).

(iv) Obtain Basic Precision (BP): BP = UEL -MLE- PGW.

(v) For the given assurance level, use the Assurance table to determine the basic precision factor.

(vi) Calculate the sampling interval by using the formula: Average Sampling Interval (ASI) = BP/BP factor.

(vii) Deduct the high value and key value items from the total population to arrive at the estimated representative population.

Risk Based Audit Approach Session 7.1

Note 7.1 4

(vii) Calculate the sample size by using the formula: Sample size = representative population/ average sampling interval.

Example 7.1.1 An auditor is performing substantive tests on the valuation of buses in the company. The total value of the fleet is Rs.25,000,000. The materiality limit established for the audit is Rs.1,000,000. The auditor anticipates an error of Rs.250,000 (based on past experience). The auditor estimates the precision gap widening of Rs.150,000. The auditor wants 90 percent assurance from the substantive tests. The auditor has identified high-value items of Rs.1,000,000 and key-value items of Rs.500,000 in the population. Using the table of basic precision and precision gap widening factors (Appendix - I), the basic precision, average sampling interval and representative sample size are calculated as explained below: Calculation of Basic Precision 1. Upper Error Limit (equal to materiality) Rs.1,000,000 2. Less: anticipated most likely errors Rs.250,000 3. Estimated precision

available (1-2) Rs.750,000 4. Less: estimated precision

gap widening Rs.150,000 5. Basic precision (3-4) Rs.600,000 (A) Calculation of Average Sampling Interval 1. Basic precision for audit Rs.600,000 2. Assurance level expected 90% 3. Basic precision factor

(from table for 90%) 2.3

3. Average sampling interval

(1/3) Rs.261,000(B) Calculation of Sample Size 1. Average sampling interval Rs.261,000(B)

as above

2. Total population Rs.25,000,000 3. Less: High-value items Rs.1,000,000 4. Less: Key-value items Rs.1,000,000 5. Representative test

population Rs.23,500,000 6. Representative sample

size (5/1) 90 Thus the representative sample size for substantive test of details will be 90. (b) Selecting the Sample Out of the four samples selection methods

is described in Session 5.1 Basic concepts of statistical sampling, the two methods that are used in MUS are systematic selection and cell selection. In systematic selection, one or two items are selected randomly and the average sampling interval is added to arrive at the other items to be selected. In cell selection, the population is divided into various cells and one item is selected from each cell. Systematic selection ensures that all items whose value is higher than the average sampling interval are automatically selected. In Cell selection all items whose value is twice more than the sampling interval are selected automatically. Thus, chances of high-value but infrequent error escaping audit scrutiny are avoided. After selecting the sample refer to Handout 7.1.3 go through it before we discuss it in detail.

(c) Evaluating Results of Substantive Test of Details

Risk Based Audit Approach Session 7.1

Note 7.1 5

The steps involved in evaluating the results of substantive test of details are: (i) Calculate the percentage of tainting

for individual items. (ii) Add the individual tainting

percentages to arrive at net tainting percentage.

(iii) Calculate the Most Likely Error (MLE) for the representative sample by using the formula: MLE = Net tainting percentage * Average Sampling Interval (ASI)

(iv) Add the errors in the high-value and key-value items to the MLE to arrive at total most likely errors as follows: Total most likely error = MLE + error in high value items + error in key value items

(v) Arrange the tainting according to the percentage of tainting, in descending order, for overstatement and understatement separately. Only tainting found out from the representative sample are to be used for this purpose and tainting in high- and key-value items is to be excluded.

(vi) Calculate the tainting adjusted PGW factor for each tainting sorted as at (v). Tainting adjusted PGW factor = Tainting percentage * PGW factor.

(vii) Total the tainting adjusted PGW factors for overstatement and understatement separately.

(viii) Calculate overstatements and understatements PGW of separately: PGW = sum of tainting adjusted PGW factors * ASI.

