ICAI Notes- Sampling | Quantitative Aptitude for CA Foundation PDF Download

 Page 1


13.43 SAMPLING
13.2.1 INTRODUCTION 
There are situations when we would like to know about a vast, innite universe or population. 
But some important factors like time, cost, efciency, vastness of the population make it almost 
impossible to go for a complete enumeration of all the units constituting the population. Instead, 
we take recourse to selecting a representative part of the population and infer about the unknown 
universe on the basis of our knowledge from the known sample. A somewhat clear picture would 
emerge out if we consider the following cases. 
In the rst example let us share the problem faced by Mr. Basu. Mr. Basu would like to put a big 
order for electrical lamps produced by Mr. Ahuja’s company “General Electricals”. But before 
putting the order, he must know whether the claim made by Mr. Ahuja that the lamps of General 
Electricals last for at least 1500 hours is justied.
Miss Manju Bedi is a well-known social activist. Of late, she has noticed that the incidence of a 
particular disease in her area is on the rise. She claims that twenty per cent of the people in her 
town have been suffering from the disease.
In both the situations, we are faced with three different types of problems. The rst problem is 
how to draw a representative sample from the population of electrical lamps in the rst case and 
from the population of human beings in her town in the second case. The second problem is to 
estimate the population parameters i.e., the average life of all the bulbs produced by General 
Electricals and the proportion of people suffering form the disease in the rst and second examples 
respectively on the basis of sample observations. The third problem relates to decision making 
i.e., is there enough evidence, once again on the basis of sample observations, to suggest that the
claims made by Mr. Ahuja or Miss Bedi are justiable so that Mr. Basu can take a decision about
buying the lamps from General Electricals in the rst case and some effective steps can be taken
in the second example with a view to reducing the outbreak of the disease. We consider tests of
signicance or tests of hypothesis before decision making.
13.2.2 BASIC PRINCIPLES OF SAMPLE SURVEY 
Sample Survey is the study of the unknown population on the basis of a proper representative 
sample drawn from it. How can a part of the universe reveal the characteristics of the unknown 
universe? The answer to this question lies in the basic principles of sample survey comprising 
the following components:
(a) Law of Statistical regularity
After reading this unit a student will learn -
? Different procedure of sampling which will be the best representative of the population;
LEARNING OBJECTIVES
UNIT 2 SAMPLING
© The Institute of Chartered Accountants of India
Page 2


13.43 SAMPLING
13.2.1 INTRODUCTION 
There are situations when we would like to know about a vast, innite universe or population. 
But some important factors like time, cost, efciency, vastness of the population make it almost 
impossible to go for a complete enumeration of all the units constituting the population. Instead, 
we take recourse to selecting a representative part of the population and infer about the unknown 
universe on the basis of our knowledge from the known sample. A somewhat clear picture would 
emerge out if we consider the following cases. 
In the rst example let us share the problem faced by Mr. Basu. Mr. Basu would like to put a big 
order for electrical lamps produced by Mr. Ahuja’s company “General Electricals”. But before 
putting the order, he must know whether the claim made by Mr. Ahuja that the lamps of General 
Electricals last for at least 1500 hours is justied.
Miss Manju Bedi is a well-known social activist. Of late, she has noticed that the incidence of a 
particular disease in her area is on the rise. She claims that twenty per cent of the people in her 
town have been suffering from the disease.
In both the situations, we are faced with three different types of problems. The rst problem is 
how to draw a representative sample from the population of electrical lamps in the rst case and 
from the population of human beings in her town in the second case. The second problem is to 
estimate the population parameters i.e., the average life of all the bulbs produced by General 
Electricals and the proportion of people suffering form the disease in the rst and second examples 
respectively on the basis of sample observations. The third problem relates to decision making 
i.e., is there enough evidence, once again on the basis of sample observations, to suggest that the
claims made by Mr. Ahuja or Miss Bedi are justiable so that Mr. Basu can take a decision about
buying the lamps from General Electricals in the rst case and some effective steps can be taken
in the second example with a view to reducing the outbreak of the disease. We consider tests of
signicance or tests of hypothesis before decision making.
