Probability sampling

Probability sampling
Probability sampling involves the selection of a sample from a population, based on the
principle of randomization or chance. Probability sampling is more complex, more time-
consuming and usually more costly than non-probability sampling. However, because units
from the population are randomly selected and each unit's probability of inclusion can be
calculated, reliable estimates can be produced along with estimates of the sampling error, and
inferences can be made about the population.
There are several different ways in which a probability sample can be selected. The method
chosen depends on a number of factors, such as the available sampling frame, how spread out
the population is, how costly it is to survey members of the population and how users will
analyse the data. When choosing a probability sample design, your goal should be to
minimize the sampling error of the estimates for the most important survey variables, while
simultaneously minimizing the time and cost of conducting the survey.
The following are the most common probability sampling methods:
simple random sampling
systematic sampling
sampling with probability proportional to size
stratified sampling
cluster sampling
multi-stage sampling
multi-phase sampling
Simple random sampling
In simple random sampling, each member of a population has an equal chance of being
included in the sample. Also, each combination of members of the population has an equal
chance of composing the sample. Those two properties are what defines simple random
sampling. To select a simple random sample, you need to list all of the units in the survey
population.
Example 1: To draw a simple random sample from a telephone book, each entry would need
to be numbered sequentially. If there were 10,000 entries in the telephone book and if the
sample size were 2,000, then 2,000 numbers between 1 and 10,000 would need to be
randomly generated by a computer. Each number will have the same chance of being
generated by the computer (in order to fill the simple random sampling requirement of an
equal chance for every unit). The 2,000 telephone entries corresponding to the 2,000
computer-generated random numbers would make up the sample.
Simple random sampling can be done with or without replacement. A sample with
replacement means that there is a possibility that the sampled telephone entry may be selected
twice or more. Usually, the simple random sampling approach is conducted without
replacement because it is more convenient and gives more precise results. For the purpose of
these descriptions, when we discuss simple random sampling, we will refer to sampling
without replacement.

Simple random sampling is the easiest method of sampling and it is the most commonly used.
Advantages of this technique are that it does not require any additional information on the
frame (such as geographic areas) other than the complete list of members of the survey
population along with information for contact. Also, since simple random sampling is a
simple method and the theory behind it is well established, standard formulas exist to
determine the sample size, the estimates and so on, and these formulas are easy to use.
On the other hand, this technique makes no use of auxiliary information present on the frame
(i.e., number of employees in each business) that could make the design of the sample more
efficient. And although it is easy to apply simple random sampling to small populations, it
can be expensive and unfeasible for large populations because all elements must be identified
and labeled prior to sampling. It can also be expensive if personal interviewers are required
since the sample may be geographically spread out across the population.
A lottery draw is a good example of simple random sampling. For example, when a sample of
6 numbers is randomly generated from a population of 49, each number has an equal chance
of being selected and each combination of 6 numbers has the same chance of being the
winning combination. Even though people tend to avoid combinations such as 1-2-3-4-5-6, it
has the same chance of being the winning set of numbers as the combination of 8-15-21-28-
32-40.
Example 2: Suppose your school has 500 students and you need to conduct a short survey on
the quality of the food served in the cafeteria. You decide that a sample of 10 students should
be sufficient for your purposes. In order to get your sample, you assign a number from 1 to
500 to each student in your school. To select the sample, you use a table of randomly
generated numbers. All you have to do is pick a starting point in the table (a row and column
number) and look at the random numbers that appear there. In this case, since the data run
into three digits, the random numbers would need to contain three digits as well. Ignore all
random numbers after 500 because they do not correspond to any of the students in the
school. Remember that the sample is without replacement, so if a number recurs, skip over it
and use the next random number. The first 10 different numbers between 001 and 500 make
up your sample.
Example 3: Imagine that you own a movie theatre and you are offering a special horror
movie film festival next month. To decide which horror movies to show, you survey
moviegoers asking them which of the listed movies are their favourites. To create the list of
movies needed for your survey, you decide to sample 100 of the 1,000 best horror movies of
all time. The horror movie population is divided evenly into classic movies (those filmed in
or before 1969) and modern movies (those filmed in or later than 1970). One way of getting a
sample would be to write out all of the movie titles on slips of paper and place them in an
empty box. Then, draw out 100 titles and you will have your sample. By using this approach,
you will have ensured that each movie had an equal chance of selection
You can also calculate the probability of a given movie being selected. Since we know the
sample size (n) and the total population (N), calculating the probability of being included in
the sample becomes a simple matter of division:
Probability of being selected (same for each horror movie)

