Section 1.2: Sampling Methods

Figure 1.5: The process of statistics emphasizing step 2.

This section focuses on the collection of data, step 2 in the process of statistics. To begin with, data can be found from many sources, including:

•a census (if this is possible)

•existing sources (previously collected data, websites, etc.)

•survey sampling (creating, administering, and analyzing surveys)

•designed experiments (compare results for groups under different conditions)

Recall that a census is a list of each individual in the population along with a measurement or observation. As we discussed in section 1.1 it is not always feasible to conduct a census. For instance, collecting information from every graduating high school senior in the United States would require an enormous effort. When conducting a census is too difficult, we instead use a sample to represent the population. If we do a good job selecting the sampled individuals, the sample will mimic the characteristics of the population, and so the results will be good approximations for the population parameters. In this section we discuss good methods for sampling.

Figure 1.6: We use sampling to represent a large population with a smaller subset. (Image credit Dan Kernler licensed under Creative Commons License 3.0) (opens in a new tab)

The most widely used and simplest sampling technique is the simple random sample. If the whole population has size N, then we select a sample of size n, where n is generally much smaller than N. A list of all individuals in the population is called a Frame. With simple random sampling, we randomly choose individuals from the frame. The important thing is that to be a simple random sample, each individual from the population must have an equal chance of being selected for inclusion in the sample, and each group of size n has an equal chance of being selected. That sounds far more complicated than it really is. You have probably used simple random sampling and not even been aware of it. Simple random sampling is like drawing names from a hat.

Okay, so how is that simple? Consider this example. Suppose that we have 12 students, numbered from 1 to 12 (see figure 1.7 below). The frame is a list of the names of all 12 students in the population. We write the numbers of all 12 students on pieces of paper and put them into a hat. Then we randomly draw 4 papers from the hat and look at the numbers. Let’s say that we randomly selected student numbers 2, 5, 8, and 10. Each student has an equal chance of being selected, and every group of 4 students has an equal chance of being selected, so this is a simple random sample.

Figure 1.7: Image of a simple random sample. (Image credit Dan Kernler licensed

under Creative Commons License 3.0) (opens in a new tab)

Solution: One option would be to have each student write their name on a slip of paper and place the papers in a hat. Then draw 4 of them from a hat. The names drawn are the students used in the sample.

Another option would be to number the students from 1 to 46. Then use a random number generator on a calculator, website, or using StatCrunch to generate 4 numbers between 1 and 46. Choose the students that correspond to the numbers to include in the sample.

Solution:

a.Select the first 4 students who show up to class. This is not a simple random sample. Students who show up later have no chance to be selected.

b.Select the first 4 students in the third row of desks. This is not a simple random sample. Students in other rows have no chance at being selected.

c.Select the first seat in each of the first four rows. This is not a random sample. Students who do not sit in the front row of desks have no chance at being selected.

d.Select 4 students whose names I know. This is not a random sample. Students whose names I do not yet know have no chance at being selected.

Although simple random sampling is the most important of the sampling techniques, there are a few others that are worth discussing. They are stratified sampling, cluster sampling, systematic sampling, and convenience sampling.

For example, strata might be gender, ethnicity, education level, etc. You might use stratified sampling when you want to make sure that you include some individuals from different groups.

Figure 1.8: Diagram of stratified sampling (Image credit Dan Kernler, licensed under

Creative Commons License 3.0) (opens in a new tab)

Returning to our previous example, see figure 1.8, let the strata be the type of student- evening, daytime, or online. Let dark blue represent an evening student, the medium blue represent daytime students, and light blue represent online students. If we want to obtain data from all three strata, we first separate them, then use a simple random sample on each of the strata.

In this case, evening students make up 25% of the population, so we randomly select 1 of the 3 evening students for our sample. That one selected student will be 1 out of 4, or 25% of the sample. So, the percent of evening students in the population matches the percent of evening students in the sample.

Similarly, daytime students make up 50% of the population, so 50% of the sample should be made up of daytime students. That means we randomly select 2 from the 6 daytime students to include. Again, the percent of daytime students in the population matches the percent of daytime students in the sample.

Finally, the Online students make up 25% of the population, and so 25% of the sample (1 out of the 3 online students) is online students.

Using stratified sampling is appropriate if you think subgroups of the population might have differing opinions, and you want to make sure you include all views. In the example above, if the question was about feelings of safety on campus, making sure you include some daytime students and some evening students is appropriate, because they may have different experiences that you want to include. If you only obtained responses from daytime students, they may feel safer because they attend in daylight. Perhaps evening students feel less safe but were not included in the survey. This could lead you to an incorrect conclusion.

On the other hand, if there is no reason to think that the strata would have different views, using a stratified sample might not be necessary. If you ask students about their favorite food, there is no reason to think that being a daytime, evening, or online student would be a factor.