13.2.2 BASIC PRINCIPLES OF SAMPLE SURVEY 
Sample Survey is the study of the unknown population on the basis of a proper representative 
sample drawn from it. How can a part of the universe reveal the characteristics of the unknown 
universe? The answer to this question lies in the basic principles of sample survey comprising 
the following components:
(a) Law of Statistical regularity
After reading this unit a student will learn -
? Different procedure of sampling which will be the best representative of the population;
LEARNING OBJECTIVES
UNIT 2 SAMPLING
© The Institute of Chartered Accountants of India
STATISTICS
13.44
(b) Principle of Inertia
(c) Principle of Optimization
(d) Principle of Validity
(a) According to the law of statistical regularity, if a sample of fairly large size is drawn from 
the population under discussion at random, then on an average the sample would posses 
the characteristics of that population.
 Thus the sample, to be taken from the population, should be moderately large. In fact larger 
the sample size, the better in revealing the identity of the population. The reliability of a 
statistic in estimating a population characteristics varies as the square root of the sample 
size. However, it is not always possible to increase the sample size as it would put an extra 
burden on the available resource. We make a compromise on the sample size in accordance 
with some factors like cost, time, efciency etc.
 Apart from the sample size, the sample should be drawn at random from the population which 
means that each and every unit of the population should have a pre-assigned probability to 
belong to the sample.
(b) The results derived from a sample, according to the principle of inertia of large numbers, are 
likely to be more reliable, accurate and precise as the sample size increases, provided other 
factors are kept constant. This is a direct consequence of the rst principle.
(c) The principle of optimization ensures that an optimum level of efciency at a minimum cost 
or the maximum efciency at a given level of cost can be achieved with the selection of an 
appropriate sampling design.
(d) The principle of validity states that a sampling design is valid only if it is possible to obtain 
valid estimates and valid tests about population parameters. Only a probability sampling 
ensures this validity.
  13.2.3  COMPARISON BETWEEN SAMPLE SURVEY AND COMPLETE 
ENUMERATION
When complete information is collected for all the units belonging to a population, it is dened as 
complete enumeration or census. In most cases, we prefer sample survey to complete enumeration 
due to the following factors:
(a) Speed: As compared to census, a sample survey could be conducted, usually, much more 
quickly simply because in sample survey, only a part of the vast population is enumerated.
(b) Cost: The cost of collection of data on each unit in case of sample survey is likely to be more 
as compared to census because better trained personnel are employed for conducting a 
sample survey. But when it comes to total cost, sample survey is likely to be less expensive 
as only some selected units are considered in a sample survey.
(c) Reliability: The data collected in a sample survey are likely to be more reliable than that in 
a complete enumeration because of trained enumerators better supervision and application 
of modern technique.
© The Institute of Chartered Accountants of India
Page 3


13.43 SAMPLING
13.2.1 INTRODUCTION 
There are situations when we would like to know about a vast, innite universe or population. 
But some important factors like time, cost, efciency, vastness of the population make it almost 
impossible to go for a complete enumeration of all the units constituting the population. Instead, 
we take recourse to selecting a representative part of the population and infer about the unknown 
universe on the basis of our knowledge from the known sample. A somewhat clear picture would 
emerge out if we consider the following cases. 
In the rst example let us share the problem faced by Mr. Basu. Mr. Basu would like to put a big 
order for electrical lamps produced by Mr. Ahuja’s company “General Electricals”. But before 
putting the order, he must know whether the claim made by Mr. Ahuja that the lamps of General 
Electricals last for at least 1500 hours is justied.
Miss Manju Bedi is a well-known social activist. Of late, she has noticed that the incidence of a 
particular disease in her area is on the rise. She claims that twenty per cent of the people in her 
town have been suffering from the disease.
In both the situations, we are faced with three different types of problems. The rst problem is 
how to draw a representative sample from the population of electrical lamps in the rst case and 
from the population of human beings in her town in the second case. The second problem is to 
estimate the population parameters i.e., the average life of all the bulbs produced by General 
Electricals and the proportion of people suffering form the disease in the rst and second examples 
respectively on the basis of sample observations. The third problem relates to decision making 
i.e., is there enough evidence, once again on the basis of sample observations, to suggest that the
claims made by Mr. Ahuja or Miss Bedi are justiable so that Mr. Basu can take a decision about
buying the lamps from General Electricals in the rst case and some effective steps can be taken
in the second example with a view to reducing the outbreak of the disease. We consider tests of
signicance or tests of hypothesis before decision making.