= (n ÷ N) x 100%
= (100 ÷ 1,000) x 100%
= 10%
This means that every movie title on the list has a 10% or a 1 in 10 chance of being selected.
You can see that that one disadvantage of simple random sampling (not the only
disadvantage, but an important one) is that even if you know that the population is made up
of 500 classic movies and 500 modern movies and you know each movie's release date from
the sampling frame, no use is made of this information. This sample might contain 77 classic
movies and 23 modern movies, which would not be representative of the whole horror movie
population.
There are ways to overcome this problem (these will be briefly discussed in the Estimation
section), but there are also ways to account for this information. (This will also be discussed
later, under the section on Stratified sampling.)
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Systematic sampling
Sometimes called interval sampling, systematic sampling means that there is a gap, or
interval, between each selected unit in the sample. In order to select a systematic sample, you
need to follow these steps:
1. Number the units on your frame from 1 to N (where N is the total population size).
2. Determine the sampling interval (K) by dividing the number of units in the population
by the desired sample size. For example, to select a sample of 100 from a population
of 400, you would need a sampling interval of 400 ÷ 100 = 4. Therefore, K = 4. You
will need to select one unit out of every four units to end up with a total of 100 units
in your sample.
3. Select a number between one and K at random. This number is called the random
start and would be the first number included in your sample. Using the sample above,
you would select a number between 1 and 4 from a table of random numbers. If you
choose 3, the third unit on your frame would be the first unit included in your sample;
if you choose 2, your sample would start with the second unit on your frame.

4. Select every Kth
(in this case, every fourth) unit after that first number. For example,
the sample might consist of the following units to make up a sample of 100: 3 (the
random start), 7, 11, 15, 19...395, 399 (up to N, which is 400 in this case).
Using the example above, you can see that with a systematic sample approach there are only
four possible samples that can be selected, corresponding to the four possible random starts:
1, 5, 9, 13... 393, 397
2, 6, 10, 14... 394, 398
3, 7, 11, 15... 395, 399
4, 8, 12, 16... 396, 400
Each member of the population belongs to only one of the four samples and each sample has
the same chance of being selected. From that, we can see that each unit has a one in four
chance of being selected in the sample. This is the same probability as if a simple random
sampling of 100 units was selected. The main difference is that with simple random
sampling, any combination of 100 units would have a chance of making up the sample, while
with systematic sampling, there are only four possible samples. From that, we can see how
precise systematic sampling is compared with simple random sampling. The population's
order on the frame will determine the possible samples for systematic sampling. If the
population is randomly distributed on the frame, then systematic sampling should yield
results that are similar to simple random sampling.
This method is often used in industry, where an item is selected for testing from a production
line to ensure that machines and equipment are of a standard quality. For example, a tester in
a manufacturing plant might perform a quality check on every 20th
product in an assembly
line. The tester might choose a random start between the numbers 1 and 20. This will
determine the first product to be tested; every 20th
product will be tested thereafter.
Interviewers can use this sampling technique when questioning people for a sample survey.
The market researcher might select, for example, every 10th person who enters a particular
store, after selecting the first person at random. The surveyor may interview the occupants of
every fifth house on a street, after randomly selecting one of the first five houses.
Example 4: Imagine you have to conduct a survey on student housing for your university or
college. Your school has an enrolment of 10,000 students and you want to take a systematic
sample of 500 students. In order to do this, you must first determine what your sampling
interval (K) would be:
Total population ÷ sample size = sampling interval
N ÷ n = K
= 10,000 ÷ 500
= 20
To begin this systematic sample, all students would have to be assigned sequential numbers.
The starting point would be chosen by selecting a random number between 1 and 20. If this
number were 9, then the 9th
student on the list would be selected along with every 20th

