Sampling methods

Here you will learn about sampling methods, including what different sampling methods are and how to use different sampling methods.

Students will first learn about sampling methods as part of statistics and probability in 7 th grade and expand upon their knowledge in high school.

What are sampling methods?

Sampling is a statistical process where researchers select a specific number of observations from a larger population to analyze and create statistical inferences about the target population.

The sampling method is the way in which the observations are chosen and collected.

Some different types of probability sampling techniques are stratified random sampling, systematic sampling and random sampling.

Below is an overview of each methodology.

Stratified random sampling is a method that divides the population into specific groups that are smaller subgroups, or strata.

A sample is then taken from each subgroup, with the size of the sample proportionate to the subgroup’s size within the overall population.

Stratified random sampling determines the number of data items in each subgroup, but a secondary sampling method is needed to select the individual items. Typically, this is done using simple random sampling.

Advantage: This method is less time-consuming than asking the whole population

Disadvantage: Another sampling method is needed to select individual items of data from a list

Step-by-step guide: Stratified random sampling

Systematic sampling is a way to pick data items at regular intervals from a population.

To make sure everything has an equal chance of being picked, the data needs to be listed in order, called a sampling frame.

First, you use a random number generator to pick a starting item. Then, you choose the rest by counting at regular intervals.

For example,

If the population size is 1,200 and you want a sample of 300 data items, you divide 1,200 by 300 to get an interval of 4.

This means you select every 4 th item from the ordered list to include in the sample.

Advantage: This method is more time-efficient than asking the whole population

Disadvantage: Every member of the entire population must be listed

Step-by-step guide: Systematic sampling

Random sampling, also called simple random sampling, is a method used to choose items from a population via random selection.

To take a random sample, you first make a list of every person or item in the population. Then, you give each one a unique number.

Next, you use a random number generator or a random number table to pick the number of items you need for your sample.

This method ensures that every member has an equal chance of being chosen, which helps reduce bias and errors in the sample.

Advantage: This method is more time-efficient than asking the whole population

Disadvantage: Large sample sizes are usually needed to ensure the desired levels of precision and accuracy

Step-by-step guide: Random sampling

What are sampling methods?

Common Core State Standards

How does this relate to 7 th grade math and high school math?

• Grade 7 – Statistic & Probability (7.SP.A.1)
  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

• Statistics & Probability – Making Inferences & Justifying Conclusions (HS.S.IC.B.3)
  Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

• Statistics & Probability – Making Inferences & Justifying Conclusions (HS.S.IC.B.4)
  Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

How to use sampling methods

There are a lot of ways to use sampling methods. For more specific step-by-step guides, check out the sampling methods pages linked in the “What are sampling methods?” section above or read through the examples below.

Sampling methods examples

Example 1: stratified sample

A gym created a questionnaire for its members. They want to collect a stratified sample of 20 \% of the members at the gym.

Calculate the number of members in each age group that will take part in the survey.

1. Calculate how many items of data will be selected for the sample.

As you know that there are 1,072 members at the gym and the question asks for a sample size of 20 \%, you need to calculate 20 \% of 1,072\text{:}

The number of people (members) selected cannot be a decimal. Round it to the nearest whole number, resulting in a required sample size of 214 members.

2Calculate how many items of data will be selected in each subcategory.

Now calculate 20 \% of each age group, round each value to the nearest integer, and write the values in the table:

3Check that the number of items of data matches the sample size.

Adding the number of students in each grade level, you get 68+86+43+17=214 which matches the sample size.

Example 2: stratified sample

Julissa, a market researcher, is studying consumer preferences for a new product. She wants to understand the responses received from a survey.

The surveys were distributed across four regions. Below is a table showing the number of responses collected from each region:

Julissa would like to take a stratified sample of 40 \% of the responses, proportional to the number of responses in each region. Determine how many responses she should collect in each region.

Calculate how many items of data will be selected in each subcategory.

Now calculate 40 \% of each region and write the values below the table:

Example 3: systematic sampling – production line

A factory produces 200 bottles of juice per minute. A quality control machine inspects the seal of 5 \% of the bottles.

The bottles pass through the machine one at a time on a conveyor belt. Use systematic random sampling to select the bottles for inspection over a period of 4 minutes.

Use a random number generator to select the first item of data.

Since you need to pick every 20 th item, the first number in the sample (starting point of the sequence) must be randomly chosen from the first 20 terms.

Using a random number generator, you get the number 5, so you choose the first item of data in the sample to be the 5 th bottle.

Below is a table used to show how the sequence develops*:

*This table only contains the first 60 bottles of the 800 in the population.

Select the remaining items of data following the given sequence.

Since you are selecting every 20 th item, you can select the following bottles from the data:

The sample will therefore contain 40 bottles with the following numbers:

5, \, 25, \, 45, \, 65, \, 85, \, 105, \, 125, \, 145, \, 165,

185, \, 205, \, 225, \, 245, \, 265, \, 285, \, 305, 325,

345, \, 365, \, 385, \, 405, \, 425, \, 445, \, 465, \, 485,

505, \, 525, \, 545, \, 565, \, 585, \, 605, \, 625, \, 645,

665, \, 685, \, 705, \, 725, \, 745, \, 765, \, 785.

Note: These numbers are in the sequence 20n-15.

Example 4: systematic sampling – smaller groups

A bookstore is analyzing the average value of online purchases over the past quarter. There were 15,320 orders placed, each with a unique transaction ID.

The bookstore decides to take a systematic sample of 3 \% of the orders. Determine which transaction IDs will be included in the sample.

Example 5: a sample of the sums of two six-sided dice

There are two fair, six-sided dice. The number rolled on the first dice and the number rolled on the second dice are added and the sum recorded. Collect a sample of 10 sums.

List every member of the population.

Since each dice has the numbers 1 to 6, you have the following sample space:

Use a random number generator to select the number of data points in the sample.

Rolling the dice together 12 times, and allowing repetitions, you obtain the following random sample:

Example 6: student survey

A school wants to investigate what students in 11 th grade eat for lunch. The school has 341 students in 11 th grade. They want to take a sample of 10 \% of the students.

Each student is ordered alphabetically by their last name and given a unique number between 1 and 341 to represent the total number of individuals. Use a random number generator to determine the students who will participate in the investigation.

Use a random number generator to select the number of data points in the sample.

You need a sample of 10 \%\text{:} \, 341\times{0.1}=34.1, rounded to a random sample of 34 students. Any duplicates are ignored.

The students involved in the investigation are:

Teaching tips for sampling methods

• Expose students to real world examples that have a variety of demographics, numbers, population size and variability.

• Offer worked examples illustrating various scenarios, including differing population sizes and sampling intervals.

• After students understand the different sampling methods and how to use them, incorporate statistical software or online tools that automate the sampling process, enabling students to engage more effectively with real-world data.

Easy mistakes to make

• Confusing sampling methods
  Using an inappropriate sampling method, such as systematic sampling or non-random sampling, to select data.

• Assuming one type of sampling method is always better than another
  For example, some students might believe that stratified sampling is always better than simple random sampling or other methods. While it can be advantageous in many situations, it may not be the best option if the population is homogeneous or the strata show minimal differences.

• Incorrect number of items in the sample / duplicates
  If a sample requires a specific number of data values, it must include exactly that number. Pay attention to whether duplicates are permitted, as some samples allow repeated values.

The next lessons are

• Probability
• Compound probability
• Probability distribution

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