13.2.2 BASIC PRINCIPLES OF SAMPLE SURVEY 
Sample Survey is the study of the unknown population on the basis of a proper representative 
sample drawn from it. How can a part of the universe reveal the characteristics of the unknown 
universe? The answer to this question lies in the basic principles of sample survey comprising 
the following components:
(a) Law of Statistical regularity
After reading this unit a student will learn -
? Different procedure of sampling which will be the best representative of the population;
LEARNING OBJECTIVES
UNIT 2 SAMPLING
© The Institute of Chartered Accountants of India
STATISTICS
13.44
(b) Principle of Inertia
(c) Principle of Optimization
(d) Principle of Validity
(a) According to the law of statistical regularity, if a sample of fairly large size is drawn from 
the population under discussion at random, then on an average the sample would posses 
the characteristics of that population.
 Thus the sample, to be taken from the population, should be moderately large. In fact larger 
the sample size, the better in revealing the identity of the population. The reliability of a 
statistic in estimating a population characteristics varies as the square root of the sample 
size. However, it is not always possible to increase the sample size as it would put an extra 
burden on the available resource. We make a compromise on the sample size in accordance 
with some factors like cost, time, efciency etc.
 Apart from the sample size, the sample should be drawn at random from the population which 
means that each and every unit of the population should have a pre-assigned probability to 
belong to the sample.
(b) The results derived from a sample, according to the principle of inertia of large numbers, are 
likely to be more reliable, accurate and precise as the sample size increases, provided other 
factors are kept constant. This is a direct consequence of the rst principle.
(c) The principle of optimization ensures that an optimum level of efciency at a minimum cost 
or the maximum efciency at a given level of cost can be achieved with the selection of an 
appropriate sampling design.
(d) The principle of validity states that a sampling design is valid only if it is possible to obtain 
valid estimates and valid tests about population parameters. Only a probability sampling 
ensures this validity.
  13.2.3  COMPARISON BETWEEN SAMPLE SURVEY AND COMPLETE 
ENUMERATION
When complete information is collected for all the units belonging to a population, it is dened as 
complete enumeration or census. In most cases, we prefer sample survey to complete enumeration 
due to the following factors:
(a) Speed: As compared to census, a sample survey could be conducted, usually, much more 
quickly simply because in sample survey, only a part of the vast population is enumerated.
(b) Cost: The cost of collection of data on each unit in case of sample survey is likely to be more 
as compared to census because better trained personnel are employed for conducting a 
sample survey. But when it comes to total cost, sample survey is likely to be less expensive 
as only some selected units are considered in a sample survey.
(c) Reliability: The data collected in a sample survey are likely to be more reliable than that in 
a complete enumeration because of trained enumerators better supervision and application 
of modern technique.
© The Institute of Chartered Accountants of India
13.45 SAMPLING
(d) Accuracy: Every sampling is subjected to what is known as sampling uctuation which is 
termed as sampling error. It is obvious that complete enumeration is totally free from this 
sampling error. However, errors due to recording observations, biases on the part of the 
enumerators, wrong and faulty interpretation of data etc. are prevalent in both sampling 
and census and this type of error is termed as non-sampling errors. It may be noted that in 
sample survey, the sampling error can be reduced to a great extent by taking several steps 
like increasing the sample size, adhering to a probability sampling design strictly and so on. 
The non-sampling errors also can be contained to a desirable degree by a proper planning 
which is not possible or feasible in case of complete enumeration.
(e) Necessity: Sometimes, sampling becomes necessity. When it comes to destructive 
sampling where the items get exhausted like testing the length of life of electrical bulbs 
or sampling from a hypothetical population like coin tossing, there is no alternative to 
sample survey.
  However, when it is necessary to get detailed information about each and every item 
constituting the population, we go for complete enumeration. If the population size is not 
large, there is hardly any merit to take recourse to sampling. If the occurrence of just one 
defect may lead to a complete destruction of the process as in an aircraft, we must go for 
complete enumeration.
13.2.4 ERRORS IN SAMPLE SURVEY 
Errors or biases in a survey may be dened as the deviation between the value of population 
parameter as obtained from a sample and its observed value. Errors are of two types.
