1.3 Sampling and Sampling Techniques

Gathering information about an entire population often costs too much, is too time consuming, or is virtually impossible. Instead, we use a sample of the population. In order to get accurate conclusions about the population from the sample, a sample should have the same characteristics as the population it represents. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods.

There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. In simple random sampling, any group of [latex]n[/latex] individuals is equally likely to be chosen as any other group of [latex]n[/latex] individuals. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members, not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in the following table.

Lisa can use a computer to generate random numbers. Suppose the computer generates the following numbers:

[latex]14 \; \; \; \; 05 \; \; \; \; 04[/latex]

The number 14 corresponds to Macierz, the number 05 corresponds to Cunningham, and the number 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Marcierz, Cuningham, and Cuarismo.

Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.

To choose a stratified sample, divide the population into groups called strata, and then take a proportionate number from each stratum. For example, a college’s student population can be stratified (grouped) by department, and then a proportionate simple random sample is chosen from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.

To choose a cluster sample, divide the population into clusters (groups), and then randomly select some of the clusters. All the members from the selected clusters are in the cluster sample. For example, divide a college’s faculty by department, so the departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers form the cluster sample.

To choose a systematic sample, randomly select a starting point and take every [latex]n[/latex]th piece of data from a listing of the population. For example, a phone book contains 20,000 residential listings, from which 400 names must be selected. Number the population from 1 to 20,000, and then use a simple random sample to pick a number that represents the first name in the sample. Then, choose every fiftieth name thereafter until a total of 400 names are selected. Systematic sampling is frequently chosen because it is a simple method.

A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. Such a sample is not random because only those customers in the store on that particular day have the opportunity to be in the sample. The results of convenience sampling may be very good in some cases and highly biased (favour certain outcomes) in others.

Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased because they may favour a certain group. It is better for the person conducting the survey to select the sample respondents.

True random sampling is done with replacement. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However, for practical reasons, in most populations, simple random sampling is done without replacement, where a member of the population may only be chosen once or not at all. Surveys are typically done without replacement. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Consequently, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.

In a college population of 10,000 people, suppose we want to randomly pick a sample of 1,000 for a survey. For any particular sample of 1,000, if we are sampling with replacement (where the person is replaced before picking the next person):

• the chance of picking the first person is 1,000 out of 10,000 (0.1000);
• the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999), and the chance of picking the same person again is 1 out of 10,000 (very low).

If we are sampling without replacement (where the person is not replaced before picking the next person):

• the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000);
• the chance of picking a different second person is 999 out of 9,999 (0.0999), and the chance of picking the same person again is 0.

Comparing the fractions [latex]\displaystyle{\frac{999}{10,000}}[/latex] and [latex]\displaystyle{\frac{999}{9,999}}[/latex] to four decimal places, these numbers are equivalent.  So we can see that the chance of selecting a small sample from a large population is basically the same, whether or not the sampling is done with replacement.

Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is 25 people, the sample is 10, and we are sampling with replacement for any particular sample, then the chance of picking the first person is 10 out of 25, and the chance of picking a different second person is 9 out of 25 (we replace the first person). If we sample without replacement, then the chance of picking the first person is still 10 out of 25, but the chance of picking the second person (who is different) is 9 out of 24. Comparing the fractions [latex]\displaystyle{\frac{9}{25}=0.36}[/latex] and [latex]\displaystyle{\frac{9}{24}=0.3750}[/latex], these numbers are not equivalent.

When we analyze data, it is important to be aware of sampling errors and non-sampling errors. The actual process of sampling causes sampling error, which is the difference between the actual population parameter and the corresponding sample statistic. In reality, a sample will never be exactly representative of the population, so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error. Factors not related to the sampling process cause non-sampling errors. For example, a defective counting device can cause a non-sampling error.

In statistics, a sampling bias is created when a sample is collected from a population, and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.

Video: “Statistics: Sources of Bias” by Mathispower4u [4:44] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.

