D1.1 Explain the importance of various sampling techniques for collecting data from a representative sample of a population.
Skill: Explaining the Importance of Various Sampling Techniques
In the junior grades, it is important to provide students with a variety of opportunities to plan a data collection. Teachers should question students throughout this stage to help them understand the importance of choosing the right type of inquiry and data to answer the question of interest, as well as the importance of identifying the population to be surveyed and, if necessary, the sample. This will encourage students to develop the critical thinking skills that will be most useful in the fourth stage of the inquiry process.
The following are examples of questions, related to sample selection, that teachers can use to guide students when planning for data collection.
Sample size:
- What will your sample size be and how did you determine it?
- With a sample of this size, will the results be representative of the target population? Why or why not?
- Do you think the results would be similar if the sample size were smaller, larger, and why?
Sample composition:
- Is the sample composition free of bias? Convince me.
- How will you randomly select your sample?
- Does your sample need to be stratified? Why of why not?
- What strata will you use in the composition of your sample? What will be the size of each stratum?
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 57-58.
Knowledge: Sampling Techniques
Selection Process
Simple Random Sampling
Students should understand that one of the best ways to have a good sample that is free of bias is to choose it randomly so that the people who make up the population have an equal chance of being in it.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 55.
It is possible, for example, to randomly select 10% of the population using a random number generator.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Systematic Random Sampling
Systematic random sampling is used when the subjects in a population are selected using a systematic approach that has been randomly determined. For example, a sample could be determined using an alphabetical list of names, using a starting name and count (for example, every fourth name) that are randomly selected.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Stratification Process
In some surveys, it is important to ensure that certain subgroups of the population are represented in the sample (for example, the subgroup of primary school students and the subgroup of junior high school students). In this case, the population is said to be stratified (divided into mutually exclusive groups). Each stratum (group) must be represented in the sample.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 56.
Stratified Random Sampling
Stratified random sampling involves dividing the population into strata and then taking a random sample from each stratum. A school population, for example, could be divided into two sub-populations (strata): one comprising students who take the bus to school and one comprising students who do not. Then, a survey could be given to 10% of the population randomly selected from each of these strata.
Source : The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Population
In statistics, the set of objects, events or persons that one wishes to study is called the population. When planning data collection, the target population must be defined. The choice of the population is partly dictated by the survey's intent and the statement of the question of interest.
Examples of Populations in Statistics
- The people of Canada
- Baseball fans
- Grade 4 students from the school
- Primary school students
- Parents of middle school students
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 52.
Knowledge: Representative Sample of a Population
In statistics, the set of objects, events or people that we want to study is called the population. The fraction of the population that is observed, measured or surveyed is called the sample. In Grades 5 and 6, surveys may focus on a population whose size makes interviewing, measuring or observing impossible. The students must then carry out their inquiry with only a part of this population. This subgroup, called the sample, should be representative of the population targeted by the inquiry. In statistics, the choice of a sample is governed by complex statistical standards based on concepts of probability, which ensure the validity and reliability of the results. In the Junior Division, students do not have to worry about these standards; they just need to develop an intuitive understanding of what could be a representative sample of the target population for the purposes of the inquiry.
The idea that the results of an inquiry within a small group can reflect the reality of a larger population is not necessarily easy to grasp. Students need to understand that the sample is part of a whole and that even a small part of the population, when properly selected, gives a good representation of the entire population. In the junior grades, students begin to learn about this concept informally. While some have an intuitive understanding, others have misconceptions that need to be addressed.
The sample allows conclusions to be drawn and generalizations to be made about the entire population without having to interview the entire population; however, these conclusions are valid only if the sample is representative of the entire target population.
Examples of Misconceptions:
- Sampling does not work because it is impossible to account for all the different characteristics of the population (variability).
- To be fair, you should always have as many Grade 4 students as Grade 5 students if you want to compare the results of the two groups.
- The results are not good because we did not interview everyone.
When planning data collection, students should have an intuitive understanding of the relationships between the representativeness of a sample relative to the population and the following three factors: sample size, selection process, and stratification process.
Sample Size
In order for the inquiry results to be representative of the population, the sample size must be considered. Teachers should help students find a balance between a sample that is too large and one that is too small.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 52-54.