I. Sampling Errors
II. Non-Sampling Errors
Sampling Errors : Since only a part of the population is investigated in a sampling, every sampling 
design is subjected to this type of errors. The factors contributing to sampling errors are listed 
below:
(a) Errors arising out due to defective sampling design: Selection of a proper sampling design 
plays a crucial role in sampling. If a non- probabilistic sampling design is followed, the bias 
or prejudice of the sampler affects the sampling technique thereby resulting some kind of 
error.
(b) Errors arising out due to substitution: A very common practice among the enumerators 
is to replace a sampling unit by a suitable unit in accordance with their convenience when 
difculty arises in getting information from the originally selected unit. Since the sampling 
design is not strictly adhered to, this results in some type of bias.
(c) Errors owing to faulty demarcation of units: It has its origin in faulty demarcation of sampling 
units. In case of an agricultural survey , the sampler has, usually , a tendency to underestimate 
or overestimate the character under consideration.
(d) Errors owing to wrong choice of statistic: One must be careful in selecting the proper statistic 
while estimating a population characteristic.
© The Institute of Chartered Accountants of India
Page 4


13.43 SAMPLING
13.2.1 INTRODUCTION 
There are situations when we would like to know about a vast, innite universe or population. 
But some important factors like time, cost, efciency, vastness of the population make it almost 
impossible to go for a complete enumeration of all the units constituting the population. Instead, 
we take recourse to selecting a representative part of the population and infer about the unknown 
universe on the basis of our knowledge from the known sample. A somewhat clear picture would 
emerge out if we consider the following cases. 
In the rst example let us share the problem faced by Mr. Basu. Mr. Basu would like to put a big 
order for electrical lamps produced by Mr. Ahuja’s company “General Electricals”. But before 
putting the order, he must know whether the claim made by Mr. Ahuja that the lamps of General 
Electricals last for at least 1500 hours is justied.
Miss Manju Bedi is a well-known social activist. Of late, she has noticed that the incidence of a 
particular disease in her area is on the rise. She claims that twenty per cent of the people in her 
town have been suffering from the disease.
In both the situations, we are faced with three different types of problems. The rst problem is 
how to draw a representative sample from the population of electrical lamps in the rst case and 
from the population of human beings in her town in the second case. The second problem is to 
estimate the population parameters i.e., the average life of all the bulbs produced by General 
Electricals and the proportion of people suffering form the disease in the rst and second examples 
respectively on the basis of sample observations. The third problem relates to decision making 
i.e., is there enough evidence, once again on the basis of sample observations, to suggest that the
claims made by Mr. Ahuja or Miss Bedi are justiable so that Mr. Basu can take a decision about
buying the lamps from General Electricals in the rst case and some effective steps can be taken
in the second example with a view to reducing the outbreak of the disease. We consider tests of
signicance or tests of hypothesis before decision making.
13.2.2 BASIC PRINCIPLES OF SAMPLE SURVEY 
Sample Survey is the study of the unknown population on the basis of a proper representative 
sample drawn from it. How can a part of the universe reveal the characteristics of the unknown 
universe? The answer to this question lies in the basic principles of sample survey comprising 
the following components:
(a) Law of Statistical regularity
After reading this unit a student will learn -
? Different procedure of sampling which will be the best representative of the population;
LEARNING OBJECTIVES
UNIT 2 SAMPLING
© The Institute of Chartered Accountants of India
STATISTICS
13.44
(b) Principle of Inertia
(c) Principle of Optimization
(d) Principle of Validity
(a) According to the law of statistical regularity, if a sample of fairly large size is drawn from 
the population under discussion at random, then on an average the sample would posses 
the characteristics of that population.
 Thus the sample, to be taken from the population, should be moderately large. In fact larger 
the sample size, the better in revealing the identity of the population. The reliability of a 
statistic in estimating a population characteristics varies as the square root of the sample 
size. However, it is not always possible to increase the sample size as it would put an extra 
burden on the available resource. We make a compromise on the sample size in accordance 
with some factors like cost, time, efciency etc.
 Apart from the sample size, the sample should be drawn at random from the population which 
means that each and every unit of the population should have a pre-assigned probability to 
belong to the sample.
(b) The results derived from a sample, according to the principle of inertia of large numbers, are 
likely to be more reliable, accurate and precise as the sample size increases, provided other 
factors are kept constant. This is a direct consequence of the rst principle.
(c) The principle of optimization ensures that an optimum level of efciency at a minimum cost 
or the maximum efciency at a given level of cost can be achieved with the selection of an 
appropriate sampling design.
