**Business
Statistics Notes**

**B.Com
2**^{nd} and 3^{rd} Semester (CBCS Pattern)

**Sampling Concepts,
Sampling Distributions and Estimation**

^{nd}and 3

^{rd}Semester (CBCS Pattern)

**Census and Sample Method**

There are basically two popular methods of collecting primary data: one is census method and another one is sample method.

**Census method or Complete Enumeration Survey**

Statistics is taken in relation to a large data. Single and unconnected data is not statistics. In the field of a statistical enquiry there may be persons, items or any other similar units. The aggregate of all such units under consideration is called “Universe or Population”. Under the census or complete enumeration survey method, data are collected for each and every unit of the population or universe which is the complete set of items which are of interest in any particular situation.

**Merits of Census Method:**

(a) Data are obtained from each and every unit of the population.

(b) Since all the individuals of the universe are investigated, highest degree of accuracy is obtained.

(c) Since there is no possibility of personal bias affecting investigation, this method is free from sampling error.

(d) It is more suitable if the field of enquiry is small.

(e) Since all the items of the universe are taken into consideration, all the characteristics of the universe are collected which can be widely used as a basis for various surveys.

**Demerits of Census Method:**

(a) It the field of enquiry is too wide, it is not suitable.

(b) Collection of primary data is costly and time consuming.

(c) Personal Bias, prejudice and whims may affect the data.

(d) If secondary data are available, then it can be avoided to save time and cost.

(e) If the population is infinite or evaluation process destroys the population unit, the method cannot be adopted.

**Sample Survey**

Sampling is simply the process of learning about the population on the basis of sample drawn from it. In sampling technique only a part of the universe is studies and conclusions are drawn on that basis for entire universe. A sample can also be called subset of population units. Since sample is a part of universe, sample survey is less expensive and less time consuming than census.

**Merits of sample survey:**

(a) Since data are limited, time and labour can be saved in sample survey method.

(b) Sample survey is suitable where highly trained personnel or specialised equipment is required for collection of data.

(c) Detailed information can be obtained under sample survey method since data are collected from a small group of respondents.

(d) Above all time, energy and money can be saved without sacrificing accuracy of result.

**Demerits of sample survey:**

(a) Sample survey is unnecessary if the universe is small.

(b) Chances of sampling errors are very high. Personal bias of the investigator can also give misleading results.

(c) If the investigators are not trained and qualified, data collected from sample survey will be misleading.

(d) If the sample is not drawn properly, the results thus obtained may be false, inaccurate and misleading.

**Difference between Census and
Sample survey:**

The following are the differences between Census and Sample method of
investigation:

(a) Under Census method, each and every individual item is investigated
whereas under sample survey only a part of universe is investigated.

(b) There is no chance of sampling error in census survey whereas
sampling error cannot be avoided under sample survey.

(c) Large number of enumerators is required in census whereas less number
of enumerators is required in sample survey.

(d) Census survey is more time consuming and costly as compared to sample
survey.

(e) Census survey is an old method and it less systematic than the sample
survey.

**Meaning,
Features and Types of Sampling**

**Meaning of Sampling:** Sampling
is simply the process of learning about the population on the basis of sample
drawn from it. In sampling technique only a part of the universe is studies and
conclusions are drawn on that basis for entire universe. In most of
the research work and surveys, the usual approach happens to be to make generalizations
or to draw inferences based on samples about the parameters of population from
which the samples are taken. The researcher quite often selects only a few
items from the universe for his study purposes. All this is done on the
assumption that the sample data will enable him to estimate the population
parameters. The items so selected constitute what is technically called a
sample, their selection process or technique is called sample design and the
survey conducted on the basis of sample is described as sample survey. Sample
should be truly representative of population characteristics without any bias
so that it may result in valid and reliable conclusions.

According
to Goode and Hatt, “A sample as the name applies, is a smaller representative
of a large whole”.

According
to Pauline V Young, “A statistical sample is a miniature of cross selection of
the entire group or aggregate from which the sample is taken”.

