Sampling

Population: An aggregate of all individuals or items (actual or possible) under study having
on some common characteristics is called a population. For example, suppose we are interested to see
the average height of the IUBAT students, only the IUBAT students will constitute the population.

Sample: A small but representative part of a population is called a sample. For example,
suppose we can select randomly some of the students, the selected students will constitute the
sample.

How do we study a population?

A population may be studied using one of two approaches: taking a census, or selecting a sample.
It is important to note that whether a census or a sample is used, both provide information that can be
used to draw conclusions about the whole population.

What is a census (complete enumeration)?

A census is a study of every unit, everyone or everything, in a population. It is known as a complete

enumeration, which means a complete count.

What is a sample (partial enumeration)?

A sample is a subset of units in a population, selected to represent all units in a population of

interest. It is a partial enumeration because it is a count from part of the population.
Information from the sampled units is used to estimate the characteristics for the entire population of
interest.

Sampling Unit

The population divided into a finite number of distinct and identifiable units is called sampling units.
The individuals whose characteristics are to be measured in the analysis are called elementary or
sampling units.

Before selecting the sample, the population must be divided into parts called sampling units or
simply sample units.

Sampling Frame

The list of all the sampling units with a proper identification (which represents the population to be
covered is called sampling frame). The frame may consist of either a list of units or a map of area (in

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case sample of area is being taken), such that every element in the population belongs to one and
only one unit.

The frame should be accurate, free from omission and duplication (overlapping), adequate, upto data
and the units must cover the whole of the population and should be well identified.

Proportion:

It is the percentage value associated with a population which is defined as,

P=X/N (for population)

p=x/n (for sample)
here, X=count of success in population
x= countof success in sample.

Sample size when Estimating a Population Proportion

In sample surveys, we are frequently encountered with the problem of estimating population
proportion , in such case, if p (sample proportion) is given, and q is the proportion not having the
attribute, such that p+q =1, then for a sufficiently large population, the formula for estimating the
sample size is

z 2 pq
n0 = 2
d

Where: no= desired sample size

z = standard normal deviate usually set at 1.96, which corresponds to the 95% confidence level.
p = assumed proportion estimated for a particular characteristic.
d = Margin of error/ degree of accuracy desired in the estimated proportion.

Example: A nutrition survey is to be conducted in a refugee camp. Assume that 40% children suffer
from malnutrition. How large sample would be needed in order to be 95% certain that the estimated
prevalence does not differ from the true prevalence by more than 0.05?

Solution: Assuming that the population is large, we employ above formula. Here z=l .96, d=0.05
and p=0.40. We now want to estimate the true proportion in the population within 5 percentage points
of p. Thus

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 z2 pq (1.96)2 (.4)(.6)
n0 = d2
= (.05)2
=396

If p is not known or difficult to assume, it will be the safest procedure to take it as 0.50 which
maximizes the expected variance and therefore indicates a sample size that is sure to be large
enough..

If N is relatively small, the formula to be used assumes the following form:

Nz2 pq
n=
Nd2 +z2pq

Definition of Sampling:

Sampling is a scientific process of selecting a smaller number of elements from a larger defined
target group of elements.

Sampling Methods can be classified into one of two categories:

 Probability Sampling: Sample has a known probability of being selected

 Non-probability Sampling: Sample does not have known probability of being selected as in
convenience or voluntary response surveys
 Probability Sampling:
1. Simple Random Sampling
2. Stratified Random Sampling
3. Systematic Sampling
4. Cluster Sampling
 Non- Probability Sampling:
1. Convenience Sampling
2. Purposive Sampling
3. Quota Sampling

Simple Random Sampling (SRS):

SRS is a method of selecting n elements from a population of size N elements in such a way that
each combination of n elements has the same chance or probability of being selected as every other
combination. The sample thus obtained is called a simple random sample.
** Sample Frame should be available to do sampling by SRS.
The selection of the simple random sample may be made with or without replacement.

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Example: We want to know the Average result of ART course of IUBAT students, and then we
have to take a sample from all students. Here, all the students have equal probability to come in
sample that is why we can precede SRS (Simple Random Sampling).