student thereafter. The sample of students would be those corresponding to student numbers
9, 29, 49, 69...9,929, 9,949, 9,969 and 9,989.
In the examples used thus far, the sampling interval K was always a whole number, but this is
not always the case. For example, if you want a sample of 30 from a population of 740, your
sampling interval (or K) will be 24.7. In these cases, there are a few options to make the
number easier to work with. You can round the number—either round it up to the nearest
whole number or round it down. Rounding down will ensure that you select at least the
number of units you originally wanted (and you can then delete some units to get the exact
sample size you wanted). Techniques exist to adapt systematic sampling to the case where N
(total population) is not a multiple of n (sample size), but still give a sample exactly the same
as the n units. These techniques will not be discussed here.
The advantages of systematic sampling are that the sample selection cannot be easier (you
only get one random number—the random start—and the rest of the sample automatically
follows) and that the sample is distributed evenly over the listed population. The biggest
drawback of the systematic sampling method is that if there is some cycle in the way the
population is arranged on a list and if that cycle coincides in some way with the sampling
interval, the possible samples may not be representative of the population. This can be seen in
the following example:
Example 5: Suppose you run a large grocery store and have a list of the employees in each
section. The grocery store is divided into the following 10 sections: deli counter, bakery,
cashiers, stock, meat counter, produce, pharmacy, photo shop, flower shop and dry cleaning.
Each section has 10 employees, including a manager (making 100 employees in total). Your
list is ordered by section, with the manager listed first and then, the other employees by
descending order of seniority.
If you wanted to survey your employees about their thoughts on their work environment, you
might choose a small sample to answer your questions. If you use a systematic sampling
approach and your sampling interval is 10, then you could end up selecting only managers or
the newest employees in each section. This type of sample would not give you a complete or
appropriate picture of your employees' thoughts.
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Sampling with probability proportional to size
Probability sampling requires that each member of the survey population have a chance of
being included in the sample, but it does not require that this chance be the same for
everyone. If there is information available on the frame about the size of each unit (e.g.,
number of employees for each business) and if those units vary in size, this information can
be used in the sampling selection in order to increase the efficiency. This is known as
sampling with probability proportional to size (PPS). With this method, the bigger the size of
the unit, the higher the chance it has of being included in the sample. For this method to bring
increased efficiency, the measure of size needs to be accurate. This is a more complex
sampling method that will not be discussed in further detail here.

Stratified sampling
Using stratified sampling, the population is divided into homogeneous, mutually exclusive
groups called strata, and then independent samples are selected from each stratum. Any of the
sampling methods mentioned in this section (and others that exist) can be used to sample
within each stratum. The sampling method can vary from one stratum to another. When
simple random sampling is used to select the sample within each stratum, the sample design
is called stratified simple random sampling. A population can be stratified by any variable
that is available for all units on the sampling frame prior to sampling (e.g., age, sex, province
of residence, income, etc.).
Why do we need to create strata? There are many reasons, the main one being that it can
make the sampling strategy more efficient. It was mentioned earlier that you need a larger
sample to get a more accurate estimation of a characteristic that varies greatly from one unit
to the other than for a characteristic that does not. For example, if every person in a
population had the same salary, then a sample of one individual would be enough to get a
precise estimate of the average salary.
This is the idea behind the efficiency gain obtained with stratification. If you create strata
within which units share similar characteristics (e.g., income) and are considerably different
from units in other strata (e.g., occupation, type of dwelling) then you would only need a
small sample from each stratum to get a precise estimate of total income for that stratum.
Then you could combine these estimates to get a precise estimate of total income for the
whole population. If you were to use a simple random sampling approach in the whole
population without stratification, the sample would need to be larger than the total of all
stratum samples to get an estimate of total income with the same level of precision.
Stratified sampling ensures an adequate sample size for sub-groups in the population of
interest. When a population is stratified, each stratum becomes an independent population
and you will need to decide the sample size for each stratum.
Example 6: Suppose you want to estimate how many high school students have part-time
jobs at the national level and also in each province. If you were to select a simple random
sample of 25,000 people from a list of all high school students in Canada (assuming such a
list was available for selection), you would end up on average with just a little over 100
people from Prince Edward Island, since they account for less than half of a percent of the
whole Canadian population. This sample would probably not be large enough for the kind of
detailed analysis you had in mind. Stratifying your list by province, again assuming that this
information is available, and then selecting a sample size for each province would allow you
to decide on the exact sample size needed for that specific province. Thus, in order to get
good representation of Prince Edward Island, you would use a larger sample than the one
allotted to it by the simple random sampling approach.
Example 7: An Ontario school board wanted to assess student opinion on dropping Grade 13
from the secondary school program. They decided to survey students from Elmsview High
School. To ensure a representative sample of students from all grade levels, the school board
used a stratified sampling technique.

