A school with students wants to know how many would use a quiet study room after classes. Asking every student takes weeks. Asking students takes an afternoon, and if those are chosen well, their answers give a good estimate for the whole school. If they are chosen badly, say only students already in the library, the estimate can be far off no matter how carefully the numbers are worked out. SAT questions on sampling ask what a sample can tell you and about whom.
Key terms
Population
Definition The population of a study is the entire group of individuals, people or objects, that the study wants to draw conclusions about. A number that describes the whole population, such as the proportion of all students who would use the room or the mean height of all trees in a forest, is a parameter. A parameter is usually unknown, because measuring every member is too costly or impossible.
In plain words The population is everyone the question is really about.
Example In the study room survey the population is all students at the school. The parameter of interest is the proportion of those students who would use the room. Students at other schools are not part of this population.
Sample
Definition A sample is a subset of the population whose members are actually measured or asked. A number computed from the sample, such as a sample proportion or a sample mean, is a statistic, and it is used to estimate the matching population parameter.
In plain words The sample is the part of the population you actually hear from. You learn about the whole by looking at the part.
Example The students who answer the study room question form the sample. If of them say they would use the room, the statistic describes these students exactly and describes the school only approximately.
Random sample
Definition A random sample is a sample chosen by a chance process from the population. In a simple random sample of size , every possible group of members of the population has the same chance of being the sample, so in particular every member has the same chance of being chosen. Random selection removes the influence of the person choosing, but it does not remove sampling variation. Two random samples from the same population usually give different statistics, and neither one is exactly the parameter.
In plain words Let chance decide who is asked, for example by drawing names from a complete list. The results still wobble from sample to sample, but they do not lean in one direction on purpose or by habit.
Example The school numbers its students from to and uses a random number generator to pick different numbers. That is a simple random sample. A second random draw might give or students who would use the room instead of . How large that wobble tends to be is the subject of Margin of Error and Interpreting Estimates.
Sample proportion
Definition If members of a sample of size have a certain property, the sample proportion is
When the sample is a random sample from a population of size , the value estimates the population proportion , and estimates the number of population members with the property. In the same way the mean of a random sample estimates the population mean. These are estimates, not the exact population values.
In plain words Find the fraction in the sample, then scale it up to the population. Say “about” when you report the result.
Example For the study room survey,
so the best estimate is that about students would use the room. The true count could be somewhat higher or lower. A sample mean works the same way. A random sample of of the households in a town has a mean of people per household, so the town’s mean household size is estimated as about and its total number of people living in households as about .
Representative sample
Definition A sample is representative of a population when the sample’s makeup, in the features that matter for the question, is close to the population’s makeup. Random sampling from the whole population is the standard way to make a representative sample likely. Results from a sample generalize only to the population it was drawn from, and not automatically to any larger or different group.
In plain words A representative sample looks like a small copy of the population. A sample drawn from one group speaks for that group and no other.
Example A random sample of students from the full list of the school can represent the school. A random sample of students from the list of seniors can represent the seniors only. Its results may say little about ninth graders, who may stay after classes more or less often, and nothing at all about students at another school.
Selection bias
Definition Selection bias is a systematic error that comes from the way a sample is chosen, when some members of the population are much more or less likely than others to end up in the sample, so that the sample tends to differ from the population in a predictable direction. Two common sources are convenience samples, made of members who are easy to reach, and voluntary response samples, made of members who choose to take part. A larger sample chosen the same biased way does not fix selection bias.
In plain words If the way you pick people leaves out a part of the population, or favors one part, the results lean that way. Asking more people the same way just repeats the lean.
Example Suppose the school surveys only students it finds in the library after classes. Those students already like to study after school, so the share who would use a study room is likely to be higher than in the whole school, and the students who leave right after the last bell are excluded completely. An online poll that students can answer if they want has the same kind of problem, since students with strong opinions are more likely to respond. The random sample drawn from the full list avoids both problems.
Check yourself
- A researcher selects dog owners at random from the list of all licensed dog owners in a city and finds that percent of them walk their dog at least twice a day. To which group can the result most reasonably be generalized? (A) All residents of the city (B) All licensed dog owners in the city (C) All dog owners in the country (D) Only the dog owners who were surveyed
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The correct choice is (B). The sample was drawn at random from the licensed dog owners of the city, so that is the population it represents. Choice (A) includes people who own no dog. Choice (C) goes beyond the population that was sampled. Choice (D) is too narrow, since a random sample is meant to support an estimate for its population.
- A park has trees. A random sample of of them is inspected, and of the sampled trees show signs of a fungus. Based on the sample, what is the best estimate of the number of trees in the park that show signs of the fungus?
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The answer is . The sample proportion is , and
This is an estimate. The actual number of affected trees in the park may differ somewhat.
- A town wants to estimate the proportion of its adult residents who support a new bus route. Which method is most likely to produce a representative sample? (A) Asking riders at the busiest bus stop one morning (B) Posting a survey online and using the first responses (C) Selecting adults at random from a complete list of the town’s adult residents (D) Asking adults who attend a meeting about the bus route
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The correct choice is (C). It is the only method in which every adult resident has the same chance of being chosen. Choice (A) is a convenience sample of people who already ride the bus. Choices (B) and (D) rely on people who choose to respond or attend, who are likely to care more about the route than residents in general.