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Sampling Methods (Independent Module)
Pri nt References
1. Introductory Statistics, Neil A. Weiss2. Introducing Statistics, Graham Upton, Ian Cook3. AP Statistics 2008- 2009, Duane C. Hinders4. A Concise Course in Advanced Level Statistics with Worked Examples,
J. Crawshaw and J. Chambers
5. Elementary Statistics: A Step by Step Approach, Allan G. BlumanNon-print References
1. Sampling Methodshttp://www.statcan.ca/english/edu/power/ch13/probability/probability.htm
2. The following link contains some interactive Excel files that allow you toexperiment and compare the various sampling methods covered in this chapter.
http://www.coventry.ac.uk/ec//~nhunt/meths/ss.html
Objective: At the end of the chapter, students should be able to
1. Understand the concepts of population, sample and random sample;2. Understand the purpose and appropriate use of the 4 sampling methods ( random,
stratified, systematic and quota samples) in practical situations;
3. Explain in simple terms why a given sampling method may be unsatisfactory, andsuggest possible improvements.
1 Introduction
1.1 The Purpose of Sampling
In many practical statistical investigations,
we are interested in gathering information
about a population based on data derived
from a sample of the population.
This enables us to draw conclusions about a
large set of data (population) by studying arelativelysmallamount of data (sample).
For the sample to be useful, it has to be a
representative sample, that is, it has the
essential characteristics of the population
being studied, and is free of any type of
systematic bias.
A sample should be large enough to give a
good representation of the population but
small enough to be manageable.
population (see 1.2.1)
What is a good sample size?
large enough for a good
representation of population
small enough to be
manageable (time & costs
constraints
sample
(see 1.2.2)
http://www.statcan.ca/english/edu/power/ch13/probability/probability.htmhttp://www.statcan.ca/english/edu/power/ch13/probability/probability.htmhttp://www.coventry.ac.uk/ec/~nhunt/meths/ss.htmlhttp://www.coventry.ac.uk/ec/~nhunt/meths/ss.htmlhttp://www.coventry.ac.uk/ec/~nhunt/meths/ss.htmlhttp://www.statcan.ca/english/edu/power/ch13/probability/probability.htm -
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1.2 Definitions
1.2.1 Population
is a complete collection of
individuals or items about which
statistical information is desired.1.2.2 Sample
is a finite subset of a population.
1.2.3 Sampli ng Frame
is a list of all members of the
population from which we can draw
a sample.
1.2.4 Random Sample
is a sample drawn from a population
such that every member of thepopulation has an equal chance of
being selected.
Subset means part of/contains some
members of.
2 Sampling Methods
In this chapter, we shall study
the following 4 sampling
methods:
(simple) random sampling
systematic sampling stratified sampling
quota sampling
The first 3 methods will produce
a random sample, whereas
quota sampling will produce a
non-random sample.
population
What methods can used to collect a sample?
(simple) random sampling quota sampling
systematic sampling
stratified sampling
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2.1 (Simple) Random Sampling
In (simple) random sampling, every
possible sample of the same size has an
equal chance of being selected.
Also, each member of the population has an
equal chance of being included in the
sample.
2.1.1 Method for (Simple) Random Sampling
Suppose we want to obtain a sample of size
n from a population of sizeN.
1. Number the list of all the members of
the population from 1 toN.
2. Make a random selection ofn of these
numbers.
(This can be done by generatingrandom numbers with a computer or
GC using the command
randInt(1, N, n).)
3. The members corresponding to these
n numbers will constitute the sample.
Example of a contextual situation where the
simple random sampling method can be
carried out:
A total of 100 students from IJC attended
an enrichment class at Science Centre, fromwhich a simple random sample of 5
students are selected to answer a
questionnaire on the effectiveness of the
programme.
1. All the 100 students could each be
assigned a number based on
alphabetical order of their first names.
2. Generate 5 random integers between 1
and 100 using the GC command
randInt(1, 100, 5).
3. The students corresponding to the
randomly selected numbers will form
the sample.
Example:
12
34
5
9899
100
population of size, N =100
population of size, N =100
1. Ang Chee Yong
2. Chan Chee Tat
3. Cheong Chong Yan
4. ..
..99. Yang Yan Yan
100. Zaina Bte Aman
1
Use computer or GC to generate
n= 5 numbers at random2
21172
101
32138
101
46316
1764
32338
11100
Some of the many samples of size n= 5.
Each of these samples has the same
probability of being selected.
