- Sampling Plans
- Cluster or Stratified — which one is better?
- Overview So Far
- Sample Size
- Let’s Summarize
As mentioned in the introduction to this unit, we will begin with the first stage of data production — sampling. Our discussion will be framed around the following examples:
Suppose you want to determine the musical preferences of all students at your university, based on a sample of students. Here are some examples of the many possible ways to pursue this problem.
So far we’ve discussed several sampling plans, and determined that a simple random sample is the only one we discussed that is not subject to any bias.
A simple random sample is the easiest way to base a selection on randomness. There are other, more sophisticated, sampling techniques that utilize randomness that are often preferable in real-life circumstances. Any plan that relies on random selection is called a probability sampling plan (or technique). The following three probability sampling plans are among the most commonly used:
- Simple Random Sampling is, as the name suggests, the simplest probability sampling plan. It is equivalent to “selecting names out of a hat.” Each individual has the same chance of being selected.
- Cluster Sampling — This sampling technique is used when our population is naturally divided into groups (which we call clusters). For example, all the students in a university are divided into majors; all the nurses in a certain city are divided into hospitals; all registered voters are divided into precincts (election districts). In cluster sampling, we take a random sample of clusters, and use all the individuals within the selected clusters as our sample. For example, in order to get a sample of high-school seniors from a certain city, you choose 3 high schools at random from among all the high schools in that city, and use all the high school seniors in the three selected high schools as your sample.
- Stratified Sampling — Stratified sampling is used when our population is naturally divided into sub-populations, which we call stratum (plural: strata). For example, all the students in a certain college are divided by gender or by year in college; all the registered voters in a certain city are divided by race. In stratified sampling, we choose a simple random sample from each stratum, and our sample consists of all these simple random samples put together. For example, in order to get a random sample of high-school seniors from a certain city, we choose a random sample of 25 seniors from each of the high schools in that city. Our sample consists of all these samples put together.
Each of those probability sampling plans, if applied correctly, are not subject to any bias, and thus produce samples that represent well the population from which they were drawn.
Comment: Cluster vs. Stratified
- Students sometimes get confused about the difference between cluster sampling and stratified sampling. Even though both methods start out with the population somehow divided into groups, the two methods are very different.
- In cluster sampling, we take a random sample of whole groups of individuals taking everyone in that group but not all groups are taken), while in stratified sampling we take a simple random sample from each group (and all groups are represented).
- For example, say we want to conduct a study on the sleeping habits of undergraduate students at a certain university, and need to obtain a sample. The students are naturally divided by majors, and let’s say that in this university there are 40 different majors.
- In cluster sampling, we would randomly choose, say, 5 majors (groups) out of the 40, and use all the students in these five majors as our sample.
- In stratified sampling, we would obtain a random sample of, say, 10 students from each of the 40 majors (groups), and use the 400 chosen students as the sample.
- Clearly in this example, stratified sampling is much better, since the major of the student might have an effect on the student’s sleeping habits, and so we would like to make sure that we have representatives from all the different majors. We’ll stress this point again following the example and activity.
Let’s go back and revisit the job satisfaction of hospital nurses example and discuss the pros and cons of the two sampling plans that are presented. Certainly, it will be much easier to conduct the study using the cluster sample, since all interviews are conducted in one hospital as opposed to the stratified sample, in which the interviews need to be conducted in 10 different hospitals. However, the hospital that a nurse works in probably has a direct impact on his/her job satisfaction, and in that sense, getting data from just one hospital might provide biased results. In this case, it will be very important to have representation from all the city hospitals, and therefore the stratified sample is definitely preferable. On the other hand, say that instead of job satisfaction, our study focuses on the age or weight of hospital nurses.
In this case, it is probably not as crucial to get representation from the different hospitals, and therefore the more easily obtained cluster sample might be preferable.
- Another commonly used sampling technique is multistage sampling, which is essentially a “complex form” of cluster sampling. When conducting cluster sampling, it might be unrealistic, or too expensive to sample all the individuals in the chosen clusters. In cases like this, it would make sense to have another stage of sampling, in which you choose a sample from each of the randomly selected clusters, hence the term multistage sampling.
