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Lesson Explainer: Types of Nonrandom Sampling Mathematics

In this explainer, we will learn how to apply nonrandom sampling methods such as quota sampling and opportunity sampling.

We first recall that taking a sample of a population means taking a subset of the population. The process of choosing the sample is called the sampling method, and the sampling method we use can have major effects on the bias and reliability of the data we measure from the sample.

Often, to help remove possible bias in the sample, we will use a random sampling method. However, it is not always possible or necessary to use any of the random sampling methods. For example, if there is no available sampling frame or no easy way to obtain a sampling frame, then it will not be possible to use a random sampling method.

In these cases, we can use a nonrandom sampling method, which means that there is bias in the selection process for the sample. Thus, every member of the population will not have the same probability of being selected for the sample.

An example of nonrandom sampling is taking a survey of anyone who is available at the time. This might be asking people on the street at a specific time of the day or sending emails to everyone who has signed up to a service. This is called opportunity sampling, and it is an easy way of collecting data and is usually very cost-effective. The problem is that it is likely to cause bias since the sample is unlikely to be representative.

Another example of nonrandom sampling involves splitting the population into mutually exclusive groups first and then sampling each group relative to its size. This is called quota sampling. In order to split the population into groups, we first need to estimate the proportion of the population that represents each group. This step usually takes time and money and is often inaccurate. However, once it is done, the rest of the method is straightforward as we ignore nonresponders and just survey additional members until the quota for the sample is reached. On the whole, this is an easy and cost-effective method of sampling that gives proportional representation to the different groups.

We can define nonrandom sampling methods formally as follows.

Definition: Nonrandom Sampling Method

A sampling method is called nonrandom if members of the population have different probabilities of being chosen for the sample.

  • Opportunity sampling involves sampling any member of the population that is available at the time.
  • Quota sampling involves splitting the population into mutually exclusive groups and then proportionally sampling each group.

In our first example, we will identify the type of the sampling method used in a given scenario.

Example 1: Identifying the Sampling Method Used

A market researcher asks the first 50 people they see in a shopping mall to fill in a survey about their spending habits. Which sampling method did the researcher use?

  1. Opportunity sampling
  2. Quota sampling

Answer

We first note that the sampling method involves only asking people who are currently in the shopping mall and only surveying the first 50 people. So, from the population, not every person has an equal chance of being selected for the sample. Thus, this is not a random sample.

In particular, we can recall that opportunity sampling is sampling any member of the population that is available at the time. An example of this is selecting the first members of the population you can find.

Hence, asking the first 50 people seen in a shopping mall to fill in a survey is an example of opportunity sampling.

In our next example, we will identify which of five given options is an advantage of quota sampling over opportunity sampling.

Example 2: Identifying an Advantage of Quota Sampling

Which of the following is an advantage of quota sampling over opportunity sampling?

  1. There is less data to process.
  2. It is easier to extend the scope of the study.
  3. The population is proportionally represented.
  4. A sampling frame is required for opportunity sampling.

Answer

We first recall that quota sampling is when we split the population into mutually exclusive groups and then proportionally sample each group, whereas opportunity sampling is when we sample any members of the population that are available at the time.

We can note that both methods are quick and easy to form a sample; however, we can also deduce that opportunity sampling does not proportionally represent the population since we are just sampling the most available members of the population. However, quota sampling does take this into consideration. Thus, we can conclude that the population being proportionally represented is an advantage of quota sampling.

We can also check the other options to see if there are multiple correct answers. We note that quota sampling and opportunity sampling will yield a similar amount of data to process and that a sampling frame is not needed for either sampling method. We can also deduce that there is a little extra difficulty extending the scope of a study when using opportunity sampling since we do not distinguish between members of the sample. However, when extending the scope of a study that used quota sampling, we will likely be splitting the sample into more groups. This means that extending the scope of the study will take more resources when using quota sampling.

Hence, we can conclude that the population being proportionally represented is the only listed advantage of quota sampling, which is answer option C.

In our next example, we will identify which type of sampling is not opportunity sampling in a given scenario.

Example 3: Identifying Sampling Methods in a Scenario

A market researcher at a clothing company wants to determine the average amount spent by a person on clothes in a year. Which of the following sampling methods does not describe opportunity sampling?