A simple random
sample of size, n =5
4. Cheong Chong Yan
25. Feng Tian
78. Lee Mary
89. Tan Mei Mei
99. Yang Yan Yan
3
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2.1.2 Use of the GC to generate random integers
The syntax for generating n random integers between m andNinclusive is as
follows:
randInt(m,N, n)
Example:
Generate 5 random integers between 1 and 100 inclusive
Keystrokes/Steps Screen Display
Step 1:
Start with the home screen.
Press .
Step 2:
Scroll right to PRBby pressing .
Press to select the function
5:randInt(.
Step 3:
Key in 1 for the lowest value of the
integers to be randomly generated, 100 for
the highestvalue, and 5 for the number of
integers to be randomly generated.
Press to generate the 5 random
integers.
Hence the 5 random integers generated are95, 91, 15, 52 and 41.
Lowest value of the
integers to be
randomly generated.
Highest value of the
integers to be
randomly generated.
Numberof integers
to be randomly
generated.
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2.1.3 Advantages and Disadvantages of Simple Random Sampling
Advantages Disadvantages
Method can be carried out
quickly and easily with the help
of a computer or GC to generaterandom numbers.
Sample obtained may happen to
be one that is not representative
of the population (see example1(iii) for illustration).
Suitable only if the sampling
frame is complete and up-to-
date.
Example 1:
In a large country ASU, an opinion poll which concerned the support for the
two presidential candidates Macane and Obaba was carried out before the
presidential election. A simple random sample of 1000 people was interviewed
and based on the results of the poll, an easy win was predicted for Macane.
(i) Describe how a simple random sample of 1000 can be selected.
(ii) Give one disadvantage of simple random sampling in the context of the
question.
(iii) When the actual election result was out, it turned out that Obaba won the
election. Give one possible reason for this difference from the predicted
result.
Solution
(i) Obtain the sampling frame of all
eligible voters from the Electoral
Board.Assign a number to each person in
the sampling frame. Generate 1000
random numbers with a computer.
The people corresponding to these
1000 numbers will constitute a
simple random sample of 1000.
Need to mention what the sampling frame
(all eligible voters in country ASU) is.
Need to assign a number to each member in
the sampling frame before generating n
(sample size) random (must mention this
word) numbers from the list of assigned
numbers.
Need to mention that the people that have
been assigned the chosen numbers make up
the selected sample.
(ii) Possible disadvantages:
As the country is large, it maybe expensive to travel to the
different parts of the country to
interview those sampled.
Some of those selected may not
be available for interview.
You may write any one of the possible
disadvantage BUT must ensure that you
write in context, and not the theoretical
answer found in 2.1.3.
(iii) Since the population is very large, a
sample size of 1000 may not result in
a representative sample. In fact, the
sample chosen could happen to have
been made up of a greater proportionof Macanes supporters.
This is an important disadvantage of simple
random sampling.
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2.2 Systematic Sampling
Systematic sampling is carried out by systematically drawing the sample from
the population.
2.2.1 Method for Systematic SamplingSuppose we want to obtain a sample of size n from a population of sizeN.
1. Listall members of the population in some order (usually with reference to
context and/or purpose of study).
For example, we can have
the pupils in a school arranged according to the school register if we want
to choose a sample to interview them for their opinion on the school
canteen food,
the order in which customers enter a particular store if we want to survey
how long they spend in the shop.
2. Determine the sampling in tervalN
kn
(Round down to the nearest integer if
necessary)
3. Choose the first member of the sample at random from the first kmembers,
i.e., choose an integer randomly from the numbers 1 to k.
4. Choose every other kth
member after the first member is chosen.
Illustration with an example:
Use systematic sampling to choose a sample of size 30 from the 400 pupils in a school
to interview them for their opinion on the school canteen food.
List all the students according to the school register.
Sampling interval, k =400
13.3 1330
Suppose the random number selected in is 2.
131 2 3 400..391
1
14 15 16 26
2
272829 39
3
378379380 390
30..
1
2
4 3
3Choose a
number
from 1 to 13at random.
131 2 3 400..391
1 3
378379380 390
30
14 15 16 26
2
272829 39
..
215 28
379
..
A systematicsample of
size 30
13 13 13
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2.2.2 Advantages and Disadvantages of Systematic Sampling
Advantages Disadvantages
Sample is more evenly spread
over the population.
Sample may be biased when the
order in which members of the
population are lined up have acyclic pattern.
Example 2:
A student councilor in a junior college is to conduct a survey of the graduating
class concerning plans for Graduation Night. There are 800 students in the
graduating class and he needs a sample of 80.
(i) Describe how a systematic sample may be taken.
(ii) Explain why the sample selected in (i) is a random sample.