Exercises

1. A study was done to determine the age, number of times per week, and the duration (amount of time) residents use a local park. The first house in the neighbourhood around the park was selected randomly, and then the resident of every eighth house in the neighbourhood around the park was interviewed.
   1. What sampling method was used?
   2. “Duration (amount of time)” is what type of data?
   3. The colours of the houses around the park are what kind of data?
   Click to see Answer

   1. Systematic.
   2. Quantitative, continuous.
   3. Qualitative.

    

2. For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience).
   1. A group of test subjects is divided into twelve groups; then, four of the groups are chosen at random.
   2. A market researcher polls every tenth person who walks into a store.
   3. The first [latex]50[/latex] people who walk into a sporting event are polled on their television preferences.
   4. A computer generates [latex]100[/latex] random numbers, and [latex]100[/latex] people whose names correspond with the numbers on the list are chosen.
   Click to see Answer

   1. Cluster.
   2. Systematic.
   3. Convenience.
   4. Simple Random.

    

3. Identify the sampling method used in each of the following situations.
   1. A woman in the airport is handing out questionnaires to travellers, asking them to evaluate the airport’s service. She does not ask travellers who are hurrying through the airport with their hands full of luggage but instead asks all travellers who are sitting near gates and not taking naps while they wait.
   2. A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class.
   3. The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, [latex]100[/latex] randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest.
   4. The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books.
   5. A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls [latex]1,200[/latex] randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates.
   6. The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. What type of sampling was used?
   Click to see Answer

   1. Convenience.
   2. Cluster.
   3. Stratified.
   4. Systematic.
   5. Simple Random.
   6. Stratified.

    

4. Suppose you want to determine the mean number of students per statistics class in your college. Describe a possible sampling method in three to five complete sentences. Make the description detailed.
   
   Click to see Answer

   (Answers will vary.) You could use a cluster sampling method. Each statistics class is a cluster. Randomly select some of the clusters/classes and record the number of students in each of the selected classes.

    

5. Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed.
   
   Click to see Answer

   (Answers will vary.) You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then, stop the tenth person as they leave a different building on campus at 1:50 in the afternoon.

    

6. List some practical difficulties involved in getting accurate results from a telephone survey.
   
   Click to see Answer

   (Answers will vary.) Many people will simply hang up. Many people will not pick up the phone if they do not recognize the caller ID. If they do respond to the surveys, you cannot be sure who is responding. Many people will not be called at all because they only have cell phones and phone lists generally only contain landline numbers.

    

7. List some practical difficulties involved in getting accurate results from a mailed survey.
   
   Click to see Answer

   (Answers will vary.) Many people will not respond to mail surveys. If they do respond to the surveys, you cannot be sure who is responding. In addition, mailing lists can be incomplete.

    

8. Airline companies are interested in the consistency of the number of babies on each flight so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study.
   1. Using complete sentences, list three things wrong with the way the survey was conducted.
   2. Using complete sentences, list three ways that you would improve the survey if it were to be repeated.
   Click to see Answer

   1. The survey was conducted using six similar flights. The survey would not be a true representation of the entire population of air travellers.
   2. Conduct the survey during different times of the year. Conduct the survey using flights to and from various locations.
      Conduct the survey on different days of the week.

    

9. In advance of the 1936 Presidential Election, a magazine titled Literary Digest released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent postcards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards.
   1. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time.
   2. What effect does the low response rate have on the reliability of the sample?
   3. Are these problems examples of sampling error or nonsampling error?
   4. During the same year, George Gallup conducted his own poll of 30,000 prospective voters. His researchers used a method they called “quota sampling” to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this section?
   Click to see Answer

   1. This 1936 poll was conducted during the Great Depression. Most of the population at that time could not afford magazines, cars, phones or club memberships, and so would not receive the postcards. These people had no chance of being included in the poll.
   2. The low response rate means the sample may not be representative of the entire population.
   3. Non-sampling.
   4. Stratified.

    

“1.3 Sampling and Data” and “1.6 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.