Solution:

a.If you were concerned about a difference in opinion due to gender identification, you could divide the population into strata based on gender identity. This would allow you to compare the data from each gender identification subgroup. Are the results between the gender identities different?

b.If you want to know how residents feel toward the police, you could divide the population into strata based on race or ethnicity. Then you could sample from those groups proportionally. Are there differences in attitudes toward police among different ethnicities?

If it is not possible to create a frame for the data, you can use systematic sampling. In this method we choose a starting point, then we follow a pattern, such as selecting every 100^th visitor to a website. Since we don’t know who might eventually visit the site, we can’t make a frame. So, we just include data from every 100^th visitor systematically.

In our example, if we were to put the 12 students in order, we can select every 3^rd student starting from a random place between 1 and 3. For this simple example, say we start at the 2^nd student and choose every 3^rd student after that. See figure 1.9 below.

Figure 1.9: Diagram of systematic sampling. (Image credit Dan Kernler, licensed under

Creative Commons License 3.0) (opens in a new tab)

We would include student numbers 2, 5, 8, and 11. Note that this sample does not include any of the dark blue students. If we had a larger population and sample size, we would expect to have some representation from all groups.

Another useful sampling technique is cluster sampling. Cluster sampling is similar to stratified sampling, and they are often confused. Stratified sampling divides the population into groups, then randomly selects some individuals from those groups. Cluster sampling uses naturally divided groups, then randomly selects some of those groups to include in the sample. Every individual in the selected cluster is included in the sample.

Cluster sampling is useful when you have groups that are already formed, for instance classes at your college. An administrator might want to give a survey to the student population. One easy way to do that is to go to a random selection of classes and give the survey. Here, the classes are the clusters. Each student that is in a selected class takes the survey, while those in other clusters do not.

If our 12 students have been paired up, perhaps for a group project, then we have clusters created. We have 6 clusters of 2 students apiece. If we randomly select two clusters, we have a sample of 4 students. Here we randomly selected the clusters including student numbers 5, 6, 11, and 12. See figure 1.10.

Figure 1.10: Diagram of a cluster sample. (Image credit Dan Kernler, licensed

under Creative Commons License 3.0) (opens in a new tab)

Solution:Assume that classes have an average of 30 students enrolled. Using the class schedule, randomly select ten courses. Visit those classes and survey all students regarding satisfaction with public transportation and/or parking on campus. This approach will result in about 10 classes × 30 students=300 students included in the sample.

Solution:Since a sample of 240 electronics is desired, randomly select 20 boxes of electronics and test all the devices in each box. That will result in 20×12=240 electronic devices tested. In this case, each box of 12 is a cluster.

Sometimes we don’t have the time or energy to conduct a scientifically sound survey or experiment. In those cases, we can use what we refer to as a convenience sample. A convenience sample is like asking your friends their opinion or asking classmates if they have seen a particular movie. You are collecting data, but you may not be getting a representative sample. Sure, it is convenient, but the results will not be reliable.

For example, if I were to ask my statistics class students if they intend to transfer, that would be convenient. It would also be likely that 100% would intend to transfer. One of the requirements for transfer is to take a quantitative reasoning course, which a statistics course satisfies. But there may be other students who don’t intend to transfer and so do not enroll in statistics. I have not included those students in my data collection. This is a biased sample, meaning that my sample overcounted some important traits. The problem is that the people who it is convenient for me to sample do not fairly represent the whole population.

A particularly tricky convenience sample is called a self-selected survey. These are samples where the individuals decide whether to participate. Any email or online survey that you see is self-selected, meaning that only the individuals who care about the topic are likely to participate. And that means that you only obtain information from the extremes. Often response rates to online surveys are less than 10%, meaning that 90% of people did not fill out the survey. Only the 10% who had the strongest opinions were included. In general, you should always view convenience samples and self-selected surveys with skepticism.

Solution:

a.Create a list of all students who have attended tutoring at the center. Divide the students by the class they received tutoring in (Algebra, Statistics, Calculus, etc.). Then select simple random samples from each group.

This is a stratified sample. The strata are the classes Algebra, Statistics, Calculus, etc.

b.Randomly select 10 days from the semester and survey all students who attended tutoring on those days.

This is a cluster sample. The clusters are the days that were selected. Every student who received tutoring on the selected days is included.

c.Create a list of all students who have attended tutoring at the center. Randomly select students from the list to survey.

This is a simple random sample.

d.Send an email to all students who have received tutoring and ask them to complete the survey.

This is a convenience sample. More specifically, it is a self-selected survey. Only students with strong opinions about their experience will likely complete the survey. It is likely that the results will not be reliable.

e.Create a list of all students who have attended tutoring at the center. Number the students starting with 1, then choose a random number between 1 and 49, select that student and every 50^th student that follows on the list.

This is a systematic sample.