(ix) Calculate UELs for overstatement and understatement separately: UEL(overstatement) = MLE + BP + PGW UEL(understatement) = MLE - BP – PGW

Example 7.1.2

In the example 7.1.1 discussed earlier, assume that the auditor has identified the following errors: Item . Book Audited No. value value Representative sample: 14 Rs.5,000 Rs.3,000 24 Rs.7,500 Rs.1,500 16 Rs.4,000 Rs.6,000 High value: 28 Rs.30,000 Rs.21,000 Using the basic precision and precision gap widening factors table (Appendix 7.1-A), the total most likely error, the upper error limits for overstatement and understatement and audit conclusion can be arrived at as explained below: Calculation of Total Most Likely Error 1. Net sample error tainting will be: Item Book Audit number value value Error Tainting 14 Rs.5,000 Rs.3,000 Rs.2,000 40% 24 Rs.7,500 Rs.1,500 Rs.6,000 80% 16 Rs.4,000 Rs.6,000(Rs.2,000) (50%) Net tainting 70% 2. Net most likely error for representative

sample will be: Sum of net tainting percentage % x

Average sampling interval We know that average sampling interval is Rs. 261,000 (B) in Example 7.1.1. 70% X Rs.261,000 = Rs.182,700 3. Error in high-value item= Rs.30,000-

Rs.21,000=Rs.9,000

Risk Based Audit Approach Session 7.1

Note 7.1 6

4. Total most likely error 2+3 =Rs.191,700(X)

Calculation of Upper Error Limits 1. Tainting adjusted Precision Gap

Widening factors

(a) Overstatements

Taintings Ranked PGW Tainting by size factors adjusted (from table) PGW factors 40% 2nd 0.43 40%X0.43=0.172 80 1st 0.59 80%X0.59=0.472

(b) Understatements 50% 1st 0.59 50%X0.59=0.295 2. Precision Gap Widening PGW = Sum of tainting adjusted PGW

factors x Average sampling interval (a) Overstatements = 0.644 X

261,000=Rs.167,040 (Y)

(b) Understatements =0.295 X 261,000=Rs.76,995 (Z)

3. Upper Error Limits

Basic Precision is Rs.600,000 as worked out at (A) in Example 7.1.1

Overstatements Understatements Rs. Rs. Basic Precision 600,000 (600,000) PGW 167,040 (76,995) (Y) as above (Z) as above Total Precision 767,040 (676,995) Add total most likely error 191,700 191,700 (X) as above

Upper Error Limits 958,740 (485,295) Audit Conclusions The most likely error in the population is Rs.191,700 overstatement. The UELs at 90 percent confidence are that the overstatement is at most Rs.958,740 and the understatement at the most is Rs.485,295. As the UELs are less than the materiality limit the auditor can conclude that there is no material error. If the UEL is more than the materiality, then the Auditor will have to conclude either that there is a material misstatement in the accounts or increase the quantum of substantive test of details. Thus MUS is used, both at the stage of planning the sample size and subsequently at the stage of evaluation of sample results of substantive test of details. We will now discuss Handout 7.1.3 performing substantive test of details Summary The key points that were discussed in the session are: • Monetary Unit Sampling (MUS) –

Advantages, limitations and relevance of MUS for substantive test of details

• Steps involved in MUS • Determining the sample size for

substantive test of details • Selecting the sample for performing

substantive test of details and • Evaluation of results of substantive

test of details.

Application of Statistical

Sampling in

Performance Audit of

National Rural Employment

Guarantee ACT

(MGNREGA)

1

2

NREGA Background information MGNREGA: All-India coverage of rural areas;

Administrative set up in the country: State level

District Level Block Level GP level

Beneficiary Level.

Through MGNREGA, the govt. was committed to

provide employment to every rural family (at min.

wages) which demand such work and whose adult

members volunteer to do such work.

