Populations, Samples and the Census

Taste the soup

When you cook a big pot of soup, you do not eat the whole pot to check the seasoning. You stir it well and taste one spoonful. If the soup is well stirred, one spoonful tells you about the whole pot.

That is the idea of sampling. The pot is the population, the spoonful is the sample, and stirring is random selection. If you forget to stir, your spoonful from the top might be all broth and no vegetables: a biased sample.

Remember from Grade 9: you met populations, samples and some sampling methods when you planned data collection in MTH1W. This lesson and the next go deeper: the sampling frame, parameters and statistics, why we sample, what makes a sample good, and how to carry out each method precisely.

Population, sampling frame and sample

• The population is everyone (or everything) the question is about: all secondary students in a school board, all bags of chips made today, all maple trees in Algonquin Park.
• The sampling frame is the list or source you actually choose from: the board's enrolment list, the production log, a map divided into plots. A good frame matches the population. A poor frame leaves people out or includes people who should not be there.
• The sample is the individuals you actually collect data from.

Remember from the lesson on variability: a number that describes the population is a parameter (usually unknown); a number calculated from the sample is a statistic (calculated, then used to estimate the parameter). Using sample data to draw conclusions about a population is called inference.

Census or sample?

A census collects data from every member of the population. Canada's Census of Population is held every five years (for example 2016, 2021 and 2026). In 2021, most households answered a short-form questionnaire and about one in four households received the longer, more detailed long-form questionnaire, so even the census uses sampling for its detailed questions.

Why not always use a census?

• Cost and time. Contacting millions of people is enormously expensive and slow. The monthly unemployment rate comes from a sample survey (the Labour Force Survey) because a monthly census would be impossible.
• Destructive testing. If testing destroys the item (crash-testing car seats, measuring how long a battery lasts), a census would leave nothing to sell.
• Unreachable or unlimited populations. All the fish in Lake Superior, all future customers, all possible rolls of a die.
• Accuracy. A small, carefully run sample with well-trained interviewers can be more accurate than a rushed census with tired, careless data collection.

A census makes sense when the population is small and easy to reach (the 28 students in your class), or when you need information about every individual (a teacher needs every student's mark, not an estimate).

What makes a good sample?

1. Representative. It looks like the population in the ways that matter for the question (ages, regions, grades).
2. Randomly selected. Chance, not the researcher's judgment or the participants' choice, decides who is in the sample. Randomness protects against bias you did not even think of.
3. Large enough. Larger samples have less sampling variability. The right size depends on how much the population varies and how precise the answer must be. (You will calculate sample sizes from a margin of error in One-Variable Data Analysis.)
4. Drawn from a good sampling frame. If the list leaves people out, so will the sample.
5. High response. If many chosen people do not respond, the people who do respond may be different.

A surprising fact: for a large population, the size of the sample matters much more than the fraction of the population sampled. A random sample of 1000 people tells you about as much about all of Canada as it would about a single large city.