According
to Bogrdus, “Sampling is the selection of certain percentage of a group of
items according to a predetermined plan”.

**NEED FOR SAMPLING**

Sampling
is used in practice for a variety of reasons such as:

1.
Sampling can save time and money. A sample study is usually less expensive than
a census study and produces results at a relatively faster speed.

2.
Sampling may enable more accurate measurements for a sample study is generally
conducted by trained and experienced investigators.

3.
Sampling remains the only way when population contains infinitely many members.

4.
Sampling remains the only choice when a test involves the destruction of the
item under study.

5.
Sampling usually enables to estimate the sampling errors and, thus, assists in
obtaining information concerning some characteristic of the population.

**Feature of Sampling Techniques**

The
sampling techniques have following good features and these bring into relief
its value and significance:

1.
Scientific Base: It is a scientific because the conclusion
derived from the study of certain units can be verified from other units. By taking random sample, we can determine the
amount of deviation from the norm.

2.
Economy:
The sampling technique is much less expensive, much less time consuming than the census
technique.

3.
Reliability:
If the choice of sample unit is made with due care and the matter under
survey is not heterogeneous, the conclusion of the sample survey can have
almost the same reliability as those of census survey.

4.
Detailed study: Since the number of sample
units is fairly small, these can be studied intensively and elaborately. They
can be examined from multiple of views.

5.
Greater Suitability in most Situations: Most of the surveys are made by the
techniques of sample survey, because whenever the matter is of homogeneous
nature, the examination of few units suffices.
This case in majority of situations.

**Methods
of Sampling**

Sampling
methods are classified into two categories: Probability sampling methods and
non-probability sampling methods.

Probability
sampling methods are those in which every item in the universe has a known
chance or probability of being chosen for the sample. Non-probability sampling
methods are those which do not provide every item in the universe with a chance
of being selected. These two methods are further divided into:

Probability Sampling |
Non-Probability Sampling |

1.
Simple random sampling 2.
Stratified random sampling 3.
Cluster sampling 4.
Systematic Sampling 5.
Multi-stage sampling |
1.
Quota Sampling 2. Judgmental or Purposive Sampling 3.
Convenience Sampling 4.
Extensive Sampling |

**(1) Simple Random Sampling: **Off all
the methods of selecting sample, random sampling technique is made maximum use
of and it is considered as the best method of sample selection. Random sampling is made in following ways:

(i)
Lottery Method: In this the numbers of data are written on sheet of paper and
they are thrown into a box. Now a casual
observer selects the number of item required in the sample. For this method it is necessary that sheet of
paper should be of equal dimensions.

(ii)
By Rotating the Drum: In this method,
piece of wood, tin or cardboard of equal length and breadth, with number 0,1 or
2 printed on them, are used. The pieces are rotated in a drum and then
requisite numbers are drawn by an impartial person.

(iii)
Selecting from Sequential List: In this
procedure units are broken up in numerical, alphabetical or geography
sequence. Now we may decide to choose 1,
5, 10 and so on , if the division is alphabetical order we decide to choose
every item starting from a, b, c and so on.

(iv)
Tippet’s Number: On the basis of
population statistics, Tippet has constructed a random list of four digits each
of 10, 400 institutions. These numbers
are the result of combining 41,600 population statistics reports.

Advantages;

1. Due to impartiality, there is possibility of selecting any unit as sample.

2. Units have the characteristic of universe, hence units are more representative.

3. Simplicity of method makes no possibility of error.

4. Error can be known easily

5. It saves money, time and labour.

Disadvantages:

1.
The selector has no control over the selection
of units. The researcher cannot contact the far situated units.

2.
He cannot prepare the whole field when the
universe is vast.

3.
If units have no homogeneity, the method is
not appropriate.

4.
There is no question of alternatives. The selected units cannot be replaced or
changed.

**(2) Stratified Sampling:** This method of selecting samples is a mixture
of both purposive and random sampling techniques. In this all the data in a
domain is spilt into various classes on the basis of their characteristics and
immediately thereafter certain items are selected from these classes by the
random sampling technique. This
technique is suitable in those cases in which the data has sub data and having
special characteristics. For example if we wish to collect information
regarding income expenditure of the male population strata on the basis of
shopkeeper, workers, etc. From these we
shall select randomly some units for study of income-expenditure statistics.

