D1.1 Explain the importance of various sampling techniques for collecting a sample of data that is representative of a population.
Activity 1: Sample Size
In the junior grades, the choice of sample size is made intuitively. The following two examples illustrate how teachers can help students understand the relationship between sample size and representativeness.
Example 1
If the population size is relatively small, for example, 25 students in a class, it is difficult to select an appropriate number of students to represent the whole. In such a case, it is best to use the entire population. To convince students of this, teachers may ask them to conduct a survey of all the students in the class and then ask them to take a random sample of 10 responses and compare the results to those of the entire population. Students may find that there are significant differences in the results.
Example 2
If the population size is moderately large, for example, the 130 students in Grades 5 and 6 in the school, it is conceivable that a sample size of 10 students would be too small and a sample size of 100 students would be too large. In this case, a sample size of about 50 students would be sufficient. To convince students that a sample size of 10 students is too small, teachers can ask them to conduct a survey of all 130 students in Grades 5 and 6. Then, they can ask them to take two samples of 10 randomly selected responses and another sample of 50 responses. In comparing the results, students should note that the 50-response survey sample is a better representation of the results of the entire population than the two 10-response surveys. Teachers can then get them to recognize that conducting a survey with a sample of 100 students requires almost as much work as conducting it with the entire population of 130 students, which is obviously not very practical.
Note: Students may tend to conclude that the larger the population size, the larger the sample size must be; for example, a population of 300 requires a larger sample size than a population of 130. Teachers should then help them recognize that for very large populations, such as a city, province or Canada, it is not always necessary to increase the sample size. In reality, surveys of the Canadian population use samples of 1000 to 2000 people, or less than one-hundredth of one percent of the population. However, these surveys are subject to very strict rules to ensure their validity. These rules are more appropriate for university-level statistical studies.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 54-55.
Activity 2: Simple Random Sampling
Through questioning, teachers can help students identify potential biases in the selection process and better understand the importance of choosing their sample randomly. For example, if students want to select a sample of 40 students from the 120 students in Grades 5 and 6 to learn about their favourite hobbies, teachers can ask questions such as:
- Would it be fair to choose all of Luke and Peter's friends as a sample? There would be about 40 names right away. (No, because Luke and Peter are hockey fans and it is likely that their friends are too. This would not be a good representation of the entire population.)
- Would it be fair to select, as a sample, the first 40 students in grades 5 and 6 who arrive at school in the morning? (No, because these students are likely to be walking to school. Only students who live near the school would be represented.)
- There are 42 students in the two Grade 5 class groups. Would it be fair to choose them as a sample? (No, because Grade 6 students would not be represented in the sample, and students' hobbies may vary by age.)
Teachers also need to help students devise different strategies for choosing a random sample. For example, to select the sample of 40 students in the previous example, students could write the names of 120 Grade 5 and 6 students on slips of paper and randomly select 40 of them. Alternatively, they could line up the 120 pieces of paper on a table in no particular order and choose every third name (for example, the 1st, 4th, 7th, and 10th names). (systematic random sample)
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 55-56.
Activity 3: Stratified Random Sampling
Teachers may suggest that students use stratified sampling when a sample does not seem fair to them. Suppose, for example, that a survey is conducted with 120 students in Grades 5 and 6, and that 40% of the students are students in Grade 5 and 60% are students in Grade 6. A random sample of 40 students was selected. It can be seen that it contains 35 Grade 6 students and 5 Grade 5 students. Some Grade 5 students protest because they believe the sample chosen does not represent them well and that this could bias the results of the survey. A discussion ensues. Is stratification necessary? If the survey is about the distance from home to school, it is probably not necessary to change the sample, but if the survey is about the choice of end-of-year activity, a stratified sample might be needed. In the sample of 40 students, we would therefore like to have approximately 40% of them (16) be Grade 5 students and approximately 60% of them (24 students) from Grade 6. The selection of the 16 Grade 5 students and the 24 Grade 6 students would then be made randomly within each grade group. This stratified sample is "proportional" because it respects the proportion of students in Grades 5 and 6 in the population.
Note: Statisticians sometimes use non-proportional stratified samples; their choice is driven by statistical considerations beyond the scope of the surveys that junior students conduct.
Source: translated from Guide d’enseignement efficace des mathématiques, de la 4e à la 6e année, Traitement des données et probabilité, p. 56-57.