  1. Selecting 50 customers randomly
  2. Asking people who are in the store
  3. Asking people at the checkout to fill out a survey
  4. Requiring any online shopper for the day to fill out a survey

Answer

We first recall that opportunity sampling is sampling any member of the population that is available at the time. This can involve asking people in a list until a quota is reached, asking available people in a store, asking people at the checkout, or requiring online shoppers to fill out a survey.

The only method listed that is not opportunity sampling is selecting 50 customers randomly since we are not taking the first 50 available customers but instead randomly selecting them from a population.

Hence, the only method listed that is not opportunity sampling is answer option D, selecting 50 customers randomly from a list of people who shopped online on that day.

In our next example, we will find an opportunity sample of a given example and determine if the results from this opportunity sample could be improved by taking a simple random sample of the same size.

Example 4: Opportunity Sample on Random Die Rolls

The results of 10 throws of a fair random die are listed in order: 1,1,3,6,5,1,4,6,2,4.

  1. By using an opportunity sample of size 5, estimate the average die roll.
  2. Would the result have been fairer if a simple random sample of size 5 reduced the bias?

Answer

Part 1

We first recall that opportunity sampling is sampling any member of the population that is available at the time. This means we will just take the first 5 rolls to be our sample: 1,1,3,6,5.

We find the average of this sample by adding the rolls and dividing by the sample size: 𝑎=1+1+3+6+55=165.

Part 2

A simple random sample would involve randomly choosing 5 of the rolls for our sample. In most studies, this can reduce bias; however, we are told that the die is a fair random die. This means that there will be no relationship between each roll and the next since the die rolls are random. Therefore, it does not matter which 5 rolls we choose for our sample; the bias will be the same since the sampling frame is randomized.

Hence, we can say that no, a simple random sample of size 5 would not have reduced the bias.

In our next example, we will use a quota sample to compare the average difference in jump distances between professional athletes and average people. We will also identify which of four options will improve the sampling method.

Example 5: Nonrandom Sampling

A study at an athletics event is being carried out to determine the difference in the average jump distances between professional athletes and average people. A researcher compares the first 5 athletes’ and average people’s jump distances. They find and compile the results in the table given below.

Athlete5 m5.5 m4.4 m5.2 m5.1 m
Average Person3.2 m2.7 m2.5 m4.8 m3.7 m
  1. On average, how much further did the professional athletes jump?
  2. Which of the following methods would improve the sampling method?
    1. Paying money to the participants
    2. Offering a reward for jumping past a certain distance
    3. Only choosing athletes who specialize in long jumps
    4. Sampling the local population for the average person rather than the athletics event

Answer

Part 1

We can determine the average of each group by adding the jump distances together and dividing by the size of the samples. We have the following: athleteaveragemaveragepersonaveragem=5+5.5+4.4+5.2+5.15=5.04,=3.2+2.7+2.5+4.8+3.75=3.38.

Thus, the difference between the average jump distances is 5.043.38=1.66.m

Part 2

We can check each of the options separately to determine if they improve the sample. However, we should first note any problems with the sampling method used. We see that the researcher chose the first 5 available people from each category who agreed to take part. This means that quota sampling was chosen.

We can note that this is a nonrandom sample, so there is likely a bias in the population chosen. We can note that this is exemplified by the fact that the average people were chosen at an athletics event, and spectators at these events are more likely to be interested in athletics.

If we were to pay money to the participants, this would likely make more people willing to take part; however, this could bias the sample. Similarly, offering a reward for jumping past a certain distance would likely make people who consider themselves good jumpers to participate. Once again, this would bias the sample. The same is true if we only choose athletes who specialize in long jumps; this would be a less fair representation of the average jump distance of an athlete. Finally, we can note that spectators at an athletics event are more likely to be interested in athletics. This means that sampling the local population for the average person rather than the athletics event will give a better representation of the average person’s jump distance.

Hence, the answer is to sample the local population for the average person rather than the athletics event. The answer is option D.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • We do not need a sampling frame to carry out nonrandom sampling; however, there is usually bias introduced with these sampling methods.
  • Opportunity sampling is sampling any member of the population that is available at the time.
  • Quota sampling involves first splitting the population into mutually exclusive groups and then proportionally sampling each group.
  • Opportunity sampling is an easy way of collecting data and is usually very cost-effective. The problem is that it is likely to cause bias since the sample is unlikely to be representative.
  • Quota sampling involves estimating the proportion of the population that represents each group and splitting the population using these criteria, which might take time and money and is often inaccurate. On the whole, however, it is an easy and cost-effective method of sampling that gives proportional representation to the different groups.

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