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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

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:

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.)

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:

- Number the units on your frame from 1 to
**N**(where**N**is the total population size).

- 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.

- 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.

- Select every
**K**(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^{th}**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 20^{th} 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 20^{th} 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** ÷

= 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 9^{th} 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.

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.

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.

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.

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.

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.

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.