Process
of Stratifying: The stratification of
domain or data should be with great care, because the success of the technique
depends upon successful stratification.
Following points should be born in mind:

1.
We should process extensive information of all
items including in a domain and should know which item make a coherent whole on
the basis of similar traits and which others are different from them and why?

2.
The size of each stratum should be large to
enable use of random sampling technique.

3.
In stratifying it must be kept in mind that
various strata should have similar relation to the domain and should be
themselves homogeneous.

4.
The various strata should differ from each
other should be the same as the proportion of stratum from the domain. Suppose a domain has four strata, accordingly
the proportion of each stratum of domain is ¼.
Now if the number of total items of the sample is 64, we shall select 16
items from each stratum and thus the proportion of selected items from each stratum
will be ¼.

Advantage:

1.
Neither group nor class of importance is
totally neglected as units of each are represented in the sample.

2.
If different classes are divided properly,
selection of few units represents the whole group.

3.
On the classification of regional basis, units
are not in contact easily. This leads to economy of time and money.

4.
There is a facility in substitution of units.
If someone is not contacted easily, the other person of the same class can be
substituted for him. Such inclusion result
will not show any contradicting.

Disadvantages:

1.
The sample does not become representative if
selected sample has more or less units of a class.

2.
If the sizes of different group are different,
no equal proportional quality can be viewed.

3.
Non-proportional selection leads to more
emphasis in the end. During such time
researcher can be biased, hence samples will not accurate.

4.
If group is not expressed properly, the
difficulty is seen about the unit to be kept under which group or class.

**(3) ****Cluster
Sampling: **In this method of sampling, the population is divided into
clusters or groups and then Random Sampling is done for each cluster. In some
instances the sampling unit consists of a group or cluster of smaller units
that we call elements or sub-units. Cluster Sampling is different Stratified sampling. In
the case of stratified sampling the elements of each stratum are homogeneous while
in cluster sampling each cluster is heterogeneous within and a representative
of the population.

**(4) ****Systematic
Sampling: **This
method of sampling is at first glance very different from random
sampling.
In practice, it is a variant of simple random sampling that involves some
listing of elements. In systematic sampling each element has an equal chance of
being selected, but each sample does not have the same chance of being
selected. Here, the first element of the population is randomly selected to
begin the sampling. But thereafter the elements are selected according to a
systematic plan. Systematic sampling proceeds by picking up one element after a
fixed interval.

**(5) Multi-Stage sampling:** This is not a favoured procedure of
sampling. In this items are selected in
different stages at random. For example,
if we wish to know per acre yield of various crops in U.P., we shall begin by
studying a single crop in one study.
Here we shall begin by making at random selection of 5 districts in the
first instance, and then of these 5 districts, 10 villages per districts will
be chosen in the same manner. Now in the final stage, again by random selection
5 fields out of every village. Thus we shall examine per acre yield in 250
farms all over U.P. this number can increased or decreased depending upon the
opinion of experts.

**(6) Quota Sampling:** This method of study is not much used. In
this method entire data is spilt into as many as there are investigators and
each investigator is asked to select certain items from his block and
study. The success of this method
depends upon the integrity and professional competence of investigators. If some investigators are competent and
others are not so competent, serious discrepancies will appear in the study.

**(7) Purposive or Judgmental or
Selective sampling:** In
this method the investigator has complete freedom to choose his sample according
to his wishes and desire. To choose or
leave an item for the purpose of study depends entirely upon the wishes of
investigator and he will chose items or units which in his judgment are
representative of the whole data. This
is a very simple technique of choosing the samples and is useful in cases where
the whole data is homogeneous and the investigator has full knowledge of the
various aspects of the problem.