Objectives : (i) enhancement of livelihood security of

households (hhs) in rural areas by providing guaranteed

wage employment (ii) creation of durable assets;

Principle implementing agency is Gram Panchayat (GP)

Audit Objectives

Structural mechanisms were in place and adequate

capacity building measures were taken by the Central

and state govts. for implementation of the Act;

Procedures for preparing perspective (long term) and

annual plan at different levels for estimating the likely

demand for work were adequate and effective;

Funds were released, accounted for and utilised by the

governments in compliance with the provisions of the

Act and other extant/existing rules;

Process of registration of households, allotment of job

cards and allocation of employment in compliance

with the Act and rules was effective5

Audit Objectives – Cont.• livelihood security was provided by giving 100 days of emp. to

hhs in rural areas on demand & wages as declared, were paid;

• MGNREGS works were efficiently and effectively executed in a

time-bound manner and in compliance with the Act and Rules,

and durable assets were created, maintained and accounted for;

• Convergence of the Scheme with other rural development

programmes as envisaged was effectively achieved in enhancing

the employment opportunities under MGNREGS;

• All required records at various levels were properly maintained

and MGNREGS MIS data was accurate, reliable and timely;

• Transparency was maintained by involving all stakeholders in

various stages of its implementation;

• Effective mechanism at Centre and state level existed to assess

the impact of MGNREGS on individual households, local labour

market, migration cycle and efficacy of assets created. 6

7

Sampling plan

Selection of Districts (1st Stage Unit): Each state was

stratified into 2-5 strata depending on geographical

contiguity; within a strata 25% districts selected with

SRSWOR subject to minimum of 2 Districts.

Selection of Rural Blocks (2nd SU): Within each

selected district, 2 -3 blocks were selected again with

SRSWOR. (2 blocks if no. of blocks in selected district

are < 10.)

Selection of GPs (3rd SU): Within each selected block

25% of GPs (max. 10) were selected by PPSWOR with

size measure as Number of job cards or any other

similar proxy parameter like no. of applicants under

NREGA or No. of BPL population or Population size.

8

Sampling Plan Selection of Works: (Final SU - I): For each selected

GP 10 works (including incomplete works and

sanctioned in different years) using SRSWOR were

selected; care was taken to select different types of

works like rural connectivity, afforestation, canal

works, wasteland development etc.

Selection of beneficiaries: (Final SU - II): For each

selected GP, 10 beneficiaries were to be selected

using systematic sampling. (min. 2 SC/ST)

Thus a Multi stage sampling plan was used for each state.

Scope for Improvement

Observations by EPoD [Evidence for

Policy Design] Centre for International

Development (CID), Harvard Univ., USA

after finalisation of audit

9

Scope for Improvement

• Selection Process used (i) ensured equal

probability to each district (ii) Neglected wide

variation in NREGA spending (varies b/w Rs.

0 to 338 crores) & rural population (varies b/w

5620 to 61 lakhs) of Districts

• FAAM guidelines indicate, “Where either

monetary values or assessed risk vary widely,

alternative methods are preferred to SRS”.

10

Approaches discussed by EPoD, CID

• 3 Indicators viz. (i) NREGA Exp. - MUS (ii) Rural

Population and (iii) Person Days Worked could have

been used for selection of Districts.

• They would have ensured – (i) Each Rupee of

Spending becomes equally likely to be selected (ii)

Potential population which qualifies for NREGA and

(iii) Intensity of implementation in a district.

• Districts could have been stratified based on one of

these 3 variables and 4th quartile (top 25% values)

assigned high probability - up-to 100%

11

Approaches discussed by CID – Cont.

• Selecting one District from states where no District

is selected by above technique so that each state is

represented in the sample.

• Rural population approach would have uncovered a

greater number of areas with high rural population

but low MNREGA coverage.

• Person days sample has 2 shortcomings as compared

to total Exp. Approach (i) It may cover less districts

where material expenses exceeded 40% of total

NREGA Exp. (ii) Proportions of households

reaching 100 days of work is artificially high. 12

Potential Extensions of methodology

• 100% Sampling in Key Areas like districts

with highest exp.

• Sampling on Multiple variables like Rural

population and NREGA Exp. taken together

• Using flags to sample Key Districts like

districts where material expenses exceed

40%

• Purposeful selection of works near Pradhan’s

residence.

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