1.2: Sampling Methods
- Page ID
- 58244
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Introduction of various sampling methods used for effective data collection.
- Understand and apply simple random, stratified, systematic, cluster, and convenience sampling techniques.
- Select appropriate sampling methods based on population structure and accessibility.
Introduction
As stated before, if you want to know something about a population, it is often impossible or impractical to examine the whole population. It might be too expensive in terms of time or money. It might be impractical – you can’t test all batteries for their length of lifetime because there wouldn’t be any batteries left to sell. You need to look at a sample. Hopefully, the sample behaves the same as the population.
When you choose a sample, you want it to be as similar to the population as possible. If you want to test a new painkiller for adults, you would want the sample to include people who are fat, skinny, old, young, healthy, not healthy, male, female, etc.
There are many ways to collect a sample. None are perfect, and you are not guaranteed to collect a representative sample. That is, unfortunately, the limitations of sampling. However, several techniques can result in samples that give you a semi-accurate picture of the population. Just remember to be aware that the sample may not be representative. As an example, you can take a random sample of a group of people that are equally males and females, yet by chance, everyone you choose is female. If this happens, it may be a good idea to collect a new sample if you have the time and money.
There are many sampling techniques, though only four will be presented here. The simplest, and the type that is strived for, is a simple random sample. This is where you pick the sample such that every sample has the same chance of being chosen. This type of sample is hard to collect, since it is sometimes difficult to obtain a complete list of all individuals. There are many cases where you cannot conduct a truly random sample. However, you can get as close as you can. Now, we suppose you are interested in what type of music people like. It might not make sense to try to find an answer for everyone in the U.S. You probably don’t like the same music as your parents. The answers vary so much that you probably couldn’t find an answer for everyone all at once. It might make sense to look at people in different age groups or people of different ethnicities. This is called a stratified sample. The issue with this sample type is that sometimes people subdivide the population too much. It is best to just have one stratification. Also, a stratified sample has similar problems that a simple random sample has. If your population has some order in it, then you could do a systematic sample. This is popular in manufacturing. The problem is that it is possible to miss a manufacturing mistake because of how this sample is taken. If you are collecting polling data based on location, then a cluster sample that divides the population based on geographical means would be the easiest sample to conduct. The problem is that if you are looking for the opinions of people, the people who live in the same region may have similar opinions. As you can see, each of the sampling techniques has pluses and minuses.
Simple Random Sample
A simple random sample (SRS) of size \(n\) is a sample that is selected from a population in a way that ensures that every different possible sample of size \(n\) has the same chance of being selected. Also, every individual associated with the population has the same chance of being selected
Examples of Simple Random Samples:
- Put all names in a hat and draw a certain number of names out.
- Assign each individual a number and use a random number table or a calculator, or a computer to randomly select the individuals that will be measured.
In the simple random sampling, there is an original population of 12 balls. We conduct a random sample by randomly selecting three balls from the population to be part of the sample. In this case, we have picked the ones that have the numbers 2, 9, and 12.
Describe how to take a simple random sample from a classroom.
Solution
Give each student in the class a number. Using a random number generator, you could then pick the number of students you want to pick.
You want to choose \(5\) students out of a class of \(20\). Give some examples of samples that are not simple random samples:
Solution
Choose \(5\) students from the front row. The people in the last row have no chance of being selected.
Choose the \(5\) shortest students. The tallest students have no chance of being selected.
Stratified Sampling
Stratified sampling is where you break the population into groups called strata, and then take a simple random sample from each strata.
Examples of Stratified Sampling:
- If you want to look at musical preference, you could divide the individuals into age groups and then conduct simple random samples inside each group.
- If you want to calculate the average price of textbooks, you could divide the individuals into groups by major and then conduct simple random samples inside each group.
The illustration above is an example of a stratified sample. There is an original population of 12 balls. They are grouped based on a common characteristic, in this case, color. We have 4 dark grey balls in the first group, 5 blue balls in the second group, and 3 purple balls in the third group. From all groups, some members are randomly selected to be part of the sample. From the first group, the balls that are numbered 1 and 12 are selected to be part of the sample. From the second group, the balls that are numbered 4 and 11 are selected to be part of the sample. From the third group, the balls that are numbered 2 and 10 are selected to be part of the sample. Based on stratified sampling, we ended up with a sample of 6 balls numbered 1, 2, 4, 10, 11, and 12.
Systematic Sampling
Systematic sampling is where you randomly choose a starting place and then select every \(k\)th individual to measure.
Examples of Systematic Sampling:
- You select every 5th item on an assembly line
- You select every 10th name on the list
- You select every 3rd customer that comes into the store.
The diagram above represents a systematic sample. We have a population of 12 balls inside a curvy road. The first one is picked randomly. In this example, it will be the second one from above. Then, the rest of them are selected below the first one in a certain pattern. The diagram shows that every third one is selected to be part of the sample. Based on systematic sampling, the ones selected to be in the sample are the ones numbered 2, 5, 8, and 11.
Cluster Sampling
Cluster sampling is where you break the population into groups called clusters. Randomly pick some clusters then poll all individuals in those clusters.
Examples of Cluster Sampling:
- A large city wants to poll all businesses in the city. They divide the city into sections (clusters), maybe a square block for each section, and use a random number generator to pick some of the clusters. Then, they poll all businesses in each chosen cluster.