Solution
(i) Get a list of all students from the
graduating class arranged according
to the school register (by classes).
Sampling interval k
Choose the first student at random
from the first 10 names.
Choose every 10th
name after the first
student has been chosen until a sample
of 80 is obtained.
It is meaningful to list the students by
classes in the school register order as
the information needed concerns the
plans for Grad Night, and generally
students from a more Science based
subject combination may have different
preference from that of a more
Humanities based subject combination.
With systematic sampling performed on
this list, the sample will be one that ismore representative of the whole
graduating batch of students.
(ii) Since the first member is selected at
random from the first 10 names, the
first 10 students in the register have a
probability of1
10of being selected.
The subsequent members of the
sample are fixed once the first member
has been selected. So every member inthe population has an equal chance,
which is1
10, of being selected to be in
the sample. Thus a systematic sample
is a random sample
Recall that
simple random sampling
systematic sampling
stratified sampling (section 2.3)
produce random samples.
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Example 3:
A company has 20 departments, each of which has 20 employees. A list of
employees is ordered by department, with the manager listed first, and then the
other employees in descending order of seniority.
A survey is conducted to find out about the employees opinions on their workenvironment. A systematic sampling approach was used to obtain a sample of
size 20. Give one disadvantage of such an approach in this case.
Solution
As the sample size is 20,
sampling interval is
k .
Thus every department
will have an employee
being selected.However, due to the
cyclic pattern in which
the employees are listed,
it will result in a sample
consisting of employees
of the same seniority(e.g. only managers or
only junior employees
are chosen).
The sample will thennot be representativeof
all the employees
opinions on their work
environment.
Suppose 19is the number obtained from the random selection
of numbers from 1 to 20 (first twenty numbers)
Note that you must give the disadvantage in context (see words
in bold in solution) and not merelythe technical answer cyclic
pattern as stated in the 2.2.2.
20
20
M AM supv grp L JJ
Department 1
M AM supv grp L JJ
Department 2
M AM supv grp L JJ
Department 3
M AM supv grp L JJ
De artment 19
M AM supv grp L JJ
De artment 20
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2.3 Stratified Sampling
Many populations contain identifiable
strata, which are distinctive non-
overlapping subsets of the
population.
Strata might be based on gender, age
groups, ethnic groups etc.
Stratified sampling ensures that the
proportions of the population falling
into these strata are reproduced by
the sample. Thus a stratified sample is
likely to be a representative sample of
its population.
However, a simple random sample is
unlikely to reproduce these proportions
and thus may provide a biased and
non-representative view of the
population.
Gender, age groups, ethnic groups can be strata
as a person can only be in 1 strata. For example,
a person is either a male or female and not
both!
You may view a stratified sample as a
miniature version of the population as the
proportions are the same.
A sample obtained from simple random sampling
may happen to be one with a greater
proportion of females which is not a good
representation of the population.
2.3.1 Method for Stratified Sampling
1. Divide the population intomeaningful strata.
Note: This method requires the
sampling frame, together with
additional information on the
chosen strata, which will give the
proportion for each stratum.
2. Draw random samples separately
from each stratum, with samplesize proportional to the relativesize of the stratum.
3. These separate random samples are
put together to form the sample of
the population.
The strata must be meaningful to the purposeof the study.
For example, the strata for survey of
preference for type of movies could be gender
as boys and girls generally prefer different
genre.
sample
male female
populationmale female
Sample (size = 12)
S1 S2 S3
435
population (size = 120 )strata
2 (S2)
strata 1
(S1)
50 30
strata 3
S3
40
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2.3.2 Advantages and Disadvantages of Stratified Sampling
Advantages Disadvantages
Likely to give a good
representative sample (for the
purpose of study) of thepopulation.
Strata may be difficult to define
clearly.
Example 4:
In a company, the proportions of staff in different age-groups are as follows:
Age under 40 between 40 and 60 over 60
Percentage 38 40 22
Describe how a stratified sample of 20 staff can be obtained.
Solution
Get a list ofall staff members. Divide them
into age-groups under 40, between 40 and
60, over 60.
Age under 40between
40 and 60over 60
Number
of staff
3820
100
7.6 8
40
20 8100
22
20100
4.4 4
From each age group, draw a random
sample of the required size as shown in the
table.
The three random samples drawn will
together form a stratified sample of 20
staff.
For stratified sampling, additional info
on the chosen strata (age in this case) is
required.
You must show how the sample size in
each strata is obtained.
Note that the numbers must be rounded
to the nearest integer because number
of people must be whole numbers. Also,
the total number must be the requiredsample size.
It is important that simple random
sampling is conducted within each
stratum.