Procedure to draw a simple random sample:

Drawing a simple random sample from a population requires that in every draw, each eligible
population element be assigned equal probability of selection. In doing so, we might use our own
judgment to select the sample. But this might not be representative. Thus to ensure the randomness
in the selection the method of selection must be independent of human judgment as far as possible.
There are two basic procedures:
1. Lottery method
2. Random numbers

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The following 8 steps procedure may be following in drawing a simple random sample of n
units from a population of N units.

1. Assign serial numbers to the units in the population from 1 through N.

2. Decide on the random number table to be used.
3. Choose an N digit random number from any point in the random number table.
4. If the random number is ≤ N, this is your first selected unit.
5. Move on to the next random number not exceeding N.
6. If the random number is ≥ N, discard it and choose the next random number.
7. If any random number is repeated, it must also be discarded and be replaced by a fresh
random number.
8. The process stops once you arrive at your desired sample size.

Example: Draw a simple random sample of size 5 from a population consist of 150 units.
Solution: here, n= 5, and N= 150.
Assign serial numbers 001, 002, 003, …, 150. Since 150 is a 3 digit number, we read 3 digit random
numbers from the random number table.
11164 36318 75061 37674 26320 75100 10431 20418 19228 91792 21215 91791 76831 58678
87054 31687 93205 43685 19732 08468 10438 44482 66558 37649 08882 90870 12462 41810
01806 02977 36792 26236 33266 66583 60881 97395 20461 36742 02852 50564 73944 04773
12032 51414 82384 38370 00249 80709 72605 67497 84013
Note that we choose only those numbers, which lie in the range 001-150. Any number lying outside
this range in omitted. The process stops once we arrive at 5 numbers. The selected numbers are:
61,100, 54 , 32 and 13. All these numbers are distinct.

Stratified random sampling

Stratified random sampling is a sampling plan in which we divide the population into several groups
(known as strata) and select a random sample from each group (stratum) in such a way that units
within the strata are homogeneous but between strata are heterogeneous.

Example: Suppose a population is composed of 700 Muslims, 200Hindus, and 100 Christians.
If a simple random sample of 100 persons (10% of the total) is desired, we would
probably not get exactly 70Muslims, 20 Hindus and 10 Christians: the proportion of Chris
particular might be relatively too small. A stratified sample of 70 Muslims, 20 Hindus and 10
Christians would ensure a better representation of the groups.

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 Systematic Sampling:

Systematic sampling is a probability sampling method where the elements are chosen from a target
population by selecting a random starting point and selecting other members after a fixed ‘sampling
interval’. Sampling interval is calculated by dividing the entire population size by the desired sample
size.

Procedure to select:

 Consider, the sample interval should be 10 in a certain survey which is the result the of division of
5000 (N= size of the population) and 500 (n=size of the sample).
Systematic Sampling Formula for interval (i) = N/n = 5000/500 = 10
 The researcher needs to select these members who fit the criteria which in this case will be 1 in 10
individuals.
 A number will be randomly chosen as the starting member (r) of the sample and this interval will
be added to the random number to keep adding members in the sample. r, r+i, r+2i etc. will be the
elements of the sample.

Example: Suppose we want to choose a sample of four students from this class. We see that there are
24 students in the whole class. Thus, we calculate 24/4 = 6, to see that we want to take every sixth
student in the list. We start at any of the first six students, say we use random number table and get as
2, 8, 6, 21 th positioned students.

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Cluster Sampling

It is a sampling technique where the entire population is dividing into clusters and random sample of
these clusters are selected where clusters are internally heterogeneous but externally
homogeneous. All observations in the selected clusters are included in the sample.

When we are not able to make a list of population (sampling frame does not available), then we allow
to apply cluster sampling.

Example

In a study of homeless people across Dhaka, all the wards are selected and a significant number of
homeless people are interviewed in each one. Here, the selected wards are the clusters. So, the
selected sample is a cluster sample and the selection process is cluster sampling.

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