1.2: Random Sampling

Now that you know that you have to take samples in order to gather data, the next question is how best to gather a sample? There are many ways to take samples. Not all of them will result in a representative sample. Also, just because a sample is large does not mean it is a good sample. As an example, you can take a sample involving one million people to find out if they feel there should be more gun control, but if you only ask members of the National Rifle Association (NRA) or the Coalition to Stop Gun Violence, then you may get biased results. You need to make sure that you ask a cross-section of individuals. Let’s look at the types of samples that can be taken. Do realize that no sample is perfect, and may not result in a representation of the population.

Census: An attempt to gather measurements or observations from all of the objects in the entire population.

A true census is very difficult to do in many cases. However, for certain populations, like the net worth of the members of the U.S. Senate, it may be relatively easy to perform a census. We should be able to find out the net worth of each and every member of the Senate since there are only 100 members. But, when our government tries to conduct the national census every 10 years, you can believe that it is impossible for them to gather data on each and every American.

The best way to find a sample that is representative of the population is to use a random sample. There are several different types of random sampling. Though it depends on the task at hand, the best method is often simple random sampling which occurs when you randomly choose a subset from the entire population.

Simple Random Sample: Every sample of size n has the same chance of being chosen, and every individual in the population has the same chance of being in the sample.

An example of a simple random sample is to put all of the names of the students in your class into a hat, and then randomly select five names out of the hat.

Stratified Sampling: This is a method of sampling that divides a population into different groups, called strata, and then takes random samples inside each strata.

An example where stratified sampling is appropriate is if a university wants to find out how much time their students spend studying each week; but they also want to know if different majors spend more time studying than others. They could divide the student body into the different majors (strata), and then randomly pick a number of people in each major to ask them how much time they spend studying. The number of people asked in each major (strata) does not have to be the same.

Systematic Sampling: This method is where you pick every kth individual, where k is some whole number. This is used often in quality control on assembly lines.

For example, a car manufacturer needs to make sure that the cars coming off the assembly line are free of defects. They do not want to test every car, so they test every 100th car. This way they can periodically see if there is a problem in the manufacturing process. This makes for an easier method to keep track of testing and is still a random sample.

Cluster Sampling: This method is like stratified sampling, but instead of dividing the individuals into strata, and then randomly picking individuals from each strata, a cluster sample separates the individuals into groups, randomly selects which groups they will use, and then takes a census of every individual in the chosen groups.

Cluster sampling is very useful in geographic studies such as the opinions of people in a state or measuring the diameter at breast height of trees in a national forest. In both situations, a cluster sample reduces the traveling distances that occur in a simple random sample. For example, suppose that the Gallup Poll needs to perform a public opinion poll of all registered voters in Colorado. In order to select a good sample using simple random sampling, the Gallup Poll would have to have all the names of all the registered voters in Colorado, and then randomly select a subset of these names. This may be very difficult to do. So, they will use a cluster sample instead. Start by dividing the state of Colorado up into categories or groups geographically. Randomly select some of these groups. Now ask all registered voters in each of the chosen groups. This makes the job of the pollsters much easier, because they will not have to travel over every inch of the state to get their sample but it is still a random sample.

Quota Sampling: This is when the researchers deliberately try to form a good sample by creating a cross-section of the population under study.

For an example, suppose that the population under study is the political affiliations of all the people in a small town. Now, suppose that the residents of the town are 70% Caucasian, 25% African American, and 5% Native American. Further, the residents of the town are 51% female and 49% male. Also, we know information about the religious affiliations of the townspeople. The residents of the town are 55% Protestant, 25% Catholic, 10% Jewish, and 10% Muslim. Now, if a researcher is going to poll the people of this town about their political affiliation, the researcher should gather a sample that is representative of the entire population. If the researcher uses quota sampling, then the researcher would try to artificially create a cross-section of the town by insisting that his sample should be 70% Caucasian, 25% African American, and 5% Native American. Also, the researcher would want his sample to be 51% female and 49% male. Also, the researcher would want his sample to be 55% Protestant, 25% Catholic, 10% Jewish, and 10% Muslim. This sounds like an admirable attempt to create a good sample, but this method has major problems with selection bias.

The main concern here is when does the researcher stop profiling the people that he will survey? So far, the researcher has cross-sectioned the residents of the town by race, gender, and religion, but are those the only differences between individuals? What about socioeconomic status, age, education, involvement in the community, etc.? These are all influences on the political affiliation of individuals. Thus, the problem with quota sampling is that to do it right, you have to take into account all the differences among the people in the town. If you cross-section the town down to every possible difference among people, you end up with single individuals, so you would have to survey the whole town to get an accurate result. The whole point of creating a sample is so that you do not have to survey the entire population, so what is the point of quota sampling?

Note: The Gallup Poll did use quota sampling in the past, but does not use it anymore.

Convenience Sampling

As the name of this sampling technique implies, the basis of convenience sampling is to use whatever method is easy and convenient for the investigator. This type of sampling technique creates a situation where a random sample is not achieved. Therefore, the sample will be biased since the sample is not representative of the entire population.

For example, if you stand outside the Democratic National Convention in order to survey people exiting the convention about their political views. This may be a convenient way to gather data, but the sample will not be representative of the entire population.

Of all of the sampling types, a random sample is the best type. Sometimes, it may be difficult to collect a perfect random sample since getting a list of all of the individuals to randomly choose from may be hard to do.

Example \(\PageIndex{1}\): Which Type of Sample?

Determine if the sample type is simple random sample, stratified sample, systematic sample, cluster sample, quota sample, or convenience sample.

Solution

1. A researcher wants to determine the different species of trees that are in the Coconino National Forest. She divides the forest using a grid system. She then randomly picks 20 different sections and records the species of every tree in each of the chosen sections.

This is a cluster sample, since she randomly selected some of the groups, and all individuals in the chosen groups were surveyed.

2. A pollster stands in front of an organic foods grocery store and asks people leaving the store how concerned they are about pesticides in their food.

This is a convenience sample, since the person is just standing out in front of one store. Most likely the people leaving an organic food grocery store are concerned about pesticides in their food, so the sample would be biased.

3. The Pew Research Center wants to determine the education level of mothers. They randomly ask mothers to say if they had some high school, graduated high school, some college, graduated from college, or advance degree.

This is a simple random sample, since the individuals were picked randomly.

4. Penn State wants to determine the salaries of their graduates in the majors of agricultural sciences, business, engineering, and education. They randomly ask 50 graduates of agricultural sciences, 100 graduates of business, 200 graduates of engineering, and 75 graduates of education what their salaries are.

This is a stratified sample, since all groups were used, and then random samples were taken inside each group.

5. In order for the Ford Motor Company to ensure quality of their cars, they test every 130th car coming off the assembly line of their Ohio Assembly Plant in Avon Lake, OH.

This is a systematic sample since they picked every 130th car.

6. A town council wants to know the opinion of their residents on a new regional plan. The town is 45% Caucasian, 25% African American, 20% Asian, and 10% Native American. It also is 55% Christian, 25% Jewish, 12% Islamic, and 8% Atheist. In addition, 8% of the town did not graduate from high school, 12% have graduated from high school but never went to college, 16% have had some college, 45% have obtained bachelor’s degree, and 19% have obtained a post-graduate degree. So the town council decides that the sample of residents will be taken so that it mirrors these breakdowns.

This is a quota sample, since they tried to pick people who fit into these subcategories.