Need to say that the random samples
from the strata together form the
required sample which is a miniature
version of the population with the sameproportions.
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2.4 Quota Sampling
Quota sampling is similar to
stratified sampling in the
sense that the population is
also divided into distinctive,non-overlapping strata.
However, the selection of
respondents within the strata
is left to the interviewer, and
thus is non-random.
Stratified sampling Quota sampling
Divide population into non-overlappingstrata (which is
meaningful to the purpose of study)Requiressampling frame
and information on the
strata chosen.
Does not require a sampling
frame.
The sample size for each
stratum is proportional
to the relative size of
the stratum.
A quota for each stratum is
specified (without
reference to population) to
define the sample.
Random sample selected
within each stratum.
Selection of sample within
each stratumis
non-random, dependent oninterviewer/convenience.
2.4.1 Method for Quota Sampling
1. Divide the population into
meaningful strata.
Note: For Quota Sampling, unlike
stratified sampling, a sampling
frame is not required.
2. A quota for each stratum is
specified to define the sample.
3. The interviewer will select a
sample that meets the
requirements of the quota.
Like in stratified sampling, the strata must be
meaningful to the purpose of the study.
The quota is pre-determined.
The interviewer can choose any person, at his
discretion/convenience, that meets the criteria
stipulated by the quota.
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Example of a contextual situation where the
simple random sampling method can be carried
out:
In a market survey, an interviewer is given an
assignment to interview 20 people in the streetbased on the following quota.
Sex Age Social Class
Male 9 20 29 4 High 2Female 11 30 44 6 Middle 4
45 64 7 Low 1465+ 3
The choice of the sample is subjective. A possible
choice that meets the quota set out is as follows:
Social Class
High Middle Low
Sex
/AgeM F M F M F
20 29 1 1 1 1
30 44 1 1 3 1
45 64 1 1 2 3
65+ 1 2
This is the quota requirements set
out for the sample.
This is a possible choice because
No. of males = 1+1+1+3+2+1=9
No. of females
= 1+1+1+1+1+1+3+2=11
No. with age 20-29 =1+1+1+1=4
No. with age 30-44 =1+1+3+1=6
No. with age 45-64 =1+1+2+3=7
No. with age 65+ =1+2=3
No. of high social class =1+1=2
No. of middle social class
=1+1+1+1=4
No. of low social class=1+3+2+1+1+1+3+2=14
Note that the interviewer can have
other choices but must ensure that
the quota requirements are met.
2.4.2 Advantages and Disadvantages of Quota Sampling
Advantages Disadvantages
Does not require a sampling
frame.
Information can be collectedquickly and easily.
Lower cost of implementation as
it is up to interviewers
convenience.
Sample obtained not random.
Sample is not a good representative of
the population as compared with othertypes of sampling.
Bias as the interviewer may simply
select those who are easiest to
interview.
However, quota sampling is widely used
in market research, street interviews and
opinion polls. It is very useful and
convenient though it is non-random in
nature.
Note:Information gathered from quotasampling should be treated with caution.
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Example 5:
A market researcher has been commissioned to find out how many times per
month members of the public visit their heartland shopping malls. She intends
to take a quota sample of 80 adults in the 18 to 65 age range.
Suggest suitable strata and describe how a quota sample may be taken. Give oneadvantage and one disadvantage of quota sampling.
Solution
As shopping habits depend on the age and
gender, the strata to be considered are age group
and gender.
Divide the population into age groups of
1825, 2635, 3655, 5665.
Within each age group, take equal number of
males and females.A possible quota for each stratum is as follows:
Age group
18
25
26
35
36
55
56
65Total
Male 10 10 10 10 40
Female 10 10 10 10 40
Total 20 20 20 20 80
The market researcher selects a sample such that
the quota is met.
Possible advantages
Quota sampling does not require a sampling
frame. Therefore quota sampling is
appropriate in this case because it would be
difficult/impossible to obtain a sampling
frame of all the shoppers visiting the mall.
Quota sampling is easy to administer and the
information can be collected quickly.
Quota sampling involves lower cost as
compared to other types of sampling methods
because it only requires minimal logistical
preparation.
Possible disadvantages
The sample is biased as the interviewer may
select those who are easiest to interview.
The sample is likely not a good representative
of the population.
The strata are chosen meaningfully
so that it addresses the purpose of
the survey.
When describing the method of
conducting quota sampling, need to
state the
strata & categories quota: the number to be chosen
for each stratum
State that the market researcher
selects a sample that meets the
requirements of the quota set out.