- You want to measure whether a tree in the forest is infected with bark beetles. Instead of having to walk all over the forest, you divide the forest up into sectors, and then randomly pick the sectors that you will travel to. Then record whether a tree is infected or not for every tree in that sector.
The illustration above is an example of a cluster sample. There is an original population of 12 balls. They are grouped into clusters (blocks). Each group has two balls. Then, some groups are selected, and all members from those groups are chosen to be part of the sample. In this example, the group that only had balls numbered 3 and 4 is selected to be part of the sample. Then, a second group with only balls numbered 9 and 10 is selected to be part of the sample. Finally, a third group with only balls numbered 11 and 12 is selected to also be part of the sample. Based on cluster sampling, we ended up with three groups and all its members being part of the sample. The sample included balls with the numbers 3, 4, 9, 10, 11, and 12.
Many people confuse stratified sampling and cluster sampling. In stratified sampling, you use all the groups and some of the members in each group. Cluster sampling is the other way around. Cluster sampling uses some of the groups and all the members in each group.
The four sampling techniques that were presented all have advantages and disadvantages. There is another sampling technique that is sometimes utilized because either the researcher doesn’t know better, or it is easier to do. This sampling technique is known as a convenience sample.
Convenience Sampling
A convenience sample is one where the researcher picks individuals to be included who are easy for the researcher to collect.
Example of a Convenience Sample:
If you want to know the opinion of people about the criminal justice system, you stand on a street corner near the county courthouse, and question the first \(10\) people who walk by. The people who walk by the county courthouse are most likely involved in some fashion with the criminal justice system, and their opinions would not represent the opinions of all individuals.
The convenience sample will not result in a representative sample and should be avoided.
On a rare occasion, you do want to collect the entire population. In which case, you conduct a census.
Census
A census is when every individual of interest is measured.
Review
Banner Health is a multi-state nonprofit chain of hospitals. Management wants to assess the incidence of complications after surgery. They wish to use a sample of surgery patients. Several sampling techniques are described below. Categorize each technique as a simple random sample, stratified sample, systematic sample, cluster sample, or convenience sampling.
- Obtain a list of patients who had surgery at all Banner Health facilities. Divide the patients according to the type of surgery. Draw simple random samples from each group.
- Obtain a list of patients who had surgery at all Banner Health facilities. Number these patients, and then use a random number table to obtain the sample.
- Randomly select some Banner Health facilities from each of the seven states, and then include all the patients on the surgery lists of the states.
- At the beginning of the year, instruct each Banner Health facility to record any complications from every 100th surgery.
- Instruct each Banner Health facility to record any complications from 20 surgeries this week and send in the results.
Solution
- This is a stratified sample since the patients were separated into different strata, and then random samples were taken from each stratum. The problem with this is that some types of surgeries may have a higher chance of complications than others. Of course, the stratified sample would show you this.
- This is a random sample since each patient has the same chance of being chosen. The problem with this one is that it will take a while to collect the data.
- This is a cluster sample since all patients are questioned in each of the selected hospitals. The problem with this is that you could have, by chance, selected hospitals that have no complications.
- This is a systematic sample since they selected every 100th surgery. The problem with this is that if every 90th surgery has complications, you wouldn’t see this come up in the data.
- This is a convenience sample since they left it up to the facility on how to do it. The problem with convenience samples is that the person collecting the data will probably collect data from surgeries that had no complications.
Authors
"1.2: Sampling Methods" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY-SA 4.0
Attributions
"1.2: Sampling Methods" by Kathryn Kozak is licensed under CC BY-SA 4.0
Exercises
- To check the machine that is used to make the engine of a car is working properly, every 20th engine is selected and inspected. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- A researcher wants to know the number of hours professors sleep per week at a certain university. The researcher numbers all the professors from the university and then randomly selects 50 of them to be interviewed. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- Jane thinks the small company she works for pays male workers more than female workers. To test her assumption, she decides to randomly pick 10 female workers and 10 male workers from the company and ask for their annual salary. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- A researcher who currently lives in Glendora wants to know the average age of workers in the United States. He decides to gather the ages of 50 workers in the city of Glendora and use this information to predict the average age of workers in the United States. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- The governor of Los Angeles wants to know if he has enough support to get reelected. He decides to pick random streets in the county of Los Angeles. Then, ask all registered voters from those select streets if they will vote for him again. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- Jack thinks he is very popular but is not sure how many people know him from school. He has a list of all the students currently attending his school. He decides to number every student on the list and he picks 30 randomly. He asks those 30 students if they know his name. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- A female researcher wants to know if one ethnicity is more likely to smoke than another. He decides to separate the population based on ethnicity. She picks 25 from each group and asks if they smoked. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- Sam has a company that makes mechanical pencils. He wants to make sure his pencils are made correctly, so he decides to check every 15th mechanical pencil. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- The president wants to know the view of the American people on gun control. He decides to ask the National Rifle Association (NRA) members if they should have more gun regulations. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- The Department of Education wants to know the percentage of high school students in the United States who are interested in pursuing a bachelor's degree or higher. The department decided to randomly pick 50 high schools and ask all students from the selected high schools some questions about their future educational goals. Identify the sampling method that was used (simple random sample, systematic, stratified, cluster, convenience). Please explain your conclusion.
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- Answers
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If you are an instructor and want the solutions to all the exercise questions for each section, please email Toros Berberyan.