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Lesson Explainer: Sampling Methods Mathematics

In this explainer, we will learn how to sample using different sampling methods such as simple random, systematic, and stratified sampling.

When forming a survey on a population, we sample the population. This means that we need to choose a subset of the population to survey, and there are many different ways we could go about choosing this sample.

Let’s say we have a list of 100 pupils, each numbered 1–100, and we want to sample 10 of these pupils. This list of 100 pupils forming the population to be sampled is called a sampling frame.

The method of sampling with the least amount of bias would be to choose the 10 pupils at random so that every pupil has an equal chance of being selected. This type of sampling is called simple random sampling. Examples of a simple random sample are using a random number generator to choose the 10 pupils or putting all the numbers into a hat and selecting 10.

This is not the only way of selecting the sample. We could choose every 10th pupil from the list or in any other regular interval. This has the advantage that it is quick and easy to implement since we already have the list of pupils; however, there is a risk of bias if the sampling frame is not ordered at random. This is called systematic sampling.

There is one final sampling method we want to discuss; to do this, let’s say that of the 100 pupils, 40 are in grade 7 and 60 are in grade 8. This means that there is a ratio of 4060 or 23 of grade 7 pupils to grade 8 pupils in the population.

If we were to use a simple random sample or a systematic sample, it is likely that we would not maintain this ratio. It can be useful to keep these proportions the same in a survey so that all of the groups have representation in the survey. If we first split the population into the two different grade groups, we can then choose 4 pupils from the 40 pupils in grade 7 and 6 pupils from the 60 pupils in grade 8. This would give us a sample size of 10 and help us maintain the ratio of the pupils in the different grade groups. We call this a stratified sample, and each subset of the population is called a stratum.

We can define these sampling methods more formally as follows.

Definition: Sampling Methods

  • A simple random sample is a sample in which every member of the population has an equal chance of being selected.
  • A systematic sample is a sample in which members of the population are chosen at regular intervals from a sampling frame.
  • A stratified sample is a sample in which the population is first divided into mutually exclusive groups (called strata) and a random sample (relative to the size of the strata) is then taken from each.

There are advantages and disadvantages to each sampling method. In general, a simple random sample eliminates bias, and we can easily note the probability of any member being chosen for the sample. The downside of a simple random sample is that it is difficult and costly to apply to large population sizes.

A systematic sample is quick and easy to implement; this makes it particularly useful for large population sizes. However, since we choose our sample by choosing in regular intervals from a sampling frame, bias can arise if the sampling frame is not generated randomly.

A stratified sample is useful for guaranteeing representation from every stratum and also guaranteeing that every stratum will be accurately represented by its size relative to the population. The disadvantages are that we still need to choose a random sample from each stratum, so any disadvantages from choosing this random sample will still apply. We also need clearly defined, mutually exclusive strata to split the population into.

In our first example, we will identify a random sample from a list of five sampling methods.

Example 1: Identifying a Random Sample

For a school project, three students need to produce a sample of students at their school. Which of the following methods leads to a random sample?

  1. Writing the names of all students they know in a list
  2. Calling for volunteers at the school assembly
  3. Asking each of their friends to name three further friends
  4. Numbering all the students and producing the sample from a random list of numbers
  5. Calling for volunteers via social media

Answer

We first recall that in a random sample every member of the population has an equal chance of being selected for the sample. This means that we need to determine in which of the five given options each student has an equal chance of being selected.

Let’s start with option A: writing the names of all students they know in a list. We can note that students in the same class as the three students are more likely to be on the list, so they are more likely to be included in this sample. Thus, this cannot be a random sample.

In option B, we are told that the students will call for volunteers at the school assembly. This can cause problems since some students are more likely than others to respond, and we do not know if every student will attend the assembly. Therefore, this cannot be a random sample.

In option C, we are told that the students will ask each of their friends to name three further friends. This has a similar problem as option A, as only their friends and friends of friends will be chosen in the sample. This means that some students are more likely to be in the sample than others, so this is not a random sample.

In option D, the three students number all the students and produce the sample from a random list of numbers. Since every student is included on the list and then chosen at random, every student is equally likely to be included in the sample. Hence, this is a random sample.

In option E, we are told that the students will call for volunteers via social media. This has similar problems to asking for volunteers at the assembly. It is unlikely that every student uses social media and that every student has access to messages. Also, different students will have different likelihoods of volunteering. Thus, this not a random sample.

Therefore, the only random sample listed is to number all of the students and produce the sample from a random list of numbers; this is answer D.

In our next example, we will use the population size and the size of a stratum to determine the sample size of toys in a warehouse.

Example 2: Using the Size of a Stratum to Determine the Sample Size

Three different toys are stored in a warehouse. There are 15‎ ‎000 toys in total and the number of each is shown in the table.