For example, say you would like to study the exercise habits of college students in the state of California. You might choose 8 colleges (clusters) at random, but you are certainly not going to use all the students in these 8 colleges as your sample. It is simply not realistic to conduct your study that way. Instead you move on to stage 2 of your sampling plan, in which you choose a random sample of 100 males and a random sample of 100 females from each of the 8 colleges you selected in stage 1.
So in total you have 8 * (100+100) = 1,600 college students in your sample.
In this case, stage 1 was a cluster sample of 8 colleges and stage 2 was a stratified sample within each college where the stratum was gender.
Multistage sampling can have more than 2 stages. For example, to obtain a random sample of physicians in the United States, you choose 10 states at random (stage 1, cluster). From each state you choose at random 8 hospitals (stage 2, cluster). Finally, from each hospital, you choose 5 physicians from each sub-specialty (stage 3, stratified).
We have defined the following:
So far, we have made no mention of sample size. Our first priority is to make sure the sample is representative of the population, by using some form of probability sampling plan. Next, we must keep in mind that in order to get a more precise idea of what values are taken by the variable of interest for the entire population, a larger sample does a better job than a smaller one. We will discuss the issue of sample size in more detail in the Inference unit, and we will actually see how changes in the sample size affect the conclusions we can draw about the population.
- In practice, we are confronted with many trade-offs in statistics. A larger sample is more informative about the population, but it is also more costly in terms of time and money. Researchers must make an effort to keep their costs down, but still obtain a sample that is large enough to allow them to report fairly precise results.
Our goal, in statistics, is to use information from a sample to draw conclusions about the larger group, called the population. The first step in this process is to obtain a sample of individuals that are truly representative of the population. If this step is not carried out properly, then the sample is subject to bias, a systematic tendency to misrepresent the variables of interest in the population.
Bias is almost guaranteed if a volunteer sample is used. If the individuals select themselves for the study, they are often different in an important way from the individuals who did not volunteer.
A convenience sample, chosen because individuals were in the right place at the right time to suit the researcher, may be different from the general population in a subtle but important way. However, for certain variables of interest, a convenience sample may still be fairly representative.
The sampling frame of individuals from whom the sample is actually selected should match the population of interest; bias may result if parts of the population are systematically excluded.
Systematic sampling takes an organized (but not random) approach to the selection process, as in picking every 50th name on a list, or the first product to come off the production line each hour. Just as with convenience sampling, there may be subtle sources of bias in such a plan, or it may be adequate for the purpose at hand.
Most studies are subject to some degree of nonresponse, referring to individuals who do not go along with the researchers’ intention to include them in a study. If there are too many non-respondents, and they are different from respondents in an important way, then the sample turns out to be biased.
In general, bias may be eliminated (in theory), or at least reduced (in practice), if researchers do their best to implement a probability sampling plan that utilizes randomness.
The most basic probability sampling plan is a simple random sample, where every group of individuals has the same chance of being selected as every other group of the same size. This is achieved by sampling at random and without replacement.
In a cluster sample, groups of individuals are randomly selected, such as all people in the same household. In a cluster sample, all members of each selected group participate in the study.
A stratified sample divides the population into groups called strata before selecting study participants at random from within those groups.
Multistage sampling makes the sampling process more manageable by working down from a large population to successively smaller groups within the population, taking advantage of stratifying along the way, and sometimes finishing up with a cluster sample or a simple random sample.
Assuming the various sources of bias have been avoided, researchers can learn more about the variables of interest for the population by taking larger samples. The “extreme” (meaning, the largest possible sample) would be to study every single individual in the population (the goal of a census), but in practice, such a design is rarely feasible. Instead, researchers must try to obtain the largest sample that fits in their budget (in terms of both time and money), and must take great care that the sample is truly representative of the population of interest.
We will further discuss the topic of sample size when we cover sampling distributions and inferential statistics.
In this short section on sampling, we learned various techniques by which one can choose a sample of individuals from an entire population to collect data from. This is seemingly a simple step in the big picture of statistics, but it turns out that it has a crucial effect on the conclusions we can draw from the sample about the entire population (i.e., inference).
That being said, other (nonrandom) sampling techniques are available, and sometimes using them is the best we can do. It is important, though, when these techniques are used, to be aware of the types of bias that they introduce, and thus the limitations of the conclusions that can be drawn from the resulting samples.