In this case, the strata were the five grade levels (grades 9 to 13). The school board then
selected a sample within each stratum. The students selected in this sample were extracted
using simple random or systematic sampling, making up a total sample of 100 students.
Stratification is most useful when the stratifying variables are
simple to work with,
easy to observe, and
closely related to the topic of the survey.
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Cluster sampling
Sometimes it is too expensive to spread a sample across the population as a whole. Travel
costs can become expensive if interviewers have to survey people from one end of the
country to the other. To reduce costs, statisticians may choose a cluster sampling technique.
Cluster sampling divides the population into groups or clusters. A number of clusters are
selected randomly to represent the total population, and then all units within selected clusters
are included in the sample. No units from non-selected clusters are included in the sample—
they are represented by those from selected clusters. This differs from stratified sampling,
where some units are selected from each group.
Examples of clusters are factories, schools and geographic areas such as electoral
subdivisions. The selected clusters are used to represent the population.
Example 8: Suppose you are a representative from an athletic organization wishing to find
out which sports Grade 11 students are participating in across Canada. It would be too costly
and lengthy to survey every Canadian in Grade 11, or even a couple of students from every
Grade 11 class in Canada. Instead, 100 schools are randomly selected from all over Canada.
These schools provide clusters of samples. Then every Grade 11 student in all 100 clusters is
surveyed. In effect, the students in these clusters represent all Grade 11 students in Canada.
Example 9: Imagine that the municipal council of a small city wants to investigate the use of
health care services by residents.
First, the council requests from Statistics Canada electoral subdivision maps that identify and
label each city block. From these maps, the council creates a list of all city blocks. This list
will serve as the sampling frame.
Every household in that city belongs to a city block, and each city block represents a cluster
of households. The council randomly picks a number of city blocks. Using the simple random
sample approach, then the council creates a list of all households in the selected city blocks;
these households make up the survey sample.

As mentioned, cost reduction is a reason for using cluster sampling. It creates 'pockets' of
sampled units instead of spreading the sample over the whole territory. Another reason is that
sometimes a list of all units in the population (a requirement when conducting simple random
sample, systematic sample or sampling with probability proportional to size) is not available,
while a list of all clusters is either available or easy to create.
In most cases, the main drawback is a loss of efficiency when compared with simple random
sampling. It is usually better to survey a large number of small clusters instead of a small
number of large clusters. This is because neighbouring units tend to be more alike, resulting
in a sample that does not represent the whole spectrum of opinions or situations present in the
overall population. In the two previous examples, students in the same school tend to
participate in the same types of sports (depending on the facilities available at their school);
similarly, elderly people have a tendency to live in specific neighbourhoods and to be heavy
users of health services.
Another drawback to cluster sampling is that you do not have total control over the final
sample size. Since not all schools have the same number of Grade 11 students and city blocks
do not all have the same number of households, and you must interview every student or
household in your sample, the final size may be larger or smaller than you expected.
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Multi-stage sampling
Multi-stage sampling is like the cluster method, except that it involves picking a sample from
within each chosen cluster, rather than including all units in the cluster. This type of sampling
requires at least two stages. In the first stage, large groups or clusters are identified and
selected. These clusters contain more population units than are needed for the final sample.
In the second stage, population units are picked from within the selected clusters (using any
of the possible probability sampling methods) for a final sample. If more than two stages are
used, the process of choosing population units within clusters continues until there is a final
sample.
Example 10: In Example 8, a cluster sample would choose 100 schools and then interview
every Grade 11 student from those schools. Instead in multi-stage sampling, you could select
more schools, get a list of all Grade 11 students from these selected schools and select a
random sample (e.g., simple random sample) of students from each school. This would be a
two-stage sampling design.
You could also get a list of all Grade 11 classes in the selected schools, pick a random sample
of classes from each of those schools, get a list of all the students in the selected classes and
finally select a random sample of students from each class. This would be a three-stage
sampling design. Each time we add a stage, the process becomes more complex.
Now imagine that each school has on average 80 Grade 11 students. Cluster sampling would
then give your organization a sample of about 8,000 students (100 schools x 80 Grade 11
students). If you wanted a bigger sample, you could select schools with more students; and