Advantage:

1.
More representation is possible in this
method.

2.
As sample is small in size, the method is less
expensive and less time consuming.

3.
The utility of this method increases when few
units of universe have special importance.

4.
When units are less in number, sample is
profitable

Disadvantages:

1.
Units are selected by researcher at his
will. Hence sample is biased.

2.
The error of the sample cannot be detected.

3.
Researcher is unable to understand the whole
group.

4.
Those hypothesis on which inference of error
of sample is attributed, are less used.

**(8) Convenience Sampling:** This is hit or miss procedure of study. The investigator selects certain item from
the domain as per his convenience. No
planned efforts are made to collect information. This is method by which a
tourist studies generally the country of his visit. He comes across certain people and things,
has transaction with them and then tries to generalize about the entire
populace in his travelogue. This is
essentially unscientific procedure and has no value as a research technique.

**(9) Extensive sampling: ** This method is virtually same as census except
that irrelevant or irascible items are left out. Every other item is examined. For instance, if we are to study the
educational levels of Indians, we may leave foreigners living in India from our
study. This method has all the merits
and demerits of census survey and is very rarely used.

**Factors
to be taken into consideration while deciding the sample size:**

a) The size of the universe – larger the size of universe, larger will be the sample size.

b) The resources available with the researcher.

c) The degree of accuracy desired by the researcher. Larger the size, more will be the accuracy.

d) Small sample size if universe consists of homogenous unit and large sample size if universe consists of heterogeneous units.

e) Nature of study – For an intensive and continuous study a small sample may be suitable.

**Sampling
Errors and Non-sampling Errors**

**Sampling Errors: **The errors caused by drawing inference about
the population on the basis of samples are
termed as sampling errors. The
sampling errors result from the bias in the selection of sample units. These errors occur because the study is based on
a part of the population. If the
whole population is taken, sampling error can be eliminated. If two or more sample units are taken from a
population by random sampling method, their results need not be identical and the results of both of them may
be different from the result of
the population. This is due to the fact that the selected two sample items will not be identical. Thus, sampling
error means precisely the difference between the sample result and that of the population when both the results
are obtained by using the same
procedure or method of calculation. The exact amount of sampling error will differ from sample to sample. The
sampling errors are inevitable even if utmost care is taken in selecting the sample. However, it is possible to
minimise the sampling errors by
designing the survey appropriately. Sampling
errors are of two types:

(i) Biased sampling errors: These errors
arises from any bias in selection, estimation

(ii) Unbiased sampling errors: These errors arise due to
difference between members of population included in sample and those not
included.

**Non-sampling
Errors: **These non-sampling
errors can occur in any survey, whether it be a complete,** **enumeration or sampling. Non-sampling errors include biases as
well as mistakes.** **These are not
chance errors. Most of the factors causing bias in complete enumeration are
similar to the one described above under sampling errors. They also include
careless definition of population, a vague conception regarding the information
sought, and inefficient method** **of
interview and so on. Mistakes arise as a result of improper coding,
computations and** **processing. More specifically,
non-sampling errors may arise because of one or more of** **the following reasons:

i) Improper and ambiguous data
specifications which are not consistent with the** **census or survey objectives.

ii) Inappropriate sampling methods,
incomplete questionnaire and incorrect way of** **interviewing.

iii) Personal bias of the investigators or
informants.

iv) Lack of trained and qualified
investigators.

v) Errors in compilation and tabulation.

**Law of statistical regularity**

Law of statistical regularity is derived from the concept of probability. According to this law, if a sample is collected at random from a population, it is probable that the sample posses the same properties which the population posses. Random selection means each and every item of the population has the equal chance of selection. A sample selected at a random from a large population will represent the whole universe. There is a very less chances of bias selection. This law is very important because quick conclusion can be drawn from a large universe. It reduces the necessary work before final conclusion is drawn.

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