5.2 Probability sampling methods

Systemic Sampling

(very similar to simple random sampling.) -Systematic sampling begins by listing all the individuals in the population, then randomly picking a starting point on the list. - The sample is then obtained by moving down the list, selecting every nth name. The size of n is calculated by dividing the population size by the desired sample size -This technique is less random than simple random sampling because the principle of independence is violated.

Proportionate Stratified Random Sampling (or proportionate random sampling)

- Occasionally, researchers try to improve the correspondence between a sample and a population by deliberately ensuring that the composition of the sample matches the composition of the population. -As with a stratified sample, we begin by identifying a set of subgroups or segments in the population. -Next, we determine what proportion of the population corresponds to each subgroup. -Finally, a sample is obtained such that the proportions in the sample exactly match the proportions in the overall population.

Stratified Random Sampling

- The main advantage of a stratified random sample is that it guarantees that each of the different subgroups will be well represented with a relatively large group of individuals in the sample. -this type of sampling is appropriate when the purpose of a research study is to examine specific subgroups and make comparisons between them. -you should notice that stratified random sampling is not equivalent to simple random sampling.

The process of simple random sampling consists of the 3 steps:

-Clearly define the population from which you want to select a sample. -List all the members of the population. -Use a random process to select individuals from the list.

Concern of proportionate random sampling

-Depending on how precisely we want sample proportions to match population proportions, the proportionate stratified sample can create a lot of extra work. -In addition, a proportionate stratified sample can make it impossible for a researcher to describe or compare some subgroups or strata that exist within the population. For example, if a specific subgroup makes up only 1% of the population, they also make up only 1% of the sample. In a sample of 100 individuals, this means that there is only one person from the subgroup. It should be clear that you cannot rely on one person to adequately represent the entire subgroup.

CONCERNS ABOUT SIMPLE RANDOM SAMPLING

-In the long run, this strategy generates a balanced, representative sample. -If we toss a coin thousands of times, eventually, the results will be 50% heads and 50% tails. -In the short run, however, there are no guarantees. Because chance determines each selection, it is possible (although usually unlikely) to obtain a very distorted sample. We could, for example, toss a balanced coin and get heads 10 times in a row. Similarly, we could get a random sample of 10 males from a population that contains an equal number of men and women. To avoid this kind of nonrepresentative sample, researchers often impose additional restrictions on the random sampling procedure;

Stratified Random Sampling

-To obtain this kind of sample, we first identify the specific subgroups (or strata) to be included in the sample. -Then we select equal-sized random samples from each of the pre-identified subgroups, using the same steps as in simple random sampling. — Finally, we combine the subgroup samples into one overall sample. EX : suppose that we plan to select 50 individuals from a large introductory psychology class and want to ensure that psychology majors and nonmajors are equally represented. First, we select a random sample of 25 students from the psychology majors in the class and then a random sample of 25 students from those who are not psychology majors. Combining these two subgroup samples produces the desired stratified random sample.

Stratified Random Sampling

A population usually consists of a variety of identifiable subgroups. For example, the population of registered voters in California can be subdivided into Republicans and Democrats, different ethnic groups, different age groups, and so on. The different subgroups can be viewed as different layers or strata like the layers of rock on a cliff face (Figure 5.4). Often, a researcher's goal for a representative sample is to ensure that each of the different subgroups is adequately represented. One technique for accomplishing this goal is to use stratified random sampling.

Simple Random Sampling

A probability sampling technique in which each individual in the population has an equal and independent chance of selection.

Sampling with replacement

This method requires that an individual selected for the sample be recorded as a sample member and then returned to the population (replaced) before the next selection is made. This procedure ensures that the probability of selection remains constant throughout a series of selections. EXAMPLE : if we select from a population of 100 individuals, the probability of selecting any particular individual is 1⁄100. To keep this same probability (1⁄100) for the second selection, it is necessary to return the first individual to the pool before the next is selected. Because the probabilities stay constant, this technique ensures that the selections are independent.

The disadvantage of cluster sampling

it can raise concerns about the independence of the individual scores. A sample of 300 individuals is assumed to contain 300 separate, individual, and independent measurements. However, the individuals within a cluster often have common characteristics or share common experiences that might influence the variables being measured. In this case, a researcher must question whether the individual measurements from the cluster actually represent separate and independent individuals.

Two advantages of Cluster Sampling

First, it is a relatively quick and easy way to obtain a large sample. Second, the measurement of individuals can often be done in groups, which can greatly facilitate the entire research project.

