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Reference

Sampling Methods

Concept

Sampling

Sampling is the process of selecting a sample from a population of objects or individuals. The sample can be then examined to make conclusions about the entire population.

There are many different sampling methods, such as convenience sampling, self-selected sampling, systematic sampling, and random sampling. For example, suppose that a sample of the students in a classroom needs to be chosen to study blood sugar levels. The following applet visualizes some of these methods.

Since it may be impractical, too expensive, or even impossible to examine every member of a population, sampling is often used in real-world scenarios. Consider the following examples.

Baseball cards are divided by teams. Then, five cards from each team are randomly selected to estimate the average RBI for the players.
Every tenth person entering a music club is asked to name their favorite music genre.
A website article on the tax increase invites readers to express their opinions on the government's decisions.
A new drug is tested by a pharmaceutical company. Volunteers are divided randomly into two groups, one of which will receive the new drug, the second of which will receive a placebo.

Each sampling method influences the quality of the results of a statistical study. To make accurate and useful conclusions about a population, it is important to select an unbiased sample that is representative of the population that is examined.

Concept

Representative Sample

A sample that accurately reflects the characteristics of the population is called a representative sample.

In a representative sample, if x parts of the population have a certain characteristic, approximately x parts of the sample will share the same characteristic. As an example, consider a sample of cats and dogs selected from all the cats and dogs at a pet shop.

The figure shows cats represent 60 % of the population and 60 % of the sample. Similarly, dogs represent 40 % of the population and 40 % of the sample. In this case, the sample is then representative. Note that in a representative sample, every subgroup of the population is represented.

Extra

Relation with Bias

It is possible that samples reflect biases. In most cases, unbiased samples result in representative samples. On the other hand, biased samples usually result in samples that are not representative of the population.

Concept

Bias in a Sample

A bias in a sample is an error in sampling that results in misrepresentation of members of a population. Bias occurs when members of a population that are representing certain characteristics are more likely to be selected in a sample than others.

	Definition
Unbiased Sample	A sample that is representative of the population. Conclusions drawn from this sample can be generalized to the whole population.
Biased Sample	A sample that overrepresents or underrepresents a certain part of the population. The inferences drawn based on this sample may be invalid.

The chosen sampling method may either introduce or minimize a bias in a sample. The following real-life scenarios present examples of biased samples.

Biased Sample	Explanation
A city council asks residents whether there should be an off-leash area for dogs in a park. A hundred dog owners are surveyed at the park.	The only people asked are dog owners. This means that respondents are more likely to have a strong opinion about an off-leash area for their dogs.
To assess the experiences of customers who shop online, a company e-mails purchasers with a link to a survey.	Because this sample is self-selected, only those who are very satisfied or dissatisfied with the shopping experience are likely to respond.
Every sixth boxer at a boxing camp is asked to name their favorite brand of boxing gloves.	Not all boxers go to a boxing camp, as camps are usually sponsored by a brand and take place in a single city. Also, professional boxers often organize their own private camps with hired sparring partners.

Concept

Sample-to-Sample Variability

Sample-to-sample variability refers to the fact that different samples taken from the same population can yield different statistics, even with the same sampling method. For example, suppose the heights of students in a classroom are studied. Pick different samples in the following applet to see their different means.

It is worth noting that even though different samples have different means, they are all typically around the population mean. This is particularly useful given that it is usually not possible to survey the whole population.

Concept

Simple Random Sample

In a simple random sample, each member of a population is equally likely to be selected as part of the sample. Consider an example where a researcher performs the following procedure. |c| There are $20$ people, each assigned a unique number, who are surveyed. Of those $20$ people, $4$ need to be randomly selected as a sample. The researcher writes each unique number on a paper and places them in a bag.The researcher then blindly selects $4$ papers out of the bag.

The researcher's way of choosing a sample means that each participant is equally likely to be selected as part of the sample. The following applet shows a generated example of simple random samples.

To generate a simple random sample, each member of a population can be assigned a unique ID — for example, a whole number. Then, a random number generator can be used to generate IDs until the desired sample size is satisfied.

