1.
Sampling fundamentals

2.
INTRODUCTION
The need for adequate and reliable data is ever
increasing
for taking wise decisions in different fields of human
activity
and business. There are two ways in which the
required
information may be obtained:
1. Complete enumeration survey or census method.
2. Sampling method.
In the first case, data are collected for each and every
unit.
i.e Universe/ population (complete set of items).

3.
What is Population?
In any field of inquiry, all the items under consideration
constitute ‘population’ or ‘universe’.
A complete enumeration of all the items of ‘population’
is
known as a census inquiry. In such an inquiry it is
assumed
that highest accuracy is obtained.
But this type of inquiry involves a great deal of time,
money
and energy. Not only this, census inquiry is not
possible in

4.
Sample
Hence, quite often we select only a few items from the
universe for our study purposes. The items so
selected is
technically called sample.

5.
What is Sampling Process?
Sampling may be defined as the selection of some
part of an
aggregate or totality on the basis of which a judgment
or
inference about the population (aggregate or totality) is
made.
In other words, it is the process of obtaining
information
about an entire population by examining only a part of
it.

6.
Need for sampling
1. Sampling can save time and money.
2. Sampling may produce more accurate information
if it is conducted by trained and experienced
investigator.
3. Sampling becomes the only option when the
population size is infinite.

7.
Sample Design
 Researcher must prepare a sample design for his study i.e.,
he must plan how a sample should be selected and of what
size a sample would be.
 Large and Small Sample: Let the population size be N and a
part of size n ( which is less than N) of this population is
selected according to some rule for some characteristics of
the population. The group consisting of these n units is
known as ‘sample’. Therefore n denotes sample size. If n>30
then it is considered as large sample, otherwise it is known
as small sample.
 The selection process i.e. the way the researcher decide to
select a sample from the population is known as the ‘sample
design’.
In other words, it is a define plan ( determined by the
researcher) before any data is collected for obtaining a
sample from a given population.
Eg : Research on pharmaceutical industry.

8.
Sampling Method/Sampling Technique
1. Probability Sampling
2. Non-probability sampling

9.
Probability Sampling
 Probability sampling is also known as ‘random
sampling’ or ‘chance sampling’.
Samples selected according to some chance are
known as
random or probability samples i.e. every item in the
population has known chance of being included in the
sample.

10.
Non-probability sampling
On the other hand, non-random or non-probability
samples
are those where the selection of sample unit is based
on
the judgment of the researcher than randomness.

11.
Important Sampling Designs
Probability Sampling:
i. Simple Random Sampling
ii. Systematic Sampling
iii. Stratified Sampling
iv. Cluster and area Sampling

12.
Major non-probability sampling are:
i. Deliberate Sampling/ Purposive sampling/
Judgment sampling
ii. Quota Sampling

13.
Simple Random Sampling Method
Under this sampling design , every item of the
universe has
an equal chance of inclusion in the sample.
For example, if we have to select a sample of 300
items
from a universe of 15,000 items, then we can put the
names
or numbers of all the 15,000 items on slips of paper
and
conduct a lottery.

14.
Under this method, sampling is done without
replacement, so
that no unit can appear more than once in the sample.
Thus, if from a population consisting of 4 members A,
B, C
and D, a simple random sample of n=2 is to be drawn,
there
would be 6 possible samples without replacement.
They are
AB, AC , AD, BC, BD, CD.
Keeping in this view, we can say that a simple random
sample of size n from population N results in N C n

15.
Exercise
Take a certain finite population of six elements ( say a,
b, c,
d, e, f). Suppose that we want to take a sample of size
n=3
from it. Find out how many possible outcomes are
there?
Write the elements . Choose one sample out of it.
What is
their probabilities of getting into the sample?

16.
Systematic Sampling
In some instances, the most practical way of sampling is to
select
every ith item from the universe where ‘i’ refers to the
sampling
interval.
The sampling interval can be determined by dividing the
size of
the population by the size of the sample to be chosen.
For example, if we wish to draw 32 names out of the list of
320
names, the sampling interval will be 10. It means every
100th
name will be selected. In this process a random start is
always

17.
Example
In a class of 120 students , it was decided to constitute
an
Academic and cultural committee with 10
representatives.
Use systematic sampling method to form the
committee.
Soln:

18.
Merits and demerits
Merits :
a. It is a simple method
b. It can be taken as an improvement over a simple
random sample as it spread more evenly over the
population.
Demerits:
a. It is not truly random in the strict sense. This is
because all items selected for the sample ( except
the first term) are pre-determined by the constant
interval.
b. There are certain dangers too in using this type of
sampling. If there is a hidden periodicity in the
population, systematic sampling will prove to be
inefficient method of sampling. Example , quality

19.
Stratified Sampling
If a population from which a sample is to be drawn does
not
constitute a homogeneous group,( highly heterogeneous)
stratified sampling technique is generally applied in order
to
obtain a representative sample. Under stratified sampling
the
population is divided into several sub-populations that are
individually more homogeneous than the total population.
The
different sub-populations are called ‘strata’ . Then we
select

20.
The following three questions are highly
relevant in the context of stratified sampling
a) How to form strata?
b) How should items be selected from each stratum?
c) How many items be selected from each stratum or
how to allocate the sample size of each stratum?

