A census collects data from every member of a population, while a sample collects data from only part of the population. This distinction matters because real studies often face limits in time, money, access, and accuracy. A census can sound perfect because it includes everyone, but it can still be wrong if people are missed, counted twice, or measured poorly.
A sample can give excellent information when it is chosen carefully and represents the population well.
The main trade-off is between coverage and practicality. A census tries to remove sampling error by measuring all units, but it can increase cost, delay results, and create more chances for nonresponse or recording errors. A sample is faster and cheaper, and statistics such as sample means and proportions can estimate population values with measurable uncertainty.
In many situations, a well-designed random sample is more useful than a rushed or flawed census.
Key Facts
- Population = the entire group you want to study.
- Census = data collected from every member of the population.
- Sample = data collected from a subset of the population.
- Sample proportion: p-hat = x/n, where x is the number with the trait and n is the sample size.
- Sampling error usually decreases as sample size increases, roughly proportional to 1/sqrt(n).
- A biased sample can give worse results than a smaller random sample because increasing n does not fix bias.
Vocabulary
- Population
- The complete group of individuals, objects, or measurements that a study is trying to describe.
- Census
- A data collection method that attempts to measure every member of the population.
- Sample
- A smaller group selected from the population to provide information about the whole population.
- Bias
- A systematic error that pushes results away from the true population value.
- Random sample
- A sample chosen by a chance process so that every member of the population has a known chance of selection.
Common Mistakes to Avoid
- Assuming a census is always accurate is wrong because nonresponse, duplicate counting, and measurement mistakes can still distort the results.
- Using a convenient sample and treating it as representative is wrong because easy-to-reach people may differ from the full population in important ways.
- Thinking a larger sample automatically removes bias is wrong because bias comes from the selection or measurement method, not just from having too few people.
- Confusing population size with sample quality is wrong because a small random sample can estimate a huge population well if the sampling method is sound.
Practice Questions
- 1 A school has 1,200 students. A survey randomly selects 150 students and finds that 96 prefer online grade reports. What is the sample proportion p-hat, and how many students in the whole school would you estimate prefer online grade reports?
- 2 A city wants to estimate support for a new park plan. A census would cost 7 per household. How much money is saved by using the sample instead of the census?
- 3 A restaurant asks only customers who follow its social media page to rate a new menu item. Explain whether this is closer to a census or a sample, and identify one reason the results may be biased.