Extra

Biased or Unbiased Sampling

A simple random sample is an unbiased sample because it involves selecting members from a population randomly, guaranteeing that each member of the population has an equal chance of being included. This is usually representative as it tends to have the same characteristics as the population. 2 &Unbiased Sampling:&& Members of a population are selected randomly. &Biased Sampling:&& Members of a population are hand chosen, for example. An example of biased surveying would be a coach who asks only their favorite players about the effectiveness of their team's strategy. Just imagine if only the stars of the team are allowed to give their opinion to the coach. However, other instances might call for biased sampling such as a study about how Olympic athletes perform under pressure.

Concept

Systematic Sample

In a systematic sample, the members of a population are ordered randomly or in a random-like way. The sample size is predetermined and selected among the members in a specified interval. The intervals are determined by dividing the population size by the sample size. Consider an example. |c| Five customers are surveyed about the quality of a fragrance shampoo.

A sample of 5 people is selected from a population of 20 people using systematic sampling.

A systematic sample is often used when a complete list of the population is available. Different samples can be created by varying the starting point when using systematic sampling. This gives the surveyors the ability to check if the conclusions hold across the samples.

Extra

Biased or Unbiased Sampling

Systematic samples can be either biased or unbiased.

Biased: &If the initial order of the population is biased, &the sample will also be biased. Unbiased: &If the members of the population are ordered &randomly without a cyclic pattern, it is &expected that the systematic sampling &will also be unbiased.

An example of a bias would be if the population is arranged in a cyclical pattern that matches the sampling interval. That can lead to a group being preferred over other groups, which leads to bias in the sample. It is important to remember that unbiased samples are better suited to generate a representative sample.

Concept

Self-Selected Sample

In a self-selected sample, the members of the sample are the people who are willing to participate. Since people participate voluntarily in these samples, the samples are also called voluntary response samples. Consider an example. |c| A survey about internet shopping is posted online. People volunteer to participate or simply ignore it. The following applet shows a certain number of people who volunteer to participate in the survey.

Extra

Biased or Unbiased Sampling

A self-selected sample is a biased sample because people with strong opinions, either positive or negative, about the topic studied are more likely to volunteer. Also, people who are interested in the topic being studied may be more likely to participate, while those who are not interested may refuse to participate.

As a result, such a sample is not representative of the population because it underrepresents people with neutral opinions about the topic or who are not interested in it.

Concept

Convenience Sample

In a convenience sample, members of a population are selected to be in a sample based on convenience or their availability to the researchers. Consider an example. |c| A researcher wants to study all wolf subspecies in a forest. However, they only have the funds to study those that frequently visit a certain area. This is an example of convenience sampling because the researcher is selecting wolves who are conveniently available and in close proximity.

This type of sampling is often used when researchers have limited resources or are under certain time constraints.

Extra

Biased or Unbiased Sampling

Convenience sampling can lead to a biased sample since the researchers choose sample members that are easily available to them. Some groups of people in the population may not be represented in the sample because they are not easily accessible, causing the sample to not be representative of the population.

Concept

Stratified Sample

In a stratified sampling, members from a population are divided into subgroups. These subgroups can be formed based on factors like age, education, or health status, among others. Members of the sample are then randomly selected from each subgroup. Consider an example. |c| A researcher wants to study the academic achievements of elementary students in a local school district. The researcher plans to randomly select students in each grade to be in the sample.

Stratified sampling can be conducted as follows.

Different samples can be generated by randomly selecting people from each subgroup of the population.

Extra

Biased or Unbiased Sampling

Stratified sampling guarantees that each subgroup of a population is represented in the sample, which means that the stratified sample is a representative sample. Therefore, a stratified sample might be considered an unbiased sample.

Concept

Cluster Sample

For a cluster sample, a population is first divided into smaller groups with similar characteristics to the whole population called clusters. One or more clusters are randomly selected. All members in the selected clusters form the sample. A member of the population cannot be included in more than one cluster. Consider an example. A group of researchers conduct a study to examine the basic mathematical skills of all the eight-graders in a city. They consider each school in the city as a separate cluster. The researchers then select one cluster to collect data from. Cluster sampling can be conducted as follows.

Cluster sampling is particularly useful for populations that have a wide geographic spread where simple random sampling is difficult to apply. Note that already-existing groups such as cities and schools can be used as clusters.

Extra

Biased or Unbiased Sampling

Cluster sampling is prone to bias. When the clusters are not representative of the characteristics of a population, the conclusions about the entire population would be biased as well.

Exercises

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