Developing Critical Numeracy Across the Curriculum

Student Reasoning about Sampling

Over the primary and middle school years it is likely that student understanding of sampling will develop in the following sequence. First is the appreciation of what a sample is:

  • Single ideas or phrases: a little bit, a test, the same as something else, a portion.

  • Expressions reflecting the part-whole nature of a sample:
    “a little bit of something, not the whole thing”
    “only part of the whole thing”
    “a small piece of material”

  • Expressions reflecting the part-whole nature of the sample-population relationship, as well as the purpose of representing the whole:
    “a small portion of something larger to try”
    “a random selection from a wider group, meant to represent it”

Second, students begin to appreciate the importance of sample size and the method used to collect the sample. For sample size the progression often includes some of the following steps.

  • Very small samples accepted (e.g., 10)

  • Instance that all of the population must be sampled

  • Suggestion of a “benchmark number” for the sample size (e.g., 100 or 1000)

  • Suggestion of a percentage of the population (e.g., 10%), regardless of population size

  • Appreciation that the larger the sample size the more confident one is in the statistic calculated.

  • For method of sampling the progression often includes some of the following steps.

  • Samples picked specifically to have a certain trait (e.g., “Look average”)

  • Voluntary or convenience samples

  • Stratified samples

  • Random samples

  • Random stratified samples

Eventually students develop the habit of questioning each sample they encounter to see if it meets criteria for being unbiased.