for a smaller sample you could select schools with fewer students.
One way to control the sample size would be to stratify the schools into large, medium and
small sizes (in terms of the number of Grade 11 students) and select a sample of schools from
each stratum. This is called stratified cluster sampling.
With a three-stage design, you could select a sample of 400 schools, then select two Grade 11
classes per school (assuming that there are two or more Grade 11 classes per school). Finally,
you could select 10 students per class. This way, you still end up with a sample of about
8,000 students (400 schools x 2 classes x 10 students), but the sample will be more spread
out.
You can see from this example that with multi-stage sampling, you still have the benefit of a
more concentrated sample for cost reduction. However, the sample is not as concentrated as
other clusters and the sample size is still bigger than for a simple random sample size. Also,
you do not need to have a list of all of the students in the population. All you need is a list of
the classes from the 400 schools and a list of the students from the 800 classes. Admittedly,
more information is needed in this type of sample than what is required in cluster sampling.
However, multi-stage sampling still saves a great amount of time and effort by not having to
create a list of all of the units in a population.
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Multi-phase sampling
A multi-phase sample collects basic information from a large sample of units and then, for a
subsample of these units, collects more detailed information. The most common form of
multi-phase sampling is two-phase sampling (or double sampling), but three or more phases
are also possible.
Multi-phase sampling is quite different from multi-stage sampling, despite the similarities in
name. Although multi-phase sampling also involves taking two or more samples, all samples
are drawn from the same frame and at each phase the units are structurally the same.
However, as with multi-stage sampling, the more phases used, the more complex the sample
design and estimation will become.
Multi-phase sampling is useful when the frame lacks auxiliary information that could be used
to stratify the population or to screen out part of the population.
Example 11: Suppose that an organization needs information about cattle farmers in Alberta,
but the survey frame lists all types of farms—cattle, dairy, grain, hog, poultry and produce.
To complicate matters, the survey frame does not provide any auxiliary information for the
farms listed there.
A simple survey could be conducted whose only question is "Is part or all of your farm
devoted to cattle farming?" With only one question, this survey should have a low cost per
interview (especially if done by telephone) and, consequently, the organization should be able
to draw a large sample. Once the first sample has been drawn, a second, smaller sample can

be extracted from among the cattle farmers and more detailed questions asked of these
farmers. Using this method, the organization avoids the expense of surveying units that are
not in this specific scope (i.e., non-cattle farmers).
Multi-phase sampling can be used when there is insufficient budget to collect information
from the whole sample, or when doing so would create excessive burden on the respondent,
or even when there are very different costs of collection for different questions on a survey.
Example 12: A health survey asks participants some basic questions about their diet,
smoking habits, exercise routines and alcohol consumption. In addition, the survey requires
that respondents subject themselves to some direct physical tests, such as running on a
treadmill or having their blood pressure and cholesterol levels measured.
Filling out questionnaires or interviewing participants are relatively inexpensive procedures,
but the medical tests require the supervision and assistance of a trained health practitioner, as
well as the use of an equipped laboratory, both of which can be quite costly. The best way to
conduct this survey would be to use a two-phase sample approach. In the first phase, the
interviews are performed on an appropriately sized sample. From this sample, a smaller
sample is drawn. This second sample will take part in the medical tests.

Probability sampling

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