(d) The principle of validity states that a sampling design is valid only if it is possible to obtain 
valid estimates and valid tests about population parameters. Only a probability sampling 
ensures this validity.
  13.2.3  COMPARISON BETWEEN SAMPLE SURVEY AND COMPLETE 
ENUMERATION
When complete information is collected for all the units belonging to a population, it is dened as 
complete enumeration or census. In most cases, we prefer sample survey to complete enumeration 
due to the following factors:
(a) Speed: As compared to census, a sample survey could be conducted, usually, much more 
quickly simply because in sample survey, only a part of the vast population is enumerated.
(b) Cost: The cost of collection of data on each unit in case of sample survey is likely to be more 
as compared to census because better trained personnel are employed for conducting a 
sample survey. But when it comes to total cost, sample survey is likely to be less expensive 
as only some selected units are considered in a sample survey.
(c) Reliability: The data collected in a sample survey are likely to be more reliable than that in 
a complete enumeration because of trained enumerators better supervision and application 
of modern technique.
© The Institute of Chartered Accountants of India
13.45 SAMPLING
(d) Accuracy: Every sampling is subjected to what is known as sampling uctuation which is 
termed as sampling error. It is obvious that complete enumeration is totally free from this 
sampling error. However, errors due to recording observations, biases on the part of the 
enumerators, wrong and faulty interpretation of data etc. are prevalent in both sampling 
and census and this type of error is termed as non-sampling errors. It may be noted that in 
sample survey, the sampling error can be reduced to a great extent by taking several steps 
like increasing the sample size, adhering to a probability sampling design strictly and so on. 
The non-sampling errors also can be contained to a desirable degree by a proper planning 
which is not possible or feasible in case of complete enumeration.
(e) Necessity: Sometimes, sampling becomes necessity. When it comes to destructive 
sampling where the items get exhausted like testing the length of life of electrical bulbs 
or sampling from a hypothetical population like coin tossing, there is no alternative to 
sample survey.
  However, when it is necessary to get detailed information about each and every item 
constituting the population, we go for complete enumeration. If the population size is not 
large, there is hardly any merit to take recourse to sampling. If the occurrence of just one 
defect may lead to a complete destruction of the process as in an aircraft, we must go for 
complete enumeration.
13.2.4 ERRORS IN SAMPLE SURVEY 
Errors or biases in a survey may be dened as the deviation between the value of population 
parameter as obtained from a sample and its observed value. Errors are of two types.
I. Sampling Errors
II. Non-Sampling Errors
Sampling Errors : Since only a part of the population is investigated in a sampling, every sampling 
design is subjected to this type of errors. The factors contributing to sampling errors are listed 
below:
(a) Errors arising out due to defective sampling design: Selection of a proper sampling design 
plays a crucial role in sampling. If a non- probabilistic sampling design is followed, the bias 
or prejudice of the sampler affects the sampling technique thereby resulting some kind of 
error.
(b) Errors arising out due to substitution: A very common practice among the enumerators 
is to replace a sampling unit by a suitable unit in accordance with their convenience when 
difculty arises in getting information from the originally selected unit. Since the sampling 
design is not strictly adhered to, this results in some type of bias.
(c) Errors owing to faulty demarcation of units: It has its origin in faulty demarcation of sampling 
units. In case of an agricultural survey , the sampler has, usually , a tendency to underestimate 
or overestimate the character under consideration.
(d) Errors owing to wrong choice of statistic: One must be careful in selecting the proper statistic 
while estimating a population characteristic.
© The Institute of Chartered Accountants of India
STATISTICS
13.46
(e) Variability in the population: Errors may occur due to variability among population units 
beyond a degree. This could be reduced by following somewhat complicated sampling design 
like stratied sampling, Multistage sampling etc.
Non-sampling Errors
As discussed earlier, this type of errors happen both in sampling and complete enumeration. 
Some factors responsible for this particular kind of biases are lapse of memory, preference for 
certain digits, ignorance, psychological factors like vanity, non- responses on the part of the 
interviewees wrong measurements of the sampling units, communication gap between the 
interviewers and the interviewees, incomplete coverage etc. on the part of the enumerators also 
lead to non-sampling errors.