Kind of ToysBearsDollsCars
Number of Toys2‎ ‎0006‎ ‎0007‎ ‎000

A stratified sample of all the toys is taken and 420 cars are selected. Calculate the total number of toys in the sample.

Answer

We begin by recalling that in a stratified sample we divide the population into mutually exclusive groups (called strata) and then randomly sample each stratum, taking a sample of sizes relative to the size of the strata.

We are told that 420 cars are selected for the sample out of the 7‎ ‎000 total cars. We can determine the percentage of cars chosen for the sample: 4207000×100=6%.

Since this is a stratified sample, we must also take 6% of the bears and 6% of the dolls for our sample. In particular, this means that 6% of the population is taken for our sample. Thus, the size of the sample is 6% of 15‎ ‎000: 6100×15000=900.

Hence, the sample size is 900.

In our next example, we will use a given scenario involving a systematic sample to determine the serial number of the final unit in the sample.

Example 3: Using Systematic Sampling

A company manufactures computer components, and each unit contains a serial number starting at 1 and increasing by 1 each time a unit is manufactured. If the company wants to survey 20 units using a systematic sample starting at the third unit and the population size is 200, determine the serial number of the last unit in the sample.

Answer

We first recall that in a systematic sample each member of the population is chosen at a regular interval from a sampling frame. We are told to start with the unit numbered 3, and we want to sample 20 units out of a population of 200 units. We can determine the size of the interval needed by dividing the population size by the number of units we want to sample. We have 20020=10.

So, the interval size in our systematic sample will be 10 units.

We can add multiples of 10 on to 3 to determine the serial numbers of the units chosen for the sample: 3+0(10)=3,3+1(10)=13,3+2(10)=23.

We can continue this process to obtain 3+19(10)=193.

If we tried to add 10 on to this value, we would have 193+10=203, which is above the population size.

Hence, the last unit in the sample will have a serial number of 193.

In our next example, we will determine the method of sampling used in a given scenario.

Example 4: Identifying the Sampling Method

A school wants to survey its students. They choose a sample by putting every student’s name into a hat and then drawing names from the hat. Which of the following is the sampling method used?

  1. Systematic sampling
  2. Simple random sampling
  3. Stratified sampling

Answer

We note that every student’s name is put into the hat and then 20 names are drawn at random. Since every student’s name is in the hat, we can conclude that each student has an equal probability of having their name drawn from that hat, so they have the same chance of being chosen in the sample. This type of sampling is called a simple random sample, since every student has the same chance of being in the sample.

Hence, we can say that this is an example of simple random sampling, which is answer B.

In our final example, we will determine the best sampling method for a given study.

Example 5: Choosing the Best Sampling Method for a Study

A politician wanted to survey voters to determine whether to fund a highway construction in a city. To obtain the sample, the politician obtained a complete list of registered voters and assigned a unique number to each individual. Then, a sample was selected based on a list of random numbers generated by a computer program. Which sampling method was used for this study?

  1. Systematic sampling
  2. Simple random sampling
  3. Stratified sampling

Answer

In order to determine the best sampling method, we first need to consider the population size. The population of a city is very large; this means that a simple random sample would be difficult to implement.

We can also note that in stratified sampling we need the population to be clearly divided into the distinct strata. The politician could do it in a few different ways; for example, the strata could be vehicle owners and non-vehicle owners. However, each stratum would still be a very large population.

In order to use systematic sampling, we need a sampling frame. We can note that most cities will have a list of residents, which can be used as a sampling frame. We can also note that systematic sampling involves choosing members of the population at regular intervals from the sampling frame. This method can introduce bias but it is very quick and easy to implement, particularly for large populations like a city.

Hence, the best sampling method for the politician to determine whether to fund a highway construction in the city is systematic sampling, which is answer A.

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

Key Points

  • A simple random sample is a sample in which every member of the population has an equal chance of being selected.
  • A systematic sample is a sample in which members of the population are chosen at regular intervals from a sampling frame.
  • A stratified sample is a sample in which the population is first divided into mutually exclusive groups (called strata) and a random sample (relative to the size of the strata) is then taken from each.
  • A simple random sample eliminates bias from sampling. However, it is difficult and costly to apply this to large population sizes.
  • A systematic sample is quick and easy to implement, which makes it particularly useful for large population sizes. However, since we choose our sample by choosing in regular intervals from a sampling frame, bias can arise if the sampling frame is not generated randomly.
  • A stratified sample is useful for guaranteeing representation from every stratum and also guaranteeing that every stratum will be accurately represented by its size relative to the population. The disadvantages are that we still need to choose a random sample from each stratum, so any disadvantages from choosing this random sample will still apply. We also need clearly defined, mutually exclusive strata to split the population into.

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