21.
Regarding the first question, we can say that the items
which are homogeneous ( i.e. of common
characteristics)
should be put in the same group or strata. In other
words
strata be formed in such a way that elements are most
homogeneous within the strata and most
heterogeneous
between the different strata.

22.
In respect to 2nd question, we can say that to choose
the
items from each strata we normally adopt simple
random
sampling.

23.
To answer the 3rd question we have to
understand the following concepts:
Stratified sampling can be of two types: proportionate
and
disproportionate.
In proportionate stratified sampling the number of
sample
units in various strata are in the same proportion as
found
in the population. Thus, larger the particular stratum,
the
more weight it receives in the analysis.

24.
Example 1
A sample of 30 students is to be drawn from a
population
consisting of 300 students belonging to two colleges A
and
B. How would you draw the desired sample by using
proportionate stratified random sampling?College Number of students
A 200
B 100

25.
Disproportionate Stratified sampling
Here the strata are represented in the total sample in a
proportion other than the one with which they are
found in
the population.
In Disproportionate Stratified sampling the sample
proportion for each stratum will be determined by
using
the following rule:
The proportion of the ith stratum will be
ni = Ni . σi i=1, 2 …k
N1σ1 + N2σ2 +….. Nkσk

26.
Example 2
A population is dived into three strata with N1=5000 ,N2=2000
and N3=3000 . Respective standard deviations are σ1 =15,
σ2 =18 and σ3 =5. How should a sample of size 84 be selected
from the three strata . Use proportionate and disproportionate
Stratified sampling technique.

27.
Example
To know the customer demand for expensive and
luxurious
item (say diamond Jewelry), among the followings
which
sampling technique will you chose? Justify your
answer.
a) Simple random sampling
b) Systematic sampling
c) Stratified Sampling

28.
Cluster Sampling /Area Sampling
In Cluster sampling first we divide the population into
groups called ‘clusters’ and then select some units
from the
groups or the clusters for sample. Cluster sampling is
totally
opposite to stratified sampling in the sense that ,
a. The units within each cluster should be as
heterogeneous as possible.
b. There should be small difference between the
clusters.

29.
Ex. If a market research team is attempting to study
the preference of TV- Brand in a large city.

30.
Area Sampling
Since geographical area of interest happens to be a
big one.
Under this sampling we divide the total area into
smaller
non-overlapping areas, generally called geographical
clusters,
then certain areas are randomly selected and all
households
in the selected area are would be interviewed to get
the
information.

31.
Non-probability sampling
i. Deliberate Sampling/ Purposive sampling/
Judgment sampling: Selection made by choice not
by chance where investigator is highly experienced and
skilled.
This method is seldom used and cannot be recommended
for general use since it suffers from the drawback of
favoritism depending on the beliefs and prejudices of the
investigator.
i. Quota Sampling: Here the interviewer got some
quota based on gender, age , income ,
occupation etc. to be filled where actual selection
of units totally depends on the interviewers
judgment.

32.
iii) Convenience Sampling
Determination of sample size: ‘ smaller but properly
selected samples are superior to large but badly
selected
samples’
1. Resources available
2. Nature of study
3. Method of sampling used
4. Nature of respondents( response rate)
5. Nature of population ( existence of heterogeneity)

33.
Statistic and Parameter
 A statistic is a characteristics of a sample , whereas
a parameter is a characteristic of a population.
 Thus, when we work out certain measures such as
mean, median, mode, standard deviation from
sample, they are called statistic as they will describe
the characteristic of the population.
 Eg Sample mean= x . Sample s.d. = s
 Eg of parameter , population mean= μ ,population s.d. = σp

34.
Formula
 Sample Mean Formula: x =∑x
n
 Sample variance Formula s2= ∑(X-X)2
(n-1)
n= sample size
 Population Mean Formula: μ =∑x
N
 Population variance Formula σ2= ∑(X-X)2
N
N= population size.

35.
Sampling error

36.
The law of large numbers
Draw observations at random from any population
with finite mean μ. As the number of observations
drawn increases, the mean of the observed
values gets closer and closer to the mean μ of the
population. `
x

37.
The central limit theorem
Take a large (30 or more) random sample of size n
from any population with mean μ and standard
deviation σ. The sample mean, X is approaches the
normal distribution with mean μ and standard
deviation . n
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