 13.2.5 SOME IMPORTANT TERMS ASSOCIATED WITH SAMPLING 
Population or Universe
It may be dened as the aggregate of all the units under consideration. All the lamps produced by 
“General Electricals“ in our rst example in the past, present and future constitute the population. 
In the second example, all the people living in the town of Miss Manju form the population. The 
number of units belonging to a population is known as population size. If there are one lakh 
people in her town then the population size, to be denoted by N, is 1 lakh.
A population may be nite or innite. If a population comprises only a nite number of units, 
then it is known as a nite population. The population in the second example is obviously , nite. 
If the population contains an innite or uncountable number of units, then it is known as an 
innite population. The population of electrical lamps of General Electricals is innite. Similarly , 
the population of stars, the population of mosquitoes in Kolkata, the population of owers in 
Mumbai, the population of insects in Delhi etc. are innite population.
Population may also be regarded as existent or hypothetical. A population consisting of real objects 
is known as an existent population. The population of the lamps produced by General Electricals 
and the population of Miss Manju’s town are example of existent populations. A population that 
exists just hypothetically like the population of heads when a coin is tossed innitely is known 
as a hypothetical or an imaginary population.
Sample
A sample may be dened as a part of a population so selected with a view to representing 
the population in all its characteristics selection of a proper representative sample is pretty 
important because statistical inferences about the population are drawn only on the basis of 
the sample observations. If a sample contains n units, then n is known as sample size. If a 
sample of 500 electrical lamps is taken from the production process of General Electricals, then 
n = 500. The units forming the sample are known as “Sampling Units”. In the rst example, 
the sampling unit is electrical lamp and in the second example, it is a human. A detailed 
and complete list of all the sampling units is known as a “Sampling Frame”. Before drawing 
sample, it is a must to have a updated sampling frame complete in all respects before the 
samples are actually drawn.
© The Institute of Chartered Accountants of India
Page 5


13.43 SAMPLING
13.2.1 INTRODUCTION 
There are situations when we would like to know about a vast, innite universe or population. 
But some important factors like time, cost, efciency, vastness of the population make it almost 
impossible to go for a complete enumeration of all the units constituting the population. Instead, 
we take recourse to selecting a representative part of the population and infer about the unknown 
universe on the basis of our knowledge from the known sample. A somewhat clear picture would 
emerge out if we consider the following cases. 
In the rst example let us share the problem faced by Mr. Basu. Mr. Basu would like to put a big 
order for electrical lamps produced by Mr. Ahuja’s company “General Electricals”. But before 
putting the order, he must know whether the claim made by Mr. Ahuja that the lamps of General 
Electricals last for at least 1500 hours is justied.
Miss Manju Bedi is a well-known social activist. Of late, she has noticed that the incidence of a 
particular disease in her area is on the rise. She claims that twenty per cent of the people in her 
town have been suffering from the disease.
In both the situations, we are faced with three different types of problems. The rst problem is 
how to draw a representative sample from the population of electrical lamps in the rst case and 
from the population of human beings in her town in the second case. The second problem is to 
estimate the population parameters i.e., the average life of all the bulbs produced by General 
Electricals and the proportion of people suffering form the disease in the rst and second examples 
respectively on the basis of sample observations. The third problem relates to decision making 
i.e., is there enough evidence, once again on the basis of sample observations, to suggest that the
claims made by Mr. Ahuja or Miss Bedi are justiable so that Mr. Basu can take a decision about
buying the lamps from General Electricals in the rst case and some effective steps can be taken
in the second example with a view to reducing the outbreak of the disease. We consider tests of
signicance or tests of hypothesis before decision making.
13.2.2 BASIC PRINCIPLES OF SAMPLE SURVEY 
Sample Survey is the study of the unknown population on the basis of a proper representative 
sample drawn from it. How can a part of the universe reveal the characteristics of the unknown 
universe? The answer to this question lies in the basic principles of sample survey comprising 
the following components:
(a) Law of Statistical regularity
After reading this unit a student will learn -
? Different procedure of sampling which will be the best representative of the population;
LEARNING OBJECTIVES
UNIT 2 SAMPLING
© The Institute of Chartered Accountants of India
STATISTICS
13.44
(b) Principle of Inertia
(c) Principle of Optimization
(d) Principle of Validity
(a) According to the law of statistical regularity, if a sample of fairly large size is drawn from 
the population under discussion at random, then on an average the sample would posses 
the characteristics of that population.
