Stratified random sampling

Here you will learn about stratified random sampling, including what stratified random sampling is, how to take a stratified sample, and the advantages and disadvantages of stratified random sampling.

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

What is stratified random sampling?

Stratified random sampling is a sampling method using proportional representation. The population is divided into smaller subgroups (strata) with the number taken from each subgroup proportional to the size of the subgroup.

The term stratification means to arrange something into groups. The larger the group, the more data entries will exist in the sample for that group.

To find a stratified sample, you need to know how many data entries are in each subgroup and the total sample size.

Stratified random sampling determines the number of items of data in each subgroup, so it requires a secondary method of sampling to select the individual items of data. This is usually done using a simple random sampling technique (using a random number generator).

Advantages and disadvantages of stratified random sampling

Following a stratified random sampling methodology has advantages and disadvantages:

Stratified random sampling is used more frequently than other sampling techniques including cluster sampling and systematic sampling (on its own) because the dataset of the sample is proportional to the population size.

This stratifying allocation means that minority groups can be represented whereas they may not be with other probability sampling methods.

Sampling error

If every member of the population is in the sample, there is no sampling error.

As the sample gets smaller, or the methodology has introduced a selection bias, the sampling error becomes more significant as this means that the sample may not be representative of the total population.

The more random a sample is, the smaller the sampling error.

Data collection

Once a stratified random sample is chosen, the next step is data collection where respondents offer data to fulfill the requirements of the questionnaire or survey (for example). The collected values within the sample then go through data analysis to find generalized results for the overall population.

For example, you may want to calculate the average reading age of books in a library. Rather than picking up 100 random books from any shelf (this would be random sampling), you find out how the books are categorized (fiction, non-fiction, autobiography, children’s, reference, etc.)

Then, to get a representative sample, you would calculate the number of books from each sub-category that will be needed for the sample, and then use a random sample or systematic sample to select the books from each subgroup.

As the reference for each book in a library is usually electronically stored, the sample can be generated quickly.

What is stratified random sampling?

Common Core State Standards

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

• Grade 7 – Statistics & 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.

• High School – 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.

• High School – 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 take a stratified sample

In order to take a stratified sample:

1. Calculate how many items of data will be selected for the sample.
2. Calculate how many items of data will be selected in each subcategory.
3. Check that the number of items of data matches the sample size.
4. Select each individual data items using a random / systematic sampling technique.

Stratified random sampling examples

Example 1: stratified sample

The student council is carrying out a survey. They want to collect a stratified sample of 10\% of students in grades 7-11. Calculate the number of students in each grade level 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 1,650 students in the school and the question asks for a sample size of 10\%, you need to calculate 10\% of 1,650\text{:}

As this number is a decimal, you will round the value to the nearest integer, so you need a sample size of 165 students.

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

You will now calculate 10\% of each grade level and 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:

34+33+32+34+32=165 which matches the sample size.

Example 2: stratified sample

Georgia wants to select some pick ‘n’ mix from the sweet shop to share with her friends. The sweets are sold from boxes with the following weights:

Georgia wants to take 15\% of sweets, proportional to their weight. Calculate how many grams of each sweet she will choose. Assume that each sweet is whole.

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

You now need to calculate 15\% of each type of sweet.

Example 3: stratified sample

An insurance company is completing some market research. They want to ask 60\% of their staff about their current working hours. Below is a table of the number of staff in each department across the business:

Use a stratified random sampling technique to find the number of staff in each category for the sample.

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

You now need to calculate the number of staff in each department for the sample:

Example 4: stratified sample

Julie is a lexicographer. She is curious about the average length of words in a book. The book has 4 chapters. Below is a table showing the number of words in each chapter:

Julie would like to take a stratified sample of 20\% of the number of words in the book, proportional to the number of words in each chapter. Determine how many words she should collect in each chapter.

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

You will now calculate 20\% of each chapter and round each value to the nearest integer, and write the values below the table:

Example 5: stratified sample

Craig would like to survey 100 people from the population. He has grouped the population into a two way table:

Calculate how many people should be chosen from each group.

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

As you need 100 people, you need to determine what proportion from each category you are selecting. This is found by dividing the number of people in the sample by the number of people in the population.

The number of people in the population is equal to 143+152+15+17=327, so you can now calculate the proportion of each category:

\cfrac{\text{Sample size}}{\text{Population size}}=\cfrac{100}{327}

By multiplying each value in the two way table by the fraction \cfrac{100}{327}, you will calculate the number of people in the sample proportional to the target population:

Teaching tips for stratified random sampling

• Use diagrams or charts to represent a population divided into different strata (subgroups or subpopulations). Show how random samples are taken from each group. Visual aids make it easier for students to understand how the method works.

• Highlight the difference between stratified and simple random sampling. In simple random sampling, every individual has an equal chance of being selected, but this can lead to some groups being over- or under-represented (disproportionate sampling). Explain how stratified random sampling helps ensure representation from all parts of the entire population.

• Explain that stratified sampling ensures all subgroups are represented. This is especially important when a population’s characteristics are diverse, like income levels, regions, or age groups.

• If available, introduce statistical software or online tools that automate the stratified sampling process, so students can explore real-life data more effectively.

Easy mistakes to make

• Mixing up a sampling method
  Using the incorrect sampling method to select data (such as using systematic sampling or non random sampling).

• Incorrect percentages / fraction of an amount
  When finding a sample of 60\% of the population, you need to find 60\% of each category, and not 60\% of the population and divide it equally between the number of categories. The larger the frequency in the category, the larger the sample taken. This is proportional representation.

• Incorrect number of items in the sample
  Remember to check that the sample size matches the total sum of the samples from each category in the population.

• Assuming it’s the best method to use
  Some students may assume that stratified sampling is always superior to simple random sampling or other sampling methods. While it’s helpful in many cases, it’s not always the best choice if the population consists of homogeneous groups or the strata don’t significantly differ.

Related sampling methods lessons

• Sampling methods
• Systematic sampling
• Random sampling

The next lessons are

• Probability
• Compound events
• Probability distribution

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