L4 theory of sampling

Sampling fundamentals

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).

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

practice under many circumstances.

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.

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.

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.

Sample Design

Researcher must prepare a sample design for his study i.e.,he must plan how a sample should be selected and of whatsize a sample would be.

Large and Small Sample: Let the population size be N and apart of size n ( which is less than N) of this population isselected according to some rule for some characteristics ofthe population. The group consisting of these n units isknown as ‘sample’. Therefore n denotes sample size. If n>30then it is considered as large sample, otherwise it is knownas 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.

Sampling Method/Sampling Technique

1. Probability Sampling

2. Non-probability sampling

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.

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.

Important Sampling Designs

Probability Sampling:

i. Simple Random Sampling

ii. Systematic Sampling

iii. Stratified Sampling

iv. Cluster and area Sampling

Major non-probability sampling are:

i. Deliberate Sampling/ Purposive sampling/

Judgment sampling

ii. Quota Sampling

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.

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

possible outcomes in such that each has the same

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?

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

Preferable, i. e. a start is determined by chance. If the

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:

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 checking of 4% sample.

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

items from each ‘strata’ to constitute a sample. Since

each

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?

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.

In respect to 2nd question, we can say that to choose

the

items from each strata we normally adopt simple

random

sampling.

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.

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

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

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.

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

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.

Ex. If a market research team is attempting to study

the preference of TV- Brand in a large city.

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.

Non-probability sampling

i. Deliberate Sampling/ Purposive sampling/Judgment sampling: Selection made by choice notby chance where investigator is highly experienced andskilled.

This method is seldom used and cannot be recommendedfor general use since it suffers from the drawback offavoritism depending on the beliefs and prejudices of theinvestigator.

i. Quota Sampling: Here the interviewer got somequota based on gender, age , income ,occupation etc. to be filled where actual selectionof units totally depends on the interviewersjudgment.

Eg Often used in market and Public opinion polls.

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)

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

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.

Sampling error

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

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

nNX

,~