 Thus the sample, to be taken from the population, should be moderately large. In fact larger 
the sample size, the better in revealing the identity of the population. The reliability of a 
statistic in estimating a population characteristics varies as the square root of the sample 
size. However, it is not always possible to increase the sample size as it would put an extra 
burden on the available resource. We make a compromise on the sample size in accordance 
with some factors like cost, time, efciency etc.
 Apart from the sample size, the sample should be drawn at random from the population which 
means that each and every unit of the population should have a pre-assigned probability to 
belong to the sample.
(b) The results derived from a sample, according to the principle of inertia of large numbers, are 
likely to be more reliable, accurate and precise as the sample size increases, provided other 
factors are kept constant. This is a direct consequence of the rst principle.
(c) The principle of optimization ensures that an optimum level of efciency at a minimum cost 
or the maximum efciency at a given level of cost can be achieved with the selection of an 
appropriate sampling design.
(d) The principle of validity states that a sampling design is valid only if it is possible to obtain 
valid estimates and valid tests about population parameters. Only a probability sampling 
ensures this validity.
  13.2.3  COMPARISON BETWEEN SAMPLE SURVEY AND COMPLETE 
ENUMERATION
When complete information is collected for all the units belonging to a population, it is dened as 
complete enumeration or census. In most cases, we prefer sample survey to complete enumeration 
due to the following factors:
(a) Speed: As compared to census, a sample survey could be conducted, usually, much more 
quickly simply because in sample survey, only a part of the vast population is enumerated.
(b) Cost: The cost of collection of data on each unit in case of sample survey is likely to be more 
as compared to census because better trained personnel are employed for conducting a 
sample survey. But when it comes to total cost, sample survey is likely to be less expensive 
as only some selected units are considered in a sample survey.
(c) Reliability: The data collected in a sample survey are likely to be more reliable than that in 
a complete enumeration because of trained enumerators better supervision and application 
of modern technique.
© The Institute of Chartered Accountants of India
13.45 SAMPLING
(d) Accuracy: Every sampling is subjected to what is known as sampling uctuation which is 
termed as sampling error. It is obvious that complete enumeration is totally free from this 
sampling error. However, errors due to recording observations, biases on the part of the 
enumerators, wrong and faulty interpretation of data etc. are prevalent in both sampling 
and census and this type of error is termed as non-sampling errors. It may be noted that in 
sample survey, the sampling error can be reduced to a great extent by taking several steps 
like increasing the sample size, adhering to a probability sampling design strictly and so on. 
The non-sampling errors also can be contained to a desirable degree by a proper planning 
which is not possible or feasible in case of complete enumeration.
(e) Necessity: Sometimes, sampling becomes necessity. When it comes to destructive 
sampling where the items get exhausted like testing the length of life of electrical bulbs 
or sampling from a hypothetical population like coin tossing, there is no alternative to 
sample survey.
  However, when it is necessary to get detailed information about each and every item 
constituting the population, we go for complete enumeration. If the population size is not 
large, there is hardly any merit to take recourse to sampling. If the occurrence of just one 
defect may lead to a complete destruction of the process as in an aircraft, we must go for 
complete enumeration.
13.2.4 ERRORS IN SAMPLE SURVEY 
Errors or biases in a survey may be dened as the deviation between the value of population 
parameter as obtained from a sample and its observed value. Errors are of two types.
I. Sampling Errors
II. Non-Sampling Errors
Sampling Errors : Since only a part of the population is investigated in a sampling, every sampling 
design is subjected to this type of errors. The factors contributing to sampling errors are listed 
below:
(a) Errors arising out due to defective sampling design: Selection of a proper sampling design 
plays a crucial role in sampling. If a non- probabilistic sampling design is followed, the bias 
or prejudice of the sampler affects the sampling technique thereby resulting some kind of 
error.
(b) Errors arising out due to substitution: A very common practice among the enumerators 
is to replace a sampling unit by a suitable unit in accordance with their convenience when 
difculty arises in getting information from the originally selected unit. Since the sampling 
design is not strictly adhered to, this results in some type of bias.
(c) Errors owing to faulty demarcation of units: It has its origin in faulty demarcation of sampling 
units. In case of an agricultural survey , the sampler has, usually , a tendency to underestimate 
or overestimate the character under consideration.
