Introducing Statistics: What Can We Learn from Data?

A city library has 38,000 cardholders and wants to know whether opening on Sundays would be worth the staff cost. Nobody can ask all 38,000 people in time to decide, so the library asks 400 of them and uses their answers to say something about everyone. Learning about a large group from data on part of it, and knowing how far to trust what you learned, is what statistics is for.

The parts of a statistical study

A statistical study starts with a question about a large group and answers it with data from a smaller group drawn from it. Every study has the same parts.

• The investigative question: what the study is trying to find out. For the library: "What proportion of the library's cardholders would visit on a Sunday at least once a month?"
• The population: every individual or item the question is about. Here, all 38,000 cardholders. The number of individuals in the population is written $N$, so $N = 38{,}000$.
• The sample: the part of the population selected for study, from which the data are obtained. Here, the 400 cardholders the library chose to ask. The sample size is written $n$, so $n = 400$.
• The data: the information recorded. One piece of information about one individual is a datum (one cardholder's answer, "yes"). All the recorded values together form a data set (the list of 400 answers).

The sample is always a subset of the population. In practice, not everyone selected gives data. If the library picks 400 cardholders and only 300 reply, the sample is still the 400 selected (so $n = 400$), the response rate is $\frac{300}{400} = 75\%$, and the data describe only the 300 who replied. Say both numbers when you describe such a study. The 100 who never answered may differ from those who did (people who never use the library are less likely to bother replying), which is nonresponse, a source of bias that topic 1.12 covers.

Why study a sample instead of everyone

Recording data on every member of a population is called a census (topic 1.10 returns to it). A census is often impossible or pointless, for a few concrete reasons:

• Size and cost. Surveying 38,000 cardholders, or weighing every one of the roughly 3.6 million babies born in the United States in a year, takes money and time that a sample of a few hundred does not.
• Reach. Some members can't be found or measured. A city can't survey "everyone who will ride the new bus line next year," because many of them haven't moved to the city yet.
• Destruction. Measuring sometimes uses up the thing measured. To find how long a batch of batteries lasts, you run batteries until they die. Testing all of them leaves nothing to sell.

The price of sampling is that the sample's answer will usually differ somewhat from the population's. If 62% of the 400 sampled cardholders say yes, the true figure for all 38,000 might be 59% or 65%. A different 400 cardholders would give a slightly different percentage. Much of AP Statistics is about measuring that sample-to-sample variation so you can say how close the sample's answer is likely to be. Topic 1.2 names the two numbers involved: the population's value (a parameter) and the sample's value (a statistic).

Investigative questions

An investigative question is the question a study is built to answer. It has to come first, and it has to be answerable with data.

It is fixed before the data come in. Suppose the library asks its 400 cardholders about Sunday hours, sees that the result is unimpressive, then notices that cardholders over 65 answered yes more often and announces "the study shows seniors want Sunday hours." The problem is that a data set split enough ways will always contain some group that looks unusual by chance. If you search the results for the most striking pattern and then pretend that was the question, the pattern is likely to be a fluke that would not show up again in a new sample. A question chosen after seeing the data can't be tested fairly by the same data. A new pattern noticed this way is a reason to run a new study with that question set in advance.

It can be answered with data you can collect and analyze. Compare these:

A good investigative question expects variation: different individuals give different answers, and the question asks about the pattern across the whole population. If the answer is a single fact you could look up or count directly, it isn't a statistical question. Topic 1.10 adds more structure: what an investigative question must say for a test or an interval, and what kind of conclusion it allows.

Answering in context

Every part of a study, and every number calculated from it, stands for something real: a group of people, a measurement with units, an answer to a survey question. Saying what it stands for is what "in context" means, and on the AP exam an answer without context often earns nothing.

Compare "the sample proportion is 0.62" with "62% of the 400 cardholders surveyed said they would visit on a Sunday at least once a month." The first is a number. The second says which individuals, which question, and which group the number describes. Use the study's own nouns: name the population ("the 38,000 cardholders of the city library"), not "the people."

Worked examples

Where students lose points