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Example 6:
There are 29 students in class A, 32 students in class B, 34 students in class C
and 25 students in classD. The teacher wants to choose a sample of 20 students
from these 4 classes. He decides to select 5 students at random from each class.
(i) Explain why the sample formed is not a random sample.(ii) Suggest and describe a method to obtain a random sample of 20 students
from all the 4 classes.
Solution
(i) The sample obtained is not a random sample because
each student from the 4 classes does not stand an equal
chance of being selected to be in the sample.
The probability of choosing a student from class A, B,
CandD are5
29
,5
32
,5
34
and5
25
respectively.
The probability of choosing
5 students at random from
a class of 29 students is
5
29.
(ii) The teacher may want to use stratified sampling to
obtain the required random sample.
Get a list of all the 120 students from the 4 classes.
Categorize them according to theirclasses.Class A B C D
No. of
students
2920
120
4.83
5
3220
120
5.33
5
3420
120
5.67
6
2520
120
4.17
4
From each class, draw a random sample of the
required size as shown in the table.
The four samples drawn will together form a
stratified sample of 20 students from the 4 classes.
Recall that a stratified
sample is a random sample.
The strata used here is
the class.
Summary
Sample
frame
Random Advantage Disadvantage
Random
Sampling
Yes Yes Quick and easy
to carry out
Difficult to get
sample frame
Systematic
Sampling
No Yes More evenly
spread
Maybe Cyclic, not
representative
Stratified
Sampling
Yes Yes Good
representative
Difficult to define
and obtain strata
Quota
Sampling
No No Easy and
economical
Non-random,
maybe biased
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Sampling Methods (Tutorial)
1 (2010/Prelim/MI/10)
A group of 10 students is to be selected for an overseas exchange programme
from an institute of 3 pre-university (PU) levels, with a total enrolment of 1000
students.
(i) Describe how the selection may be done using simple random sampling
and suggest an advantage of this sampling method in the context of the
question. [3]
(ii) Write down one disadvantage of this method of selection, citing an
example in the context of the question. [2]
(iii) State and describe a sampling method that gives a better representation
of each pre-university level in the selection process, given that there are
500 PU1 students, 300 PU2 students and 200 PU3 students.
[3]
2 (H2 Maths/9740/N08/II/5)
A school has 950 pupils. A sample of 50 pupils is to be chosen to take part in a
survey. Describe how the sample could be chosen using systematic sampling.
[2]
The purpose of the survey is to investigate pupils opinions about the spo rts
facilities available at the school. Give a reason why a stratified sample might be
preferable in this context. [2]
3 (H2 Maths/9740/N09/II/5)
A cinema manager wishes to take a survey of opinions of cinema-goers.
Describe how a quota sample of size 100 might be obtained, and state one
disadvantage of quota sampling. [3]
4 (H1 Maths/8863/N09/I/11(a))
An insurance company receives a large number of claims for flood damage. On
a particular day the company receives 72 such claims. Because of staff
shortages, it is only possible to process 8 of these claims.
(i) Describe how you would choose a systematic random sample of size 8
from the received claims. [2]
(ii) Comment on whether this method of sampling gives a better indication
of the value of the 72 claims as compared to simply choosing as the
sample the first 8 claims received. [1]
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5 An employment agency wants to estimate the number of unemployed people in
an HDB new town, and so decide to obtain information from a sample of its
adults who are of working age.
(i) Give one reason why it would not be appropriate to obtain the sample by
stopping adults at the town centres supermarket during one workingday.
(ii) The new town has 10 000 households and they can be classified as
follows.
Housing type No. of household
3-room 1250
4-room 4500
5-room 2000
Executive 1750
Others 500
Explain, using suitable calculations, how the agency could use this
information to obtain a representative sample of 1000 households.
(iii) In an attempt to obtain a representative sample of people, the agency
considers two options:
(A) to interview all the working age adults in the chosen household;
(B) to interview one such person from each household.
The agency decided on option A. Give one reason in favour of this
decision.
6 (H2 Maths/9740/N07/II/5)
(i) Give a real-life example of a situation in which quota sampling could be
used. Explain why quota sampling would be appropriate in this
situation, and describe briefly any disadvantage that quota sampling has.
[4]
(ii) Explain briefly whether it would be possible to use stratified sampling
the situation you have described in part (i). [1]
7 A certain private estate contains many small houses (with small gardens) and a
few large houses (with large gardens). A sample survey of all houses is to be
carried out in this estate. A student suggests that the sample could be selected
by sticking a pin into a map of the estate the requisite number of times, while
blindfolded.
(i) Give two reasons why this method does not produce a random sample.(ii) Describe a better method.