Combined-Strategy Sampling

Occasionally, researchers combine two or more sampling strategies to select participants. Selection strategies are commonly combined to optimize the chances that a sample is representative of a widely dispersed or broad-based population such as in a wide market survey or a political poll. For example, a superintendent of schools may first divide the district into regions (e.g., north, south, east, and west), which involves stratified sampling. From the different regions, the superintendent may then select two third-grade classrooms, which involves cluster sampling

proportionate stratified sampling example

suppose that the college administration wants a sample of students that accurately represent the distribution of freshmen, sophomores, juniors, and seniors in the college population. If the overall population contains 30% freshmen, 26% sophomores, 25% juniors, and 19% seniors, then the sample is selected so that it has exactly the same percentages for the four groups. First, determine the desired size of the sample, then randomly select from the freshmen in the population until you have a number corresponding to 30% of the sample size. Next, select sophomores until you have a number equal to 26% of the sample size. Continue this process with the juniors and seniors to obtain the full sample.

Sampling without replacement

this method removes each selected individual from the population before the next selection is made. This method guarantees that no individual appears more than once in a single sample. However, each time an individual is removed, the probability of selection changes for the remaining individuals. Example : if the population has 100 people, then the probability of being selected starts at 1/100. After the first selection, only 99 people are left and the probability of selection changes to 1/99. Because the probabilities change with each selection, this technique does not produce independent selections.

Concern of Stratified Random Sampling

-stratified random sampling tends to produce a distorted picture of the overall population. -Suppose, for example, that we are taking a stratified random sample of 50 children from an elementary school with a total population consisting of 50 students who are an only child and 250 who have siblings. -When using stratified sampling the two subgroups would be represented equally in the sample (with 25 children from each group) -but the subgroups are not equally represented in the population. Only children, for example, represent less than 17% of the population but make up 50% of the sample. - Specifically, every individual in the population does not have an equal chance of being selected for the sample. In our example, 25 of the 50 only-child students are selected so the probability of selecting an only child is 25/50 or ½. By comparison, 25 of the 250 children with siblings are selected so their probability is 25/250 or 1 out of 10. Thus, stratified random sampling does not produce a true random sample because all individuals in the population are not equally likely to be selected.

The obvious goal of a simple random sample is to ensure that the selection procedure cannot discriminate among individuals and thereby result in a nonrepresentative sample. The two principal methods of random sampling are:

1 - sampling with replacement : 2- Sampling without replacement

cluster sampling

All of the sampling techniques we have considered so far are based on selecting individual participants, one at a time, from the population. Occasionally, however, the individuals in the population are already clustered in preexisting groups, and a researcher can randomly select groups instead of selecting individuals. For example, a researcher may want to obtain a large sample of third-grade students from the city school system. Instead of selecting 300 students one at a time, the researcher can randomly select 10 classrooms (each with about 30 students) and still end up with 300 individuals in the sample. This procedure is called cluster sampling and can be used whenever well-defined clusters exist within the population of interest.

Systemic sampling example

For example, suppose a researcher has a population of 100 third-grade students and would like to select a sample of 25 children. Each child's name is put on a list and assigned a number from 1 to 100. Then, the researcher uses a random process such as a table of random numbers to select the first participant, for example, participant number 11. With a population of 100 children and a desired sample size of 25, the size of n in this example is 100/25+ 4 . Therefore, every fourth individual after participant 11 (15, 19, 23, and so on) is selected. -Note that systematic sampling is identical to simple random sampling, however, after the first individual is selected, the researcher does not continue to use a random process to select the remaining individuals for the sample. -This technique is less random than simple random sampling because the principle of independence is violated. Specifically, if we select participant number 11, we are biased against choosing participants number 12, 13, and 14, and we are biased in favor of choosing participant number 15. However, as a probability sampling method, this method ensures a high degree of representativeness.

Summary of probability Sampling Methods

Probability sampling techniques have a very good chance of producing a representative sample because they tend to rely on a random selection process. However, as we noted earlier, simple random sampling by itself does not guarantee a high degree of representativeness. To correct this problem, researchers often impose restrictions on the random process. Specifically, stratified random sampling can be used to guarantee that different subgroups are equally represented in the sample, and proportionate stratified sampling can be used to guarantee that the overall composition of the sample matches the composition of the population. However, probability sampling techniques can be extremely time consuming and tedious (i.e., obtaining a list of all the members of a population and developing a random, unbiased selection process). These techniques also require that researchers "know" the whole population and have access to it. For these reasons, probability sampling techniques are rarely used except in research involving small, contained populations (e.g., students at a school or prisoners at one correctional facility) or large-scale surveys.