(d) Errors owing to wrong choice of statistic: One must be careful in selecting the proper statistic 
while estimating a population characteristic.
© The Institute of Chartered Accountants of India
STATISTICS
13.46
(e) Variability in the population: Errors may occur due to variability among population units 
beyond a degree. This could be reduced by following somewhat complicated sampling design 
like stratied sampling, Multistage sampling etc.
Non-sampling Errors
As discussed earlier, this type of errors happen both in sampling and complete enumeration. 
Some factors responsible for this particular kind of biases are lapse of memory, preference for 
certain digits, ignorance, psychological factors like vanity, non- responses on the part of the 
interviewees wrong measurements of the sampling units, communication gap between the 
interviewers and the interviewees, incomplete coverage etc. on the part of the enumerators also 
lead to non-sampling errors.
 13.2.5 SOME IMPORTANT TERMS ASSOCIATED WITH SAMPLING 
Population or Universe
It may be dened as the aggregate of all the units under consideration. All the lamps produced by 
“General Electricals“ in our rst example in the past, present and future constitute the population. 
In the second example, all the people living in the town of Miss Manju form the population. The 
number of units belonging to a population is known as population size. If there are one lakh 
people in her town then the population size, to be denoted by N, is 1 lakh.
A population may be nite or innite. If a population comprises only a nite number of units, 
then it is known as a nite population. The population in the second example is obviously , nite. 
If the population contains an innite or uncountable number of units, then it is known as an 
innite population. The population of electrical lamps of General Electricals is innite. Similarly , 
the population of stars, the population of mosquitoes in Kolkata, the population of owers in 
Mumbai, the population of insects in Delhi etc. are innite population.
Population may also be regarded as existent or hypothetical. A population consisting of real objects 
is known as an existent population. The population of the lamps produced by General Electricals 
and the population of Miss Manju’s town are example of existent populations. A population that 
exists just hypothetically like the population of heads when a coin is tossed innitely is known 
as a hypothetical or an imaginary population.
Sample
A sample may be dened as a part of a population so selected with a view to representing 
the population in all its characteristics selection of a proper representative sample is pretty 
important because statistical inferences about the population are drawn only on the basis of 
the sample observations. If a sample contains n units, then n is known as sample size. If a 
sample of 500 electrical lamps is taken from the production process of General Electricals, then 
n = 500. The units forming the sample are known as “Sampling Units”. In the rst example, 
the sampling unit is electrical lamp and in the second example, it is a human. A detailed 
and complete list of all the sampling units is known as a “Sampling Frame”. Before drawing 
sample, it is a must to have a updated sampling frame complete in all respects before the 
samples are actually drawn.
© The Institute of Chartered Accountants of India
13.47 SAMPLING
Parameter
A parameter may be dened as a characteristic of a population based on all the units of the 
population. Statistical inferences are drawn about population parameters based on the sample 
observations drawn from that population. In the rst example, we are interested about the 
parameter “Population Mean”. If x a denotes the a th member of the population, then population 
mean m, which represents the average length of life of all the lamps produced by General Electricals 
is given by
 
=
µ=
?
n
a
a1
x
N
 (13.2.1)
Where N denotes the population size i.e. the total number of lamps produced by the company . In 
the second example, we are concerned about the population proportion P , representing the ratio 
of the people suffering from the disease to the total number of people in the town. Thus if there 
are X people possessing this attribute i.e. suffering from the disease, then we have
 P = 
X
N
 (13.2.2)
Another important parameter namely the population variance, to be denoted by s
2
 is given by 
  
() -µ
s=
?
2
a
2
X
N
 (13.2.3)
Also we have SD = 
() -µ
s=
?
2
a
X
N
 (13.2.4)
Statistics 
A statistic may be dened as a statistical measure of sample observation and as such it is a function 
of sample observations. If the sample observations are denoted by x
1
, x
2
, x
3
, ……….. x
n
, then a 
statistic T may be expressed as T = f(x
1
, x
2
, x
3
, ……….. x
n
)
A statistic is used to estimate a particular population parameter. The estimates of population 
mean, variance and population proportion are given by 
 
i
x
x
n
-?
= µ=
?
  (13.2.5)
2
i
2
2
xx
S
n
-
?
??
-
??
??
=s=
?
  (13.2.6)
and p = 
x
pP
n
?
= =
  (13.2.7